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<title>Integrated Encryption Scheme</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Integrated Encryption Scheme</span></span>

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<p><b>Integrated Encryption Scheme</b> (<b>IES</b>) is a <a href="Hybrid_encryption" class="mw-redirect" title="Hybrid encryption">hybrid encryption</a> scheme which provides <a href="Semantic_security" title="Semantic security">semantic security</a> against an <a href="Adversary_(cryptography)" title="Adversary (cryptography)">adversary</a> who is able to use <a href="Chosen-plaintext_attack" title="Chosen-plaintext attack">chosen-plaintext</a> or <a href="Chosen-ciphertext_attack" title="Chosen-ciphertext attack">chosen-ciphertext</a> attacks. The security of the scheme is based on the computational <a href="Diffie%E2%80%93Hellman_problem" title="Diffie–Hellman problem">Diffie–Hellman problem</a>.<br>

Two variants of IES are specified: <a href="Discrete_logarithm" title="Discrete logarithm">Discrete Logarithm</a> Integrated Encryption Scheme (DLIES) and <a href="Elliptic_curve" title="Elliptic curve">Elliptic Curve</a> Integrated Encryption Scheme (ECIES), which is also known as the Elliptic Curve Augmented Encryption Scheme or simply the Elliptic Curve Encryption Scheme. These two variants are identical up to the change of an underlying group.

</p>

<div class="mw-heading mw-heading2"><h2 id="Informal_description_of_DLIES">Informal description of DLIES</h2></div>

<p>As a <i>brief and informal</i> description and overview of how IES works, a Discrete Logarithm Integrated Encryption Scheme (DLIES) is used, focusing on illuminating the reader's understanding, rather than precise technical details.

</p>

<ol><li><a href="Alice_and_Bob" title="Alice and Bob">Alice</a> learns <a href="Alice_and_Bob" title="Alice and Bob">Bob's</a> public key <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d7ba7b5f807c49eb4aecce4653d08dbb5ec2b2c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:2.676ex;" alt="{\displaystyle g^{x}}" loading="lazy"></span> through a public key infrastructure or some other distribution method.<br>Bob knows his own private key <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>.</li>

<li>Alice generates a fresh, ephemeral value <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>, and its associated public value <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{y}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./93a0e40f1301fb58b35d9930253427825b68b177.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.168ex; height:2.676ex;" alt="{\displaystyle g^{y}}" loading="lazy"></span>.</li>

<li>Alice then computes a symmetric key <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

</semantics>

</math></span><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"></span> using this information and a <a href="Key_derivation_function" title="Key derivation function">key derivation function</a> (KDF) as follows: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\textrm {KDF}}(g^{xy})}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k={\textrm {KDF}}(g^{xy})}</annotation>

</semantics>

</math></span><img src="./6de698cd307cf43098f4cb9272c764242dc29575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.328ex; height:2.843ex;" alt="{\displaystyle k={\textrm {KDF}}(g^{xy})}" loading="lazy"></span></li>

<li>Alice computes her ciphertext <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> from her actual message <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> (by symmetric encryption of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>) encrypted with the key <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> (using an <a href="Authenticated_encryption" title="Authenticated encryption">authenticated encryption scheme</a>) as follows: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=E(k;m)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=E(k;m)}</annotation>

</semantics>

</math></span><img src="./f5f430f82eb3cf6264622b3a8568d338d82664d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.976ex; height:2.843ex;" alt="{\displaystyle c=E(k;m)}" loading="lazy"></span></li>

<li>Alice transmits (in a single message) both the public ephemeral <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{y}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./93a0e40f1301fb58b35d9930253427825b68b177.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.168ex; height:2.676ex;" alt="{\displaystyle g^{y}}" loading="lazy"></span> and the ciphertext <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span>.</li>

<li>Bob, knowing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{y}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./93a0e40f1301fb58b35d9930253427825b68b177.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.168ex; height:2.676ex;" alt="{\displaystyle g^{y}}" loading="lazy"></span>, can now compute <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\textrm {KDF}}(g^{xy})}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k={\textrm {KDF}}(g^{xy})}</annotation>

</semantics>

</math></span><img src="./6de698cd307cf43098f4cb9272c764242dc29575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.328ex; height:2.843ex;" alt="{\displaystyle k={\textrm {KDF}}(g^{xy})}" loading="lazy"></span> and decrypt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span>.</li></ol>

<p>Note that the scheme does not provide Bob with any assurance as to who really sent the message: This scheme does nothing to stop anyone from pretending to be Alice.

</p>

<div class="mw-heading mw-heading2"><h2 id="Formal_description_of_ECIES">Formal description of ECIES</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Required_information">Required information</h3></div>

<p>To send an encrypted message to Bob using ECIES, Alice needs the following information:

</p>

<ul><li>The cryptography suite to be used, including a <a href="Key_derivation_function" title="Key derivation function">key derivation function</a> (e.g., <i>ANSI-X9.63-KDF with SHA-1 option</i>), a <a href="Message_authentication_code" title="Message authentication code">message authentication code system</a> (e.g., <i>HMAC-SHA-1-160 with 160-bit keys</i> or <i>HMAC-SHA-1-80 with 80-bit keys</i>) and a <a href="Symmetric-key_algorithm" title="Symmetric-key algorithm">symmetric encryption scheme</a> (e.g., <i><a href="TDEA" class="mw-redirect" title="TDEA">TDEA</a> in <a href="Cipher_block_chaining" class="mw-redirect" title="Cipher block chaining">CBC</a> mode</i> or <i>XOR encryption scheme</i>) — noted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>.</li>

