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Commitment scheme

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A commitment scheme is a <a href="Cryptographic_primitive" title="Cryptographic primitive">cryptographic primitive</a> that allows one to commit to a chosen value (or chosen statement) while keeping it hidden to others, with the ability to reveal the committed value later.<a href="#cite_note-Goldreich-1">[1]</a> Commitment schemes are designed so that a party cannot change the value or statement after they have committed to it: that is, commitment schemes are binding. Commitment schemes have important applications in a number of <a href="Cryptographic_protocol" title="Cryptographic protocol">cryptographic protocols</a> including secure coin flipping, <a href="Zero-knowledge_proof" title="Zero-knowledge proof">zero-knowledge proofs</a>, and <a href="Secure_computation" class="mw-redirect" title="Secure computation">secure computation</a>.

A way to visualize a commitment scheme is to think of a sender as putting a message in a locked box, and giving the box to a receiver. The message in the box is hidden from the receiver, who cannot open the lock themselves. Since the receiver has the box, the message inside cannot be changed—merely revealed if the sender chooses to give them the key at some later time.

Interactions in a commitment scheme take place in two phases:

<ol><li>the commit phase during which a value is chosen and committed to</li>

<li>the reveal phase during which the value is revealed by the sender, then the receiver verifies its authenticity</li></ol>

In the above metaphor, the commit phase is the sender putting the message in the box, and locking it. The reveal phase is the sender giving the key to the receiver, who uses it to open the box and verify its contents. The locked box is the commitment, and the key is the proof.

In simple protocols, the commit phase consists of a single message from the sender to the receiver. This message is called the commitment. It is essential that the specific value chosen cannot be extracted from the message by the receiver at that time (this is called the hiding property). A simple reveal phase would consist of a single message, the opening, from the sender to the receiver, followed by a check performed by the receiver. The value chosen during the commit phase must be the only one that the sender can compute and that validates during the reveal phase (this is called the binding property).

The concept of commitment schemes was perhaps first formalized by <a href="Gilles_Brassard" title="Gilles Brassard">Gilles Brassard</a>, <a href="David_Chaum" title="David Chaum">David Chaum</a>, and <a href="Claude_Cr%C3%A9peau" title="Claude Crépeau">Claude Crépeau</a> in 1988,<a href="#cite_note-BCC-2">[2]</a> as part of various zero-knowledge protocols for <a href="NP_(complexity)" title="NP (complexity)">NP</a>, based on various types of commitment schemes.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> But the concept was used prior to that without being treated formally.<a href="#cite_note-Naor-5">[5]</a><a href="#cite_note-Crepeau-6">[6]</a> The notion of commitments appeared earliest in works by <a href="Manuel_Blum" title="Manuel Blum">Manuel Blum</a>,<a href="#cite_note-Blum-7">[7]</a> <a href="Shimon_Even" title="Shimon Even">Shimon Even</a>,<a href="#cite_note-Even-8">[8]</a> and <a href="Adi_Shamir" title="Adi Shamir">Adi Shamir</a> et al.<a href="#cite_note-SRA81-9">[9]</a> The terminology seems to have been originated by Blum,<a href="#cite_note-Crepeau-6">[6]</a> although commitment schemes can be interchangeably called bit commitment schemes—sometimes reserved for the special case where the committed value is a <a href="Bit" title="Bit">bit</a>. Prior to that, commitment via one-way hash functions was considered, e.g., as part of, say, <a href="Lamport_signature" title="Lamport signature">Lamport signature</a>, the original one-time one-bit signature scheme.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Coin_flipping">Coin flipping</h3></div>

Suppose Alice and Bob want to resolve some dispute via <a href="Coin_flipping" title="Coin flipping">coin flipping</a>. If they are physically in the same place, a typical procedure might be:

<ol><li>Alice "calls" the coin flip,</li>

<li>Bob flips the coin,</li>

<li>If Alice's call is correct, she wins, otherwise Bob wins.</li></ol>

If Alice and Bob are not in the same place a problem arises. Once Alice has "called" the coin flip, Bob can stipulate the flip "results" to be whatever is most desirable for him. Similarly, if Alice doesn't announce her "call" to Bob, after Bob flips the coin and announces the result, Alice can report that she called whatever result is most desirable for her. Alice and Bob can use commitments in a procedure that will allow both to trust the outcome:

<ol><li>Alice "calls" the coin flip but only tells Bob a commitment to her call,</li>

<li>Bob flips the coin and reports the result,</li>

<li>Alice reveals what she committed to,</li>

<li>Bob verifies that Alice's call matches her commitment,</li>

<li>If Alice's revelation matches the coin result Bob reported, Alice wins.</li></ol>

For Bob to be able to skew the results to his favor, he must be able to understand the call hidden in Alice's commitment. If the commitment scheme is a good one, Bob cannot skew the results. Similarly, Alice cannot affect the result if she cannot change the value she commits to.

A real-life application of this problem exists, when people (often in media) commit to a decision or give an answer in a "sealed envelope", which is then opened later. "Let's find out if that's what the candidate answered", for example on a game show, can serve as a model of this system.

<div class="mw-heading mw-heading3"><h3 id="Zero-knowledge_proofs">Zero-knowledge proofs</h3></div>

One particular motivating example is the use of commitment schemes in <a href="Zero-knowledge_proof" title="Zero-knowledge proof">zero-knowledge proofs</a>. Commitments are used in zero-knowledge proofs for two main purposes: first, to allow the prover to participate in "cut and choose" proofs where the verifier will be presented with a choice of what to learn, and the prover will reveal only what corresponds to the verifier's choice. Commitment schemes allow the prover to specify all the information in advance, and only reveal what should be revealed later in the proof.<a href="#cite_note-GMW-10">[10]</a>

Second, commitments are also used in zero-knowledge proofs by the verifier, who will often specify their choices ahead of time in a commitment. This allows zero-knowledge proofs to be composed in parallel without revealing additional information to the prover.<a href="#cite_note-GK-11">[11]</a>

<div class="mw-heading mw-heading3"><h3 id="Signature_schemes">Signature schemes</h3></div>

The <a href="Lamport_signature" title="Lamport signature">Lamport signature</a> scheme is a <a href="Digital_signature" title="Digital signature">digital signature</a> system that relies on maintaining two sets of secret <a href="Network_packet" title="Network packet">data packets</a>, publishing <a href="Cryptographic_hash_function" title="Cryptographic hash function">verifiable hashes</a> of the data packets, and then selectively revealing partial secret data packets in a manner that conforms specifically to the data to be signed. In this way, the prior public commitment to the secret values becomes a critical part of the functioning of the system.

Because the Lamport signature system cannot be used more than once, a system to combine many Lamport key-sets under a single public value that can be tied to a person and verified by others was developed. This system uses trees of <a href="Cryptographic_hash_function" title="Cryptographic hash function">hashes</a> to compress many published Lamport-key-commitment sets into a single hash value that can be associated with the prospective author of later-verified data.

<div class="mw-heading mw-heading3"><h3 id="Verifiable_secret_sharing">Verifiable secret sharing</h3></div>

Another important application of commitments is in <a href="Verifiable_secret_sharing" title="Verifiable secret sharing">verifiable secret sharing</a>, a critical building block of <a href="Secure_multiparty_computation" class="mw-redirect" title="Secure multiparty computation">secure multiparty computation</a>. In a <a href="Secret_sharing" title="Secret sharing">secret sharing</a> scheme, each of several parties receive "shares" of a value that is meant to be hidden from everyone. If enough parties get together, their shares can be used to reconstruct the secret, but even a malicious cabal of insufficient size should learn nothing. Secret sharing is at the root of many protocols for <a href="Secure_computation" class="mw-redirect" title="Secure computation">secure computation</a>: in order to securely compute a function of some shared input, the secret shares are manipulated instead. However, if shares are to be generated by malicious parties, it may be important that those shares can be checked for correctness. In a verifiable secret sharing scheme, the distribution of a secret is accompanied by commitments to the individual shares. The commitments reveal nothing that can help a dishonest cabal, but the shares allow each individual party to check to see if their shares are correct.<a href="#cite_note-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="Defining_the_security">Defining the security</h2></div>

Formal definitions of commitment schemes vary strongly in notation and in flavour. The first such flavour is whether the commitment scheme provides perfect or computational security with respect to the hiding or binding properties. Another such flavour is whether the commitment is interactive, i.e. whether both the commit phase and the reveal phase can be seen as being executed by a <a href="Cryptographic_protocol" title="Cryptographic protocol">cryptographic protocol</a> or whether they are non-interactive, consisting of two algorithms Commit and CheckReveal. In the latter case CheckReveal can often be seen as a derandomised version of Commit, with the randomness used by Commit constituting the opening information.

If the commitment C to a value x is computed as C:=Commit(x,open) with open being the randomness used for computing the commitment, then CheckReveal (C,x,open) reduces to simply verifying the equation C=Commit (x,open).

Using this notation and some knowledge about <a href="Mathematical_function" class="mw-redirect" title="Mathematical function">mathematical functions</a> and <a href="Probability_theory" title="Probability theory">probability theory</a> we formalise different versions of the binding and hiding properties of commitments. The two most important combinations of these properties are perfectly binding and computationally hiding commitment schemes and computationally binding and perfectly hiding commitment schemes. Note that no commitment scheme can be at the same time perfectly binding and perfectly hiding – a computationally unbounded adversary can simply generate Commit(x,open) for every value of x and open until finding a pair that outputs C, and in a perfectly binding scheme this uniquely identifies x.

<div class="mw-heading mw-heading3"><h3 id="Computational_binding">Computational binding</h3></div>

Let open be chosen from a set of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle 2^{k}}</annotation>

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</math><img src="./2d82641ae2702b0db07dd11830af27b9ee0cd196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle 2^{k}}" loading="lazy">, i.e., it can be represented as a k bit string, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Commit}}_{k}}">

<msub>

<mtext>Commit</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Commit}}_{k}}</annotation>

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</math><img src="./74ccbeb53d623ca13d6e4f6c40d67ed1bcd6521d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.352ex; height:2.509ex;" alt="{\displaystyle {\text{Commit}}_{k}}" loading="lazy"> be the corresponding commitment scheme. As the size of k determines the security of the commitment scheme it is called the <a href="Security_parameter" title="Security parameter">security parameter</a>.

Then for all <a href="Uniformity_(complexity)" class="mw-redirect" title="Uniformity (complexity)">non-uniform</a> <a href="PP_(complexity)" title="PP (complexity)">probabilistic polynomial time algorithms</a> that output <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,x'}">

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<annotation encoding="application/x-tex">{\displaystyle x,x'}</annotation>

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</math><img src="./868106c4dab7e143519d80dfb0ce9ca91f367330.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle x,x'}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle open,open'}">

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<annotation encoding="application/x-tex">{\displaystyle open,open'}</annotation>

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</math><img src="./8ebdb7aa33a2bc284bac5a44da91b5b460a0337a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.269ex; height:2.843ex;" alt="{\displaystyle open,open'}" loading="lazy"> of increasing length k, the probability that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq x'}">

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<annotation encoding="application/x-tex">{\displaystyle x\neq x'}</annotation>

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</math><img src="./2c6cfab4b1bd197bfe3d7b6ff190cd3ae73f0368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.443ex; height:3.009ex;" alt="{\displaystyle x\neq x'}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Commit}}_{k}(x,open)={\text{Commit}}_{k}(x',open')}">

<msub>

<mtext>Commit</mtext>

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<mtext>Commit</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Commit}}_{k}(x,open)={\text{Commit}}_{k}(x',open')}</annotation>

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</math><img src="./62f1b312bce16abbebedf82512b3f2a8c35bd00c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.069ex; height:3.009ex;" alt="{\displaystyle {\text{Commit}}_{k}(x,open)={\text{Commit}}_{k}(x',open')}" loading="lazy"> is a <a href="Negligible_function" title="Negligible function">negligible function</a> in k.

This is a form of <a href="Asymptotic_analysis" title="Asymptotic analysis">asymptotic analysis</a>. It is also possible to state the same requirement using <a href="Concrete_security" title="Concrete security">concrete security</a>: A commitment scheme Commit is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t,\epsilon )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (t,\epsilon )}</annotation>

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</math><img src="./0661eb28dd8be643476824c8bdc857e8fed026dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.627ex; height:2.843ex;" alt="{\displaystyle (t,\epsilon )}" loading="lazy"> secure, if for all algorithms that run in time t and output <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,x',open,open'}">

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<annotation encoding="application/x-tex">{\displaystyle x,x',open,open'}</annotation>

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</math><img src="./1f37eefcd1071178adfd7ea83cb83d145497f14b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.681ex; height:2.843ex;" alt="{\displaystyle x,x',open,open'}" loading="lazy"> the probability that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq x'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq x'}</annotation>

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</math><img src="./2c6cfab4b1bd197bfe3d7b6ff190cd3ae73f0368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.443ex; height:3.009ex;" alt="{\displaystyle x\neq x'}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Commit}}(x,open)={\text{Commit}}(x',open')}">

<mtext>Commit</mtext>

</mrow>

<mtext>Commit</mtext>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Commit}}(x,open)={\text{Commit}}(x',open')}</annotation>

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</math><img src="./073179b70b410f98b45a88d983472dc740caed22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.892ex; height:3.009ex;" alt="{\displaystyle {\text{Commit}}(x,open)={\text{Commit}}(x',open')}" loading="lazy"> is at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation>

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</math><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Perfect,_statistical,_and_computational_hiding">Perfect, statistical, and computational hiding</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{k}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle U_{k}}</annotation>

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</math><img src="./924cf85ae48192ae1c1ce15c109032e0a69b1014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.676ex; height:2.509ex;" alt="{\displaystyle U_{k}}" loading="lazy"> be the uniform distribution over the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle 2^{k}}</annotation>

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</math><img src="./2d82641ae2702b0db07dd11830af27b9ee0cd196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle 2^{k}}" loading="lazy"> opening values for security parameter k. A commitment scheme is respectively perfect, statistical, or computational hiding, if for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq x'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq x'}</annotation>

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</math><img src="./2c6cfab4b1bd197bfe3d7b6ff190cd3ae73f0368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.443ex; height:3.009ex;" alt="{\displaystyle x\neq x'}" loading="lazy"> the <a href="Probability_ensemble" class="mw-redirect" title="Probability ensemble">probability ensembles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\text{Commit}}_{k}(x,U_{k})\}_{k\in \mathbb {N} }}">

<msub>

<mtext>Commit</mtext>

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</mrow>

</msub>

<msub>

</mrow>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \{{\text{Commit}}_{k}(x,U_{k})\}_{k\in \mathbb {N} }}</annotation>

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</math><img src="./17e720639250e8208c50233ee6c149b44ffebf13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.898ex; height:2.843ex;" alt="{\displaystyle \{{\text{Commit}}_{k}(x,U_{k})\}_{k\in \mathbb {N} }}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\text{Commit}}_{k}(x',U_{k})\}_{k\in \mathbb {N} }}">

<msub>

<mtext>Commit</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \{{\text{Commit}}_{k}(x',U_{k})\}_{k\in \mathbb {N} }}</annotation>

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</math><img src="./1189f7d2a52e3688d1d32327821e857b8da59ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.583ex; height:3.009ex;" alt="{\displaystyle \{{\text{Commit}}_{k}(x',U_{k})\}_{k\in \mathbb {N} }}" loading="lazy"> are equal, <a href="Statistically_close" class="mw-redirect" title="Statistically close">statistically close</a>, or <a href="Computationally_indistinguishable" class="mw-redirect" title="Computationally indistinguishable">computationally indistinguishable</a>.

<div class="mw-heading mw-heading3"><h3 id="Impossibility_of_universally_composable_commitment_schemes">Impossibility of universally composable commitment schemes</h3></div>

It is impossible to realize commitment schemes in the <a href="Universal_composability" title="Universal composability">universal composability</a>

(UC) framework. The reason is that UC commitment has to be extractable, as shown by Canetti and Fischlin<a href="#cite_note-13">[13]</a> and explained below.

The ideal commitment functionality, denoted here by F, works roughly as follows. Committer C sends value m to F, which

stores it and sends "receipt" to receiver R. Later, C sends "open" to

F, which sends m to R.

Now, assume we have a protocol π that realizes this functionality. Suppose that the committer C is corrupted. In the UC framework, that essentially means that C is now controlled by the environment, which attempts to distinguish protocol execution from the ideal process. Consider an environment that chooses a message m and then tells C to act as prescribed by π, as if it has committed to m. Note here that in order to realize F, the receiver must, after receiving a commitment, output a message "receipt". After the environment sees this message, it tells C to open the commitment.

The protocol is only secure if this scenario is indistinguishable from the ideal case, where the functionality interacts with a simulator S. Here, S has control of C. In particular, whenever R outputs "receipt", F has to do likewise. The only way to do that is for S to tell C to send a value to F. However, note

that by this point, m is not known to S. Hence, when the commitment is opened during protocol execution, it is unlikely that F will open to m, unless S can extract m from the messages it received from the environment before R outputs the receipt.

However a protocol that is extractable in this sense cannot be statistically hiding. Suppose such a simulator S exists. Now consider an

environment that, instead of corrupting C, corrupts R instead. Additionally it runs a copy of S. Messages received from C are fed into S, and replies from S are forwarded to C.

The environment initially tells C to commit to a message m. At some point in the interaction, S will commit to a value m′. This message is handed to R, who outputs m′. Note that by assumption we have m' = m with high probability. Now in the ideal process the simulator has to come up with m. But this is impossible, because at this point the commitment has not been opened yet, so the only message R can have received in the ideal process is a "receipt" message. We thus have a contradiction.

<div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2></div>

A commitment scheme can either be perfectly binding (it is impossible for Alice to alter her commitment after she has made it, even if she has unbounded computational resources); or perfectly concealing (it is impossible for Bob to find out the commitment without Alice revealing it, even if he has unbounded computational resources); or formulated as an instance-dependent commitment scheme, which is either hiding or binding depending on the solution to another problem.<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a> A commitment scheme cannot be both perfectly hiding and perfectly binding at the same time.

<div class="mw-heading mw-heading3"><h3 id="Bit-commitment_in_the_random_oracle_model">Bit-commitment in the random oracle model</h3></div>

Bit-commitment schemes are trivial to construct in the <a href="Random_oracle" title="Random oracle">random oracle</a> model. Given a <a href="Cryptographic_hash_function" title="Cryptographic hash function">hash function</a> H with a 3k bit output, to commit the k-bit message m, Alice generates a random k bit string R and sends Bob H(R || m). The probability that any R′, m′ exist where m′ ≠ m such that H(R′ || m′) = H(R || m) is ≈ 2−k, but to test any guess at the message m Bob will need to make 2k (for an incorrect guess) or 2k-1 (on average, for a correct guess) queries to the random oracle.<a href="#cite_note-16">[16]</a> We note that earlier schemes based on hash functions, essentially can be thought of schemes based on idealization of these hash functions as random oracle.

<div class="mw-heading mw-heading3"><h3 id="Bit-commitment_from_any_one-way_permutation">Bit-commitment from any one-way permutation</h3></div>

One can create a bit-commitment scheme from any <a href="One-way_function" title="One-way function">one-way function</a> that is <a href="Injective_function" title="Injective function">injective</a>. The scheme relies on the fact that every one-way function can be modified (via the <a href="Goldreich-Levin_theorem" class="mw-redirect" title="Goldreich-Levin theorem">Goldreich-Levin theorem</a>) to possess a computationally <a href="Hard-core_predicate" title="Hard-core predicate">hard-core predicate</a> (while retaining the injective property).

Let f be an injective one-way function, with h a hard-core predicate. Then to commit to a bit b Alice picks a random input x and sends the triple

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<annotation encoding="application/x-tex">{\displaystyle (h,f(x),b\oplus h(x))}</annotation>

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</math><img src="./cbf6699da90f362b16e81134802fe3be1d3d6379.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.95ex; height:2.843ex;" alt="{\displaystyle (h,f(x),b\oplus h(x))}" loading="lazy"></dd></dl>

to Bob, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }">

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<annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation>

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</math><img src="./8b16e2bdaefee9eed86d866e6eba3ac47c710f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }" loading="lazy"> denotes XOR, i.e., bitwise addition modulo 2. To decommit, Alice simply sends x to Bob. Bob verifies by computing f(x) and comparing to the committed value. This scheme is concealing because for Bob to recover b he must recover h(x). Since h is a computationally hard-core predicate, recovering h(x) from f(x) with probability greater than one-half is as hard as inverting f. Perfect binding follows from the fact that f is injective and thus f(x) has exactly one preimage.

<div class="mw-heading mw-heading3"><h3 id="Bit-commitment_from_a_pseudo-random_generator">Bit-commitment from a pseudo-random generator</h3></div>

Note that since we do not know how to construct a one-way permutation from any one-way function, this section reduces the strength of the cryptographic assumption necessary to construct a bit-commitment protocol.

In 1991 Moni Naor showed how to create a bit-commitment scheme from a <a href="Cryptographically_secure_pseudorandom_number_generator" title="Cryptographically secure pseudorandom number generator">cryptographically secure pseudorandom number generator</a>.<a href="#cite_note-17">[17]</a> The construction is as follows. If G is a pseudo-random generator such that G takes n bits to 3n bits, then if Alice wants to commit to a bit b:

<ul><li>Bob selects a random 3n-bit vector R and sends R to Alice.</li>

<li>Alice selects a random n-bit vector Y and computes the 3n-bit vector G(Y).</li>

<li>If b=1 Alice sends G(Y) to Bob, otherwise she sends the bitwise <a href="Exclusive_disjunction" class="mw-redirect" title="Exclusive disjunction">exclusive-or</a> of G(Y) and R to Bob.</li></ul>

To decommit Alice sends Y to Bob, who can then check whether he initially received G(Y) or G(Y) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }">

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</math><img src="./8b16e2bdaefee9eed86d866e6eba3ac47c710f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }" loading="lazy"> R.

This scheme is statistically binding, meaning that even if Alice is computationally unbounded she cannot cheat with probability greater than 2−n. For Alice to cheat, she would need to find a Y', such that G(Y') = G(Y) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }">

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</math><img src="./8b16e2bdaefee9eed86d866e6eba3ac47c710f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }" loading="lazy"> R. If she could find such a value, she could decommit by sending the truth and Y, or send the opposite answer and Y'. However, G(Y) and G(Y') are only able to produce 2n possible values each (that's 22n) while R is picked out of 23n values. She does not pick R, so there is a 22n/23n = 2−n probability that a Y' satisfying the equation required to cheat will exist.

The concealing property follows from a standard reduction, if Bob can tell whether Alice committed to a zero or one, he can also distinguish the output of the pseudo-random generator G from true-random, which contradicts the cryptographic security of G.

<div class="mw-heading mw-heading3"><h3 id="A_perfectly_binding_scheme_based_on_the_discrete_log_problem_and_beyond">A perfectly binding scheme based on the discrete log problem and beyond</h3></div>

Alice chooses a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> of prime order p, with multiplicative generator g.

Alice randomly picks a secret value x from 0 to p − 1 to commit to and calculates c = gx and publishes c. The <a href="Discrete_logarithm_problem" class="mw-redirect" title="Discrete logarithm problem">discrete logarithm problem</a> dictates that from c, it is computationally infeasible to compute x, so under this assumption, Bob cannot compute x. On the other hand, Alice cannot compute a x′ <> x, such that gx′ = c, so the scheme is binding.

This scheme isn't perfectly concealing as someone could find the commitment if he manages to solve the <a href="Discrete_logarithm_problem" class="mw-redirect" title="Discrete logarithm problem">discrete logarithm problem</a>. In fact, this scheme isn't hiding at all with respect to the standard hiding game, where an adversary should be unable to guess which of two messages he chose were committed to - similar to the <a href="IND-CPA" class="mw-redirect" title="IND-CPA">IND-CPA</a> game. One consequence of this is that if the space of possible values of x is small, then an attacker could simply try them all and the commitment would not be hiding.

A better example of a perfectly binding commitment scheme is one where the commitment is the encryption of x under a <a href="Semantically_secure" class="mw-redirect" title="Semantically secure">semantically secure</a>, public-key encryption scheme with perfect completeness, and the decommitment is the string of random bits used to encrypt x. An example of an information-theoretically hiding commitment scheme is the Pedersen commitment scheme,<a href="#cite_note-Pedersen_pp._129–140-18">[18]</a> which is computationally binding under the discrete logarithm assumption.

Additionally to the scheme above, it uses another generator h of the prime group and a random number r. The commitment is set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=g^{x}h^{r}}">

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These constructions are tightly related to and based on the algebraic properties of the underlying groups, and the notion originally seemed to be very much related to the algebra. However, it was shown that basing statistically binding commitment schemes on general unstructured assumption is possible, via the notion of interactive hashing

for commitments from general complexity assumptions (specifically and originally, based on any one way permutation) as in.<a href="#cite_note-21">[21]</a>

<div class="mw-heading mw-heading3"><h3 id="A_perfectly_hiding_commitment_scheme_based_on_RSA">A perfectly hiding commitment scheme based on RSA</h3></div>

Alice selects <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=p\cdot q}">

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</math><img src="./36e46a18a8492c3e55b7fefa9cc832a734fb413e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.08ex; height:2.509ex;" alt="{\displaystyle N=p\cdot q}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> are large secret prime numbers. Additionally, she selects a prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e>N^{2}}">

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</math><img src="./1ca537dd0626c41c9c06ac1ce877b7e78716b89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.359ex; height:2.676ex;" alt="{\displaystyle e>N^{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gcd(e,\phi (N^{2}))=1}">

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</math><img src="./3fec2e0353f0fcca6d8cb0f360cc77fa532b85c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.898ex; height:3.176ex;" alt="{\displaystyle gcd(e,\phi (N^{2}))=1}" loading="lazy">. Alice then computes a public number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{m}}">

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</math><img src="./28dbe49b383cddeba92dfccc07b285a189b443a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.784ex; height:2.009ex;" alt="{\displaystyle g_{m}}" loading="lazy"> as an element of maximum order in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{N^{2}}^{*}}">

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</math><img src="./1d19d27f5baefc3d1c3b2066b6a5f203b3d5946e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.115ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} _{N^{2}}^{*}}" loading="lazy"> group.<a href="#cite_note-menezes2018handbook-22">[22]</a> Finally, Alice commits to her secret <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> by first generating a random number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{N^{2}}^{*}}">

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</math><img src="./1d19d27f5baefc3d1c3b2066b6a5f203b3d5946e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.115ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} _{N^{2}}^{*}}" loading="lazy"> and then by computing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=m^{e}g_{m}^{r}}">

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<annotation encoding="application/x-tex">{\displaystyle c=m^{e}g_{m}^{r}}</annotation>

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</math><img src="./33f17dde19ea12cbf8e2fe8732c384c6f5fba301.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.928ex; height:2.676ex;" alt="{\displaystyle c=m^{e}g_{m}^{r}}" loading="lazy">.

The security of the above commitment relies on the hardness of the RSA problem and has perfect hiding and computational binding.<a href="#cite_note-masquerade2022-23">[23]</a>

<div class="mw-heading mw-heading3"><h3 id="Additive_and_multiplicative_homomorphic_properties_of_commitments">Additive and multiplicative homomorphic properties of commitments</h3></div>

The Pedersen commitment scheme introduces an interesting homomorphic property that allows performing addition between two commitments. More specifically, given two messages <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}}">

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</math><img src="./31aafa60e48d39ccce922404c0b80340b2cc777a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle m_{2}}</annotation>

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</math><img src="./0ecebe334d5cadc3ffcf245eb02919034d7a2ec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{2}}" loading="lazy"> and randomness <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation>

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</math><img src="./ea214f2b31fb3869344bb9311da41c5cc38a99e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation>

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</math><img src="./4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" loading="lazy">, respectively, it is possible to generate a new commitment such that: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=C(m_{1}+m_{2},r_{1}+r_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=C(m_{1}+m_{2},r_{1}+r_{2})}</annotation>

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</math><img src="./130ee16f09ab66dff808e6531f066bda46e4d847.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.077ex; height:2.843ex;" alt="{\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=C(m_{1}+m_{2},r_{1}+r_{2})}" loading="lazy">. Formally:

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<annotation encoding="application/x-tex">{\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=g^{m_{1}}h^{r_{1}}\cdot g^{m_{2}}h^{r_{2}}=g^{m_{1}+m_{2}}h^{r_{1}+r_{2}}=C(m_{1}+m_{2},r_{1}+r_{2})}</annotation>

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</math><img src="./730fc635b8c686d3d546a50840460832dee6a098.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:79.665ex; height:3.009ex;" alt="{\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=g^{m_{1}}h^{r_{1}}\cdot g^{m_{2}}h^{r_{2}}=g^{m_{1}+m_{2}}h^{r_{1}+r_{2}}=C(m_{1}+m_{2},r_{1}+r_{2})}" loading="lazy">

To open the above Pedersen commitment to a new message <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}+m_{2}}">

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</math><img src="./648effd88b01ad99e59e69e7b60809ea505ea337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.03ex; height:2.343ex;" alt="{\displaystyle m_{1}+m_{2}}" loading="lazy">, the randomness <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}">

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</math><img src="./ea214f2b31fb3869344bb9311da41c5cc38a99e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation>

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</math><img src="./4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" loading="lazy"> has to be added.

Similarly, the RSA-based commitment mentioned above has a homomorphic property with respect to the multiplication operation. Given two messages <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle m_{1}}</annotation>

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</math><img src="./31aafa60e48d39ccce922404c0b80340b2cc777a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle m_{2}}</annotation>

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</math><img src="./0ecebe334d5cadc3ffcf245eb02919034d7a2ec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{2}}" loading="lazy"> with randomness <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation>

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</math><img src="./ea214f2b31fb3869344bb9311da41c5cc38a99e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation>

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</math><img src="./4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" loading="lazy">, respectively, one can compute: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=C(m_{1}\cdot m_{2},r_{1}+r_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=C(m_{1}\cdot m_{2},r_{1}+r_{2})}</annotation>

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</math><img src="./bbd5560b112bea1adc5c6cd13df81bc3c1a8b0d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.916ex; height:2.843ex;" alt="{\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=C(m_{1}\cdot m_{2},r_{1}+r_{2})}" loading="lazy">. Formally:

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<annotation encoding="application/x-tex">{\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=m_{1}^{e}g_{m}^{r_{1}}\cdot m_{2}^{e}g_{m}^{r_{2}}=(m_{1}\cdot m_{2})^{e}g_{m}^{r_{1}+r_{2}}=C(m_{1}\cdot m_{2},r_{1}+r_{2})}</annotation>

</semantics>

</math><img src="./d79ea92ff09e20dc4e7535b8eb59928479fbab9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:80.279ex; height:3.343ex;" alt="{\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=m_{1}^{e}g_{m}^{r_{1}}\cdot m_{2}^{e}g_{m}^{r_{2}}=(m_{1}\cdot m_{2})^{e}g_{m}^{r_{1}+r_{2}}=C(m_{1}\cdot m_{2},r_{1}+r_{2})}" loading="lazy">.

To open the above commitment to a new message <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}\cdot m_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle m_{1}\cdot m_{2}}</annotation>

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</math><img src="./a7e23bafcd9afe4b7092628109999b988675d7f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.868ex; height:2.009ex;" alt="{\displaystyle m_{1}\cdot m_{2}}" loading="lazy">, the randomness <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation>

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</math><img src="./ea214f2b31fb3869344bb9311da41c5cc38a99e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation>

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</math><img src="./4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" loading="lazy"> has to be added. This newly generated commitment is distributed similarly to a new commitment to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}\cdot m_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle m_{1}\cdot m_{2}}</annotation>

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</math><img src="./a7e23bafcd9afe4b7092628109999b988675d7f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.868ex; height:2.009ex;" alt="{\displaystyle m_{1}\cdot m_{2}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Partial_reveal">Partial reveal</h2></div>

Some commitment schemes permit a proof to be given of only a portion of the committed value. In these schemes, the secret value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a vector of many individually separable values.

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<annotation encoding="application/x-tex">{\displaystyle X=(x_{1},x_{2},...,x_{n})}</annotation>

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</math><img src="./c81d3963c9f0d511ffe709f18c9d0b9e453e1884.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.407ex; height:2.843ex;" alt="{\displaystyle X=(x_{1},x_{2},...,x_{n})}" loading="lazy"></dd></dl>

The commitment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is computed from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> in the commit phase. Normally, in the reveal phase, the prover would reveal all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and some additional proof data (such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> in <a href="#Bit-commitment_in_the_random_oracle_model">simple bit-commitment</a>). Instead, the prover is able to reveal any single value from the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> vector, and create an efficient proof that it is the authentic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">th element of the original vector that created the commitment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">. The proof does not require any values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> other than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> to be revealed, and it is impossible to create valid proofs that reveal different values for any of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> than the true one.<a href="#cite_note-24">[24]</a>

<div class="mw-heading mw-heading3"><h3 id="Vector_hashing">Vector hashing</h3></div>

Vector hashing is a naive vector commitment partial reveal scheme based on bit-commitment. Values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},m_{2},...m_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle m_{1},m_{2},...m_{n}}</annotation>

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</math><img src="./a45e563c9abe587730733a81ebffc36423b11250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.618ex; height:2.009ex;" alt="{\displaystyle m_{1},m_{2},...m_{n}}" loading="lazy"> are chosen randomly. Individual commitments are created by hashing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}=H(x_{1}||m_{1}),y_{2}=H(x_{2}||m_{2}),...}">

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<annotation encoding="application/x-tex">{\displaystyle y_{1}=H(x_{1}||m_{1}),y_{2}=H(x_{2}||m_{2}),...}</annotation>

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</math><img src="./9eec2edb0000dfe60581fdc5dc33af5750e54761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.657ex; height:2.843ex;" alt="{\displaystyle y_{1}=H(x_{1}||m_{1}),y_{2}=H(x_{2}||m_{2}),...}" loading="lazy">. The overall commitment is computed as

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<annotation encoding="application/x-tex">{\displaystyle C=H(y_{1}||y_{2}||...||y_{n})}</annotation>

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</math><img src="./5839e98624516f0f5cb0436c0ccb71f5fb320381.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.465ex; height:2.843ex;" alt="{\displaystyle C=H(y_{1}||y_{2}||...||y_{n})}" loading="lazy"></dd></dl>

In order to prove one element of the vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, the prover reveals the values

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<annotation encoding="application/x-tex">{\displaystyle (i,y_{1},y_{2},...,y_{i-1},x_{i},m_{i},y_{i+1},...,y_{n})}</annotation>

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</math><img src="./693c0b656ef290bcad664c8fab5bfb7d46c3ca19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.914ex; height:2.843ex;" alt="{\displaystyle (i,y_{1},y_{2},...,y_{i-1},x_{i},m_{i},y_{i+1},...,y_{n})}" loading="lazy"></dd></dl>

The verifier is able to compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>

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</math><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle m_{i}}</annotation>

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</math><img src="./95ec8e804f69706d3f5ad235f4f983220c8df7c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}" loading="lazy">, and then is able to verify that the hash of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> values is the commitment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">.

Unfortunately the proof is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">

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<annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation>

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</math><img src="./34109fe397fdcff370079185bfdb65826cb5565a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}" loading="lazy"> in size and verification time. Alternately, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is the set of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> values, then the commitment is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation>

</semantics>

</math><img src="./34109fe397fdcff370079185bfdb65826cb5565a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}" loading="lazy"> in size, and the proof is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation>

</semantics>

</math><img src="./e66384bc40452c5452f33563fe0e27e803b0cc21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}" loading="lazy"> in size and verification time. Either way, the commitment or the proof scales with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation>

</semantics>

</math><img src="./34109fe397fdcff370079185bfdb65826cb5565a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}" loading="lazy"> which is not optimal.

<div class="mw-heading mw-heading3"><h3 id="Merkle_tree">Merkle tree</h3></div>

A common example of a practical partial reveal scheme is a <a href="Merkle_tree" title="Merkle tree">Merkle tree</a>, in which a binary hash tree is created of the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. This scheme creates commitments that are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation>

</semantics>

</math><img src="./e66384bc40452c5452f33563fe0e27e803b0cc21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}" loading="lazy"> in size, and proofs that are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\log _{2}{n})}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(\log _{2}{n})}</annotation>

</semantics>

</math><img src="./57f7ecfbc8bb146c07ac0e4c03677ceac63c5c49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.39ex; height:2.843ex;" alt="{\displaystyle O(\log _{2}{n})}" loading="lazy"> in size and verification time. The root hash of the tree is the commitment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">. To prove that a revealed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> is part of the original tree, only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}{n}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log _{2}{n}}</annotation>

</semantics>

</math><img src="./16fa1f359e8bbe6bcffefe05c75c46d5778179bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.808ex; height:2.676ex;" alt="{\displaystyle \log _{2}{n}}" loading="lazy"> hash values from the tree, one from each level, must be revealed as the proof. The verifier is able to follow the path from the claimed leaf node all the way up to the root, hashing in the sibling nodes at each level, and eventually arriving at a root node value that must equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">.<a href="#cite_note-25">[25]</a>

<div class="mw-heading mw-heading3"><h3 id="KZG_commitment">KZG commitment</h3></div>

A Kate-Zaverucha-Goldberg commitment uses <a href="Pairing-based_cryptography" title="Pairing-based cryptography">pairing-based cryptography</a> to build a partial reveal scheme with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation>

</semantics>

</math><img src="./e66384bc40452c5452f33563fe0e27e803b0cc21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}" loading="lazy"> commitment sizes, proof sizes, and proof verification time. In other words, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, the number of values in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, increases, the commitments and proofs do not get larger, and the proofs do not take any more effort to verify.

A KZG commitment requires a predetermined set of parameters to create a <a href="Pairing" title="Pairing">pairing</a>, and a trusted trapdoor element. For example, a <a href="Tate_pairing" title="Tate pairing">Tate pairing</a> can be used. Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{1},\mathbb {G} _{2}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{1},\mathbb {G} _{2}}</annotation>

</semantics>

</math><img src="./81b6e145c31fc414d8b5990a6fd6401fb16a67b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.759ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{1},\mathbb {G} _{2}}" loading="lazy"> are the additive groups, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{T}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{T}}</annotation>

</semantics>

</math><img src="./0897b2beaf6721f43453d63f48f8e5ddf706036f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{T}}" loading="lazy"> is the multiplicative group of the pairing. In other words, the pairing is the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e:\mathbb {G} _{1}\times \mathbb {G} _{2}\rightarrow \mathbb {G} _{T}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e:\mathbb {G} _{1}\times \mathbb {G} _{2}\rightarrow \mathbb {G} _{T}}</annotation>

</semantics>

</math><img src="./b0b233c58cab33f6b1f1b263299919fe0d433fed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.397ex; height:2.509ex;" alt="{\displaystyle e:\mathbb {G} _{1}\times \mathbb {G} _{2}\rightarrow \mathbb {G} _{T}}" loading="lazy">. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {F} _{p}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {F} _{p}}</annotation>

</semantics>

</math><img src="./ef4e0a0c5b5e2d32ba96e5c74dcf54f9417d80d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.16ex; height:2.843ex;" alt="{\displaystyle t\in \mathbb {F} _{p}}" loading="lazy"> be the trapdoor element (if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is the prime order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{1}}</annotation>

</semantics>

</math><img src="./726aefa2fe8762775a4d0e21d8fbc75371e631e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{2}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{2}}</annotation>

</semantics>

</math><img src="./77759bcf62cde320f8f721c589d438906aaf9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{2}}" loading="lazy">), and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> be the generators of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{1}}</annotation>

</semantics>

</math><img src="./726aefa2fe8762775a4d0e21d8fbc75371e631e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{2}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{2}}</annotation>

</semantics>

</math><img src="./77759bcf62cde320f8f721c589d438906aaf9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{2}}" loading="lazy"> respectively. As part of the parameter setup, we assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cdot t^{i}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cdot t^{i}}</annotation>

</semantics>

</math><img src="./2d27b2c7a3488e5eb89367047e7a213d0bc8d8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.145ex; height:2.676ex;" alt="{\displaystyle G\cdot t^{i}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\cdot t^{i}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\cdot t^{i}}</annotation>

</semantics>

</math><img src="./f4c3c1da94a05466fd886ce352efdd44adfe565c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.382ex; height:2.676ex;" alt="{\displaystyle H\cdot t^{i}}" loading="lazy"> are known and shared values for arbitrarily many positive integer values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, while the trapdoor value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> itself is discarded and known to no one.

<div class="mw-heading mw-heading4"><h4 id="Commit">Commit</h4></div>

A KZG commitment reformulates the vector of values to be committed as a polynomial. First, we calculate a polynomial such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(i)=x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(i)=x_{i}}</annotation>

</semantics>

</math><img src="./73f7a4c8e6fcd9b0a7f739388f37f87a81acfcbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.098ex; height:2.843ex;" alt="{\displaystyle p(i)=x_{i}}" loading="lazy"> for all values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> in our vector. <a href="Lagrange_interpolation" class="mw-redirect" title="Lagrange interpolation">Lagrange interpolation</a> allows us to compute that polynomial

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)=\sum _{i=0}^{n-1}x_{i}\prod _{0\leq j<n,j\neq i}{\frac {x-j}{i-j}}}</annotation>

</semantics>

</math><img src="./86f49d763e10e8733553176cef3e75b8c5066268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; margin-left: -0.089ex; width:28.13ex; height:7.843ex;" alt="{\displaystyle p(x)=\sum _{i=0}^{n-1}x_{i}\prod _{0\leq j<n,j\neq i}{\frac {x-j}{i-j}}}" loading="lazy"></dd></dl>

Under this formulation, the polynomial now encodes the vector, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(0)=x_{0},p(1)=x_{1},...}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(0)=x_{0},p(1)=x_{1},...}</annotation>

</semantics>

</math><img src="./37c8b8061f79285e06f86eca6b23fa942c66a32d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:24.119ex; height:2.843ex;" alt="{\displaystyle p(0)=x_{0},p(1)=x_{1},...}" loading="lazy">. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0},p_{1},...,p_{n-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{0},p_{1},...,p_{n-1}}</annotation>

</semantics>

</math><img src="./478fabc7dd97f91b948612e82359787a5e6e6dd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:15.229ex; height:2.009ex;" alt="{\displaystyle p_{0},p_{1},...,p_{n-1}}" loading="lazy"> be the coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p(x)=\sum _{i=0}^{n-1}p_{i}x^{i}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle p(x)=\sum _{i=0}^{n-1}p_{i}x^{i}}</annotation>

</semantics>

</math><img src="./4f26c91c860df1faf4368663ab0117951d7c3ab4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:17.755ex; height:3.509ex;" alt="{\textstyle p(x)=\sum _{i=0}^{n-1}p_{i}x^{i}}" loading="lazy">. The commitment is calculated as

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C=\sum _{i=0}^{n-1}p_{i}Gt^{i}}</annotation>

</semantics>

</math><img src="./0c5d935cd50f558db0fab0e77401c53de594a754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.042ex; height:7.343ex;" alt="{\displaystyle C=\sum _{i=0}^{n-1}p_{i}Gt^{i}}" loading="lazy"></dd></dl>

This is computed simply as a <a href="Dot_product" title="Dot product">dot product</a> between the predetermined values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cdot t^{i}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cdot t^{i}}</annotation>

</semantics>

</math><img src="./2d27b2c7a3488e5eb89367047e7a213d0bc8d8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.145ex; height:2.676ex;" alt="{\displaystyle G\cdot t^{i}}" loading="lazy"> and the polynomial coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation>

</semantics>

</math><img src="./5bab39399bf5424f25d957cdc57c84a0622626d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}" loading="lazy">. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{1}}</annotation>

</semantics>

</math><img src="./726aefa2fe8762775a4d0e21d8fbc75371e631e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{1}}" loading="lazy"> is an additive group with associativity and commutativity, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is equal to simply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cdot p(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cdot p(t)}</annotation>

</semantics>

</math><img src="./ab88ada53368b2d525be15d164b7f8c29ac66dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.324ex; height:2.843ex;" alt="{\displaystyle G\cdot p(t)}" loading="lazy">, since all the additions and multiplications with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> can be distributed out of the evaluation. Since the trapdoor value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is unknown, the commitment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is essentially the polynomial evaluated at a number known to no one, with the outcome obfuscated into an opaque element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{1}}</annotation>

</semantics>

</math><img src="./726aefa2fe8762775a4d0e21d8fbc75371e631e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{1}}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Reveal">Reveal</h4></div>

A KZG proof must demonstrate that the revealed data is the authentic value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> was computed. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=x_{i}}</annotation>

</semantics>

</math><img src="./480208be374db35ca5430afab6813a076ffb41ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.383ex; height:2.009ex;" alt="{\displaystyle y=x_{i}}" loading="lazy">, the revealed value we must prove. Since the vector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> was reformulated into a polynomial, we really need to prove that the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, when evaluated at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, takes on the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">. Simply, we just need to prove that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(i)=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(i)=y}</annotation>

</semantics>

</math><img src="./1c058cd022cf0f8d661deb0b1b2f094b034cc2fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.125ex; height:2.843ex;" alt="{\displaystyle p(i)=y}" loading="lazy">. We will do this by demonstrating that subtracting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> yields a root at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">. Define the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> as

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(x)={\frac {p(x)-y}{x-i}}}</annotation>

</semantics>

</math><img src="./4cc6fab66085ca7604bb7a2d4284e6574db7b640.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.447ex; height:5.843ex;" alt="{\displaystyle q(x)={\frac {p(x)-y}{x-i}}}" loading="lazy"></dd></dl>

This polynomial is itself the proof that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(i)=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(i)=y}</annotation>

</semantics>

</math><img src="./1c058cd022cf0f8d661deb0b1b2f094b034cc2fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.125ex; height:2.843ex;" alt="{\displaystyle p(i)=y}" loading="lazy">, because if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> exists, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)-y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)-y}</annotation>

</semantics>

</math><img src="./47178b352a165ce12aa32e69668186900f47e210.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.394ex; height:2.843ex;" alt="{\displaystyle p(x)-y}" loading="lazy"> is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x-i}</annotation>

</semantics>

</math><img src="./08c6415ec45d76965789b24add7cd20817617a4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.973ex; height:2.343ex;" alt="{\displaystyle x-i}" loading="lazy">, meaning it has a root at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(i)-y=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(i)-y=0}</annotation>

</semantics>

</math><img src="./18a48a96ce528086665d1544ae9d883341b19d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.127ex; height:2.843ex;" alt="{\displaystyle p(i)-y=0}" loading="lazy"> (or, in other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(i)=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(i)=y}</annotation>

</semantics>

</math><img src="./1c058cd022cf0f8d661deb0b1b2f094b034cc2fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.125ex; height:2.843ex;" alt="{\displaystyle p(i)=y}" loading="lazy">). The KZG proof will demonstrate that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> exists and has this property.

The prover computes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> through the above polynomial division, then calculates the KZG proof value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi =\sum _{i=0}^{n-1}q_{i}Gt^{i}}</annotation>

</semantics>

</math><img src="./8fefa2d81f0a365857bd322733a2e8af92eb8282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.475ex; height:7.343ex;" alt="{\displaystyle \pi =\sum _{i=0}^{n-1}q_{i}Gt^{i}}" loading="lazy"></dd></dl>

This is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cdot q(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cdot q(t)}</annotation>

</semantics>

</math><img src="./e5526669086cbe9581161f4222b8f3d798710350.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.224ex; height:2.843ex;" alt="{\displaystyle G\cdot q(t)}" loading="lazy">, as above. In other words, the proof value is the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> again evaluated at the trapdoor value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, hidden in the generator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{1}}</annotation>

</semantics>

</math><img src="./726aefa2fe8762775a4d0e21d8fbc75371e631e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{1}}" loading="lazy">.

This computation is only possible if the above polynomials were evenly divisible, because in that case the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> is a polynomial, not a <a href="Rational_function" title="Rational function">rational function</a>. Due to the construction of the trapdoor, it is not possible to evaluate a rational function at the trapdoor value, only to evaluate a polynomial using linear combinations of the precomputed known constants of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cdot t^{i}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cdot t^{i}}</annotation>

</semantics>

</math><img src="./2d27b2c7a3488e5eb89367047e7a213d0bc8d8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.145ex; height:2.676ex;" alt="{\displaystyle G\cdot t^{i}}" loading="lazy">. This is why it is impossible to create a proof for an incorrect value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Verify">Verify</h4></div>

To verify the proof, the bilinear map of the <a href="Pairing" title="Pairing">pairing</a> is used to show that the proof value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> summarizes a real polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> that demonstrates the desired property, which is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)-y}">

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<annotation encoding="application/x-tex">{\displaystyle p(x)-y}</annotation>

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</math><img src="./47178b352a165ce12aa32e69668186900f47e210.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.394ex; height:2.843ex;" alt="{\displaystyle p(x)-y}" loading="lazy"> was evenly divided by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-i}">

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<annotation encoding="application/x-tex">{\displaystyle x-i}</annotation>

</semantics>

</math><img src="./08c6415ec45d76965789b24add7cd20817617a4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.973ex; height:2.343ex;" alt="{\displaystyle x-i}" loading="lazy">. The verification computation checks the equality

<mover>

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<annotation encoding="application/x-tex">{\displaystyle e(\pi ,H\cdot t-H\cdot i)\ {\stackrel {?}{=}}\ e(C-G\cdot y,H)}</annotation>

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</math><img src="./c4fd0d08f9e0e1698fe5ca9f862432fbe6f4a89b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.454ex; height:3.843ex;" alt="{\displaystyle e(\pi ,H\cdot t-H\cdot i)\ {\stackrel {?}{=}}\ e(C-G\cdot y,H)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> is the bilinear map function as above. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\cdot t}">

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<annotation encoding="application/x-tex">{\displaystyle H\cdot t}</annotation>

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</math><img src="./c0f8e161cc6844834897caa66664d418575b28d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.582ex; height:2.176ex;" alt="{\displaystyle H\cdot t}" loading="lazy"> is a precomputed constant, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\cdot i}">

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<annotation encoding="application/x-tex">{\displaystyle H\cdot i}</annotation>

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</math><img src="./644b70a8399efbfb2643665e1fef7e430b33ffe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.545ex; height:2.176ex;" alt="{\displaystyle H\cdot i}" loading="lazy"> is computed based on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">.

By rewriting the computation in the pairing group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{T}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{T}}</annotation>

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</math><img src="./0897b2beaf6721f43453d63f48f8e5ddf706036f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{T}}" loading="lazy">, substituting in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =q(t)\cdot G}">

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<annotation encoding="application/x-tex">{\displaystyle \pi =q(t)\cdot G}</annotation>

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</math><img src="./ca6d77dd701245580e65d79c3445dc155fbc8b8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.655ex; height:2.843ex;" alt="{\displaystyle \pi =q(t)\cdot G}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=p(t)\cdot G}">

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<annotation encoding="application/x-tex">{\displaystyle C=p(t)\cdot G}</annotation>

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</math><img src="./2fc8d06e151fe4c5f3f130167b7de4a675ce2c37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.189ex; height:2.843ex;" alt="{\displaystyle C=p(t)\cdot G}" loading="lazy">, and letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (x)=e(G,H)^{x}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \tau (x)=e(G,H)^{x}}</annotation>

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</math><img src="./eebd8ef71d64cd46141ac6f6fad610d231ac0c8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.429ex; height:2.843ex;" alt="{\displaystyle \tau (x)=e(G,H)^{x}}" loading="lazy"> be a helper function for lifting into the pairing group, the proof verification is more clear.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(\pi ,H\cdot t-H\cdot i)=e(C-G\cdot y,H)}</annotation>

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</math><img src="./b9fc571ea7cef893400a53273054f05f66934fb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.583ex; height:2.843ex;" alt="{\displaystyle e(\pi ,H\cdot t-H\cdot i)=e(C-G\cdot y,H)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(G\cdot q(t),H\cdot t-H\cdot i)=e(G\cdot p(t)-G\cdot y,H)}</annotation>

</semantics>

</math><img src="./dec3d7e66c2be04e5fa9e2bebe232d80f766d221.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.034ex; height:2.843ex;" alt="{\displaystyle e(G\cdot q(t),H\cdot t-H\cdot i)=e(G\cdot p(t)-G\cdot y,H)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(G\cdot q(t),H\cdot (t-i))=e(G\cdot (p(t)-y),H)}</annotation>

</semantics>

</math><img src="./21f68b98600fcd15dbed2e4c64886b4e237fca55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.404ex; height:2.843ex;" alt="{\displaystyle e(G\cdot q(t),H\cdot (t-i))=e(G\cdot (p(t)-y),H)}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(G,H)^{q(t)\cdot (t-i)}=e(G,H)^{p(t)-y}}</annotation>

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</math><img src="./05dca2ca9a8f29be2e02f5f0f532bc1ffd1904dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.798ex; height:3.343ex;" alt="{\displaystyle e(G,H)^{q(t)\cdot (t-i)}=e(G,H)^{p(t)-y}}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau (q(t)\cdot (t-i))=\tau (p(t)-y)}</annotation>

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</math><img src="./23ccbb490f84f9fbe9163b6afe4b6026049420d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.624ex; height:2.843ex;" alt="{\displaystyle \tau (q(t)\cdot (t-i))=\tau (p(t)-y)}" loading="lazy"></dd></dl>

Assuming that the bilinear map is validly constructed, this demonstrates that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)(x-i)=p(x)-y}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle q(x)(x-i)=p(x)-y}</annotation>

</semantics>

</math><img src="./df1c5265fa800b0df3bd82b16db59068d8ba6ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.393ex; height:2.843ex;" alt="{\displaystyle q(x)(x-i)=p(x)-y}" loading="lazy">, without the validator knowing what <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> are. The validator can be assured of this because if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (q(t)\cdot (t-i))=\tau (p(t)-y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau (q(t)\cdot (t-i))=\tau (p(t)-y)}</annotation>

</semantics>

</math><img src="./23ccbb490f84f9fbe9163b6afe4b6026049420d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.624ex; height:2.843ex;" alt="{\displaystyle \tau (q(t)\cdot (t-i))=\tau (p(t)-y)}" loading="lazy">, then the polynomials evaluate to the same output at the trapdoor value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=t}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x=t}</annotation>

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</math><img src="./f6c0426ebd38c267805e4308c63f11cb976e4963.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.009ex;" alt="{\displaystyle x=t}" loading="lazy">. This demonstrates the polynomials are identical, because, if the parameters were validly constructed, the trapdoor value is known to no one, meaning that engineering a polynomial to have a specific value at the trapdoor is impossible (according to the <a href="Schwartz%E2%80%93Zippel_lemma" title="Schwartz–Zippel lemma">Schwartz–Zippel lemma</a>). If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)(x-i)=p(x)-y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(x)(x-i)=p(x)-y}</annotation>

</semantics>

</math><img src="./df1c5265fa800b0df3bd82b16db59068d8ba6ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.393ex; height:2.843ex;" alt="{\displaystyle q(x)(x-i)=p(x)-y}" loading="lazy"> is now verified to be true, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> is verified to exist, therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)-y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)-y}</annotation>

</semantics>

</math><img src="./47178b352a165ce12aa32e69668186900f47e210.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.394ex; height:2.843ex;" alt="{\displaystyle p(x)-y}" loading="lazy"> must be polynomial-divisible by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-i)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x-i)}</annotation>

</semantics>

</math><img src="./31e93651a8f09f03475c64fb06d69350527e15c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.782ex; height:2.843ex;" alt="{\displaystyle (x-i)}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(i)-y=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(i)-y=0}</annotation>

</semantics>

</math><img src="./18a48a96ce528086665d1544ae9d883341b19d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.127ex; height:2.843ex;" alt="{\displaystyle p(i)-y=0}" loading="lazy"> due to the <a href="Factor_theorem" title="Factor theorem">factor theorem</a>. This proves that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">th value of the committed vector must have equaled <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, since that is the output of evaluating the committed polynomial at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">.

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</style><div class="hidden-begin mw-collapsible mw-collapsed" style="border: 1px solid lightgray;"><div class="hidden-title skin-nightmode-reset-color" style="text-align:center; text-align:left">Why the bilinear map pairing is used</div><div class="hidden-content mw-collapsible-content" style="">

The utility of the bilinear map pairing is to allow the multiplication of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(x)}</annotation>

</semantics>

</math><img src="./c38bbafe34a043d284f19231b946a76c0a4b16b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.209ex; height:2.843ex;" alt="{\displaystyle q(x)}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x-i}</annotation>

</semantics>

</math><img src="./08c6415ec45d76965789b24add7cd20817617a4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.973ex; height:2.343ex;" alt="{\displaystyle x-i}" loading="lazy"> to happen securely. These values truly lie in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{1}}">

<msub>

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</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{1}}</annotation>

</semantics>

</math><img src="./726aefa2fe8762775a4d0e21d8fbc75371e631e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{1}}" loading="lazy">, where division is assumed to be computationally hard. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{1}}">

<msub>

</mrow>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{1}}</annotation>

</semantics>

</math><img src="./726aefa2fe8762775a4d0e21d8fbc75371e631e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{1}}" loading="lazy"> might be an <a href="Elliptic_curve" title="Elliptic curve">elliptic curve</a> over a finite field, as is common in <a href="Elliptic-curve_cryptography" title="Elliptic-curve cryptography">elliptic-curve cryptography</a>. Then, the division assumption is called the <a href="Elliptic-curve_cryptography#Rationale" title="Elliptic-curve cryptography">elliptic curve discrete logarithm problem</a>, and this assumption is also what guards the trapdoor value from being computed, making it also a foundation of KZG commitments. In that case, we want to check if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)(x-i)=p(x)-y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(x)(x-i)=p(x)-y}</annotation>

</semantics>

</math><img src="./df1c5265fa800b0df3bd82b16db59068d8ba6ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.393ex; height:2.843ex;" alt="{\displaystyle q(x)(x-i)=p(x)-y}" loading="lazy">. This cannot be done without a pairing, because with values on the curve of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cdot q(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cdot q(x)}</annotation>

</semantics>

</math><img src="./1c12c944dc0745f2814bcddfa2bf2b1efb0b90ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.714ex; height:2.843ex;" alt="{\displaystyle G\cdot q(x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cdot (x-i)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cdot (x-i)}</annotation>

</semantics>

</math><img src="./3a89b573f24fdb5ecfd685033a3340e3cfe2edda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.288ex; height:2.843ex;" alt="{\displaystyle G\cdot (x-i)}" loading="lazy">, we cannot compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cdot (q(x)(x-i))}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G\cdot (q(x)(x-i))}</annotation>

</semantics>

</math><img src="./45b26ebc233d704dd40325c90b727fb54b1cb011.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.305ex; height:2.843ex;" alt="{\displaystyle G\cdot (q(x)(x-i))}" loading="lazy">. That would violate the <a href="Computational_Diffie%E2%80%93Hellman_assumption" title="Computational Diffie–Hellman assumption">computational Diffie–Hellman assumption</a>, a foundational assumption in <a href="Elliptic-curve_cryptography" title="Elliptic-curve cryptography">elliptic-curve cryptography</a>. We instead use a <a href="Pairing" title="Pairing">pairing</a> to sidestep this problem. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle q(x)}</annotation>

</semantics>

</math><img src="./c38bbafe34a043d284f19231b946a76c0a4b16b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.209ex; height:2.843ex;" alt="{\displaystyle q(x)}" loading="lazy"> is still multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> to get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cdot q(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cdot q(x)}</annotation>

</semantics>

</math><img src="./1c12c944dc0745f2814bcddfa2bf2b1efb0b90ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.714ex; height:2.843ex;" alt="{\displaystyle G\cdot q(x)}" loading="lazy">, but the other side of the multiplication is done in the paired group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{2}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{2}}</annotation>

</semantics>

</math><img src="./77759bcf62cde320f8f721c589d438906aaf9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{2}}" loading="lazy">, so, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\cdot (t-i)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\cdot (t-i)}</annotation>

</semantics>

</math><img src="./55726f0fa39fbec250bfebc1e44175f2f85da344.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.034ex; height:2.843ex;" alt="{\displaystyle H\cdot (t-i)}" loading="lazy">. We compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(G\cdot q(t),H\cdot (t-i))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(G\cdot q(t),H\cdot (t-i))}</annotation>

</semantics>

</math><img src="./c75478b7a362c0f6380fa3927e2fc63c2e619841.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.185ex; height:2.843ex;" alt="{\displaystyle e(G\cdot q(t),H\cdot (t-i))}" loading="lazy">, which, due to the <a href="Bilinear_map" title="Bilinear map">bilinearity</a> of the map, is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(G,H)^{q(t)\cdot (t-i)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(G,H)^{q(t)\cdot (t-i)}}</annotation>

</semantics>

</math><img src="./5262da30c445b6e63cbb4c151d63cbddf792f91f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.855ex; height:3.343ex;" alt="{\displaystyle e(G,H)^{q(t)\cdot (t-i)}}" loading="lazy">. In this output group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{T}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{T}}</annotation>

</semantics>

</math><img src="./0897b2beaf6721f43453d63f48f8e5ddf706036f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{T}}" loading="lazy"> we still have the <a href="Discrete_logarithm#Cryptography" title="Discrete logarithm">discrete logarithm problem</a>, so even though we know that value and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(G,H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(G,H)}</annotation>

</semantics>

</math><img src="./234f59b180bf75400ceb09c83a92be0cd6ecb7ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.817ex; height:2.843ex;" alt="{\displaystyle e(G,H)}" loading="lazy">, we cannot extract the exponent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(t)\cdot (t-i)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(t)\cdot (t-i)}</annotation>

</semantics>

</math><img src="./bb9fb3dbeb12aa0100c24afba5a7d16452024e97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.689ex; height:2.843ex;" alt="{\displaystyle q(t)\cdot (t-i)}" loading="lazy">, preventing any contradiction with discrete logarithm earlier. This value can be compared to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(G\cdot (p(t)-y),H)=e(G,H)^{p(t)-y}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(G\cdot (p(t)-y),H)=e(G,H)^{p(t)-y}}</annotation>

</semantics>

</math><img src="./c6467f61f6f607ec35eb209557b1457e582f8738.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.063ex; height:3.343ex;" alt="{\displaystyle e(G\cdot (p(t)-y),H)=e(G,H)^{p(t)-y}}" loading="lazy"> though, and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(G,H)^{q(t)\cdot (t-i)}=e(G,H)^{p(t)-y}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(G,H)^{q(t)\cdot (t-i)}=e(G,H)^{p(t)-y}}</annotation>

</semantics>

</math><img src="./05dca2ca9a8f29be2e02f5f0f532bc1ffd1904dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.798ex; height:3.343ex;" alt="{\displaystyle e(G,H)^{q(t)\cdot (t-i)}=e(G,H)^{p(t)-y}}" loading="lazy"> we are able to conclude that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(t)\cdot (t-i)=p(t)-y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(t)\cdot (t-i)=p(t)-y}</annotation>

</semantics>

</math><img src="./c084f43c824f5828bcb37b59e2be9e120008abec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.602ex; height:2.843ex;" alt="{\displaystyle q(t)\cdot (t-i)=p(t)-y}" loading="lazy">, without ever knowing what the actual value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is, let alone <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(t)(t-i)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(t)(t-i)}</annotation>

</semantics>

</math><img src="./03bb536917e9e88a42680e6e5bbccaf0c7925b87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.01ex; height:2.843ex;" alt="{\displaystyle q(t)(t-i)}" loading="lazy">.

</div></div>

Additionally, a KZG commitment can be extended to prove the values of any arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (not just one value), with the proof size remaining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation>

</semantics>

</math><img src="./e66384bc40452c5452f33563fe0e27e803b0cc21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}" loading="lazy">, but the proof verification time scales with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(k)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(k)}</annotation>

</semantics>

</math><img src="./f5ec39041121b14e8c2b1a986c9b04547b223e3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.794ex; height:2.843ex;" alt="{\displaystyle O(k)}" loading="lazy">. The proof is the same, but instead of subtracting a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, we subtract a polynomial that causes multiple roots, at all the locations we want to prove, and instead of dividing by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x-i}</annotation>

</semantics>

</math><img src="./08c6415ec45d76965789b24add7cd20817617a4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.973ex; height:2.343ex;" alt="{\displaystyle x-i}" loading="lazy"> we divide by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{i}x-i}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \prod _{i}x-i}</annotation>

</semantics>

</math><img src="./b986715c473d365c7575efd721b8a0c023c704dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.353ex; height:3.009ex;" alt="{\textstyle \prod _{i}x-i}" loading="lazy"> for those same locations.<a href="#cite_note-26">[26]</a>

<div class="mw-heading mw-heading2"><h2 id="Quantum_bit_commitment">Quantum bit commitment</h2></div>

It is an interesting question in <a href="Quantum_cryptography" title="Quantum cryptography">quantum cryptography</a> if unconditionally secure bit commitment protocols exist on the quantum level, that is, protocols which are (at least asymptotically) binding and concealing even if there are no restrictions on the computational resources. One could hope that there might be a way to exploit the intrinsic properties of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, as in the protocols for <a href="Quantum_key_distribution" title="Quantum key distribution">unconditionally secure key distribution</a>.

However, this is impossible, as Dominic Mayers showed in 1996 <style data-mw-deduplicate="TemplateStyles:r1033199720">

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</style>(see this citation:<a href="#cite_note-27">[27]</a> for the original proof). Any such protocol can be reduced to a protocol where the system is in one of two pure states after the commitment phase, depending on the bit Alice wants to commit. If the protocol is unconditionally concealing, then Alice can unitarily transform these states into each other using the properties of the <a href="Schmidt_decomposition" title="Schmidt decomposition">Schmidt decomposition</a>, effectively defeating the binding property.

One subtle assumption of the proof is that the commit phase must be finished at some point in time. This leaves room for protocols that require a continuing information flow until the bit is unveiled or the protocol is cancelled, in which case it is not binding anymore.<a href="#cite_note-28">[28]</a> More generally, Mayers' proof applies only to protocols that exploit <a href="Quantum_physics" class="mw-redirect" title="Quantum physics">quantum physics</a> but not <a href="Special_relativity" title="Special relativity">special relativity</a>. Kent has shown that there exist unconditionally secure protocols for bit commitment that exploit the principle of <a href="Special_relativity" title="Special relativity">special relativity</a> stating that information cannot travel faster than light.<a href="#cite_note-29">[29]</a>

<div class="mw-heading mw-heading2"><h2 id="Commitments_based_on_physical_unclonable_functions">Commitments based on physical unclonable functions</h2></div>

<a href="Physical_unclonable_function" title="Physical unclonable function">Physical unclonable functions</a> (PUFs) rely on the use of a physical key with internal randomness, which is hard to clone or to emulate. Electronic, optical and other types of PUFs<a href="#cite_note-30">[30]</a> have been discussed extensively in the literature, in connection with their potential cryptographic applications including commitment schemes.<a href="#cite_note-31">[31]</a><a href="#cite_note-32">[32]</a>

<ul><li><a href="Oblivious_transfer" title="Oblivious transfer">Oblivious transfer</a></li>

<li><a href="Accumulator_(cryptography)" title="Accumulator (cryptography)">Accumulator (cryptography)</a></li>

<li><a href="Key_signing_party" title="Key signing party">Key signing party</a></li>

<li><a href="Web_of_trust" title="Web of trust">Web of trust</a></li>

<li><a href="Zerocoin" class="mw-redirect" title="Zerocoin">Zerocoin</a></li>

<li><a href="Anagram#Establishment_of_priority" title="Anagram">Anagrams</a> — used by 17th-century natural philosophers to establish priority of a discovery without revealing it to others</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><div class="reflist">

<li id="cite_note-Goldreich-1"><a href="#cite_ref-Goldreich_1-0">^</a> <a href="Oded_Goldreich" title="Oded Goldreich">Oded Goldreich</a> (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230323175232/https://www.wisdom.weizmann.ac.il/~oded/foc-book.html">Foundations of Cryptography</a>: Volume 1, Basic Tools. Cambridge University Press. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-79172-3</bdi>.: 224

</li>

<li id="cite_note-BCC-2"><a href="#cite_ref-BCC_2-0">^</a> Gilles Brassard, David Chaum, and Claude Crépeau, <a rel="nofollow" class="external text" href="http://crypto.cs.mcgill.ca/~crepeau/PDF/BCC88-jcss.pdf">Minimum Disclosure Proofs of Knowledge</a>, Journal of Computer and System Sciences, vol. 37, pp. 156–189, 1988.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFGoldreichMicaliWigderson1991" class="citation journal cs1">Goldreich, Oded; Micali, Silvio; Wigderson, Avi (1991). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F116825.116852">"Proofs that yield nothing but their validity"</a>. Journal of the ACM. 38 (3): 690–728. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.420.1478">10.1.1.420.1478</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F116825.116852">10.1145/116825.116852</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2389804">2389804</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Russell Impagliazzo, Moti Yung: Direct Minimum-Knowledge Computations. CRYPTO 1987: 40-51

</li>

<li id="cite_note-Naor-5"><a href="#cite_ref-Naor_5-0">^</a> <cite id="CITEREFNaor1991" class="citation journal cs1">Naor, Moni (1991). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00196774">"Bit commitment using pseudorandomness"</a>. Journal of Cryptology. 4 (2): 151–158. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00196774">10.1007/BF00196774</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15002247">15002247</a>.</cite>

</li>

<li id="cite_note-Crepeau-6">^ <a href="#cite_ref-Crepeau_6-0">a</a> <a href="#cite_ref-Crepeau_6-1">b</a> Claude Crépeau, <a rel="nofollow" class="external text" href="http://crypto.cs.mcgill.ca/~crepeau/PDF/Commit.pdf">Commitment</a>, Cryptography and Quantum Information Lab, <a href="McGill_University_School_of_Computer_Science" title="McGill University School of Computer Science">McGill University School of Computer Science</a>, accessed April 11, 2008

</li>

<li id="cite_note-Blum-7"><a href="#cite_ref-Blum_7-0">^</a> Manuel Blum, <a rel="nofollow" class="external text" href="https://www.cs.cmu.edu/~mblum/research/pdf/coin/in4.html">Coin Flipping by Telephone</a>, Proceedings of <a href="CRYPTO" class="mw-redirect" title="CRYPTO">CRYPTO</a> 1981, pp. 11–15, 1981, reprinted in SIGACT News vol. 15, pp. 23–27, 1983, <a href="Carnegie_Mellon_School_of_Computer_Science" title="Carnegie Mellon School of Computer Science">Carnegie Mellon School of Computer Science</a>.

</li>

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<li id="cite_note-32"><a href="#cite_ref-32">^</a> <cite id="CITEREFNikolopoulos2019" class="citation journal cs1">Nikolopoulos, Georgios M. (2019-09-30). "Optical scheme for cryptographic commitments with physical unclonable keys". Optics Express. 27 (20): 29367–29379. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1909.13094">1909.13094</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019OExpr..2729367N">2019OExpr..2729367N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FOE.27.029367">10.1364/OE.27.029367</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1094-4087">1094-4087</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/31684673">31684673</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:203593129">203593129</a>.</cite>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=355717">Quantum bit commitment on arxiv.org</a></li>

<li><a rel="nofollow" class="external text" href="https://alinush.github.io/2020/05/06/kzg-polynomial-commitments.html">Kate-Zaverucha-Goldberg (KZG) Constant-Sized Polynomial Commitments - Alin Tomescu</a></li>

<li><a rel="nofollow" class="external text" href="https://dankradfeist.de/ethereum/2020/06/16/kate-polynomial-commitments.html">Kate polynomial commitments</a></li></ul></div><div><div class="zim-footer">

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