<li>The elliptic curve domain parameters: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,a,b,G,n,h)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (p,a,b,G,n,h)}</annotation>

</semantics>

</math></span><img src="./855fbb66e0a1bf31aa7f20678c781678270bb231.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.936ex; height:2.843ex;" alt="{\displaystyle (p,a,b,G,n,h)}" loading="lazy"></span> for a curve over a prime field or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m,f(x),a,b,G,n,h)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./199946400bece04d8379df40a7fb043f5b02310d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.259ex; height:2.843ex;" alt="{\displaystyle (m,f(x),a,b,G,n,h)}" loading="lazy"></span> for a curve over a binary field.</li>

<li>Bob's public key <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{B}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./041770ed83382d595875532cc29acd938c39a74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.453ex; height:2.509ex;" alt="{\displaystyle K_{B}}" loading="lazy"></span>, which Bob generates it as follows: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{B}=k_{B}G}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./eebdd9dd3b5ff978d086f71bc455ef62ee83c030.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.069ex; height:2.509ex;" alt="{\displaystyle K_{B}=k_{B}G}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{B}\in [1,n-1]}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{B}\in [1,n-1]}</annotation>

</semantics>

</math></span><img src="./2fb6f96114c8a3030bb5510ac07394174e746972.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.419ex; height:2.843ex;" alt="{\displaystyle k_{B}\in [1,n-1]}" loading="lazy"></span> is the private key he chooses at random.</li>

<li>Some optional shared information: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math></span><img src="./1143e284d5f25cef778ab482edf6617a523ddd9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}" loading="lazy"></span></li>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" loading="lazy"></span> which denotes the <a href="Elliptic_curve#The_group_law" title="Elliptic curve">point at infinity</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Encryption">Encryption</h3></div>

<p>To encrypt a message <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> Alice does the following:

</p>

<ol><li>generates a random number <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in [1,n-1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\in [1,n-1]}</annotation>

</semantics>

</math></span><img src="./78c9dbe6ded14037a70d1ee4fc94a00af6ec70eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.777ex; height:2.843ex;" alt="{\displaystyle r\in [1,n-1]}" loading="lazy"></span> and calculates <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=rG}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./735a66dfdbbf409e75289bf4e8c78ea07e82f009.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle R=rG}" loading="lazy"></span></li>

<li>derives a shared secret: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=P_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a070a1b20035f67f8e6e3d904cf8ce3e1d53e5d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.262ex; height:2.509ex;" alt="{\displaystyle S=P_{x}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(P_{x},P_{y})=rK_{B}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P=(P_{x},P_{y})=rK_{B}}</annotation>

</semantics>

</math></span><img src="./ac1c9a10afb41d30098cb4cb609b31715155abfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.493ex; height:3.009ex;" alt="{\displaystyle P=(P_{x},P_{y})=rK_{B}}" loading="lazy"></span> (and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\neq O}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9f05e38836e6e4be9ac73f0ed60a70309f18ae9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.617ex; height:2.676ex;" alt="{\displaystyle P\neq O}" loading="lazy"></span>)</li>

<li>uses a <a href="Key_derivation_function" title="Key derivation function">KDF</a> to derive symmetric encryption keys and <a href="Message_authentication_code" title="Message authentication code">MAC</a> keys: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{E}\|k_{M}={\textrm {KDF}}(S\|S_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{E}\|k_{M}={\textrm {KDF}}(S\|S_{1})}</annotation>

</semantics>

</math></span><img src="./008282fa2694eb52a8bc1a2a4c68501d254f967b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.182ex; height:2.843ex;" alt="{\displaystyle k_{E}\|k_{M}={\textrm {KDF}}(S\|S_{1})}" loading="lazy"></span></li>

<li>encrypts the message: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=E(k_{E};m)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=E(k_{E};m)}</annotation>

</semantics>

</math></span><img src="./7992c64f9447d44686954506ac8ceda65b638ffa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.464ex; height:2.843ex;" alt="{\displaystyle c=E(k_{E};m)}" loading="lazy"></span></li>

<li>computes the tag of encrypted message and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1143e284d5f25cef778ab482edf6617a523ddd9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}" loading="lazy"></span>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\textrm {MAC}}(k_{M};c\|S_{2})}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d={\textrm {MAC}}(k_{M};c\|S_{2})}</annotation>

</semantics>

</math></span><img src="./1aee9c12bab96be882c360bd22edcd869fa2cba5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.529ex; height:2.843ex;" alt="{\displaystyle d={\textrm {MAC}}(k_{M};c\|S_{2})}" loading="lazy"></span></li>

<li>outputs <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\|c\|d}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a2e09e28268abdb4182ecc846407c1da5f550774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.312ex; height:2.843ex;" alt="{\displaystyle R\|c\|d}" loading="lazy"></span></li></ol>

<div class="mw-heading mw-heading3"><h3 id="Decryption">Decryption</h3></div>

<p>To decrypt the ciphertext <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\|c\|d}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a2e09e28268abdb4182ecc846407c1da5f550774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.312ex; height:2.843ex;" alt="{\displaystyle R\|c\|d}" loading="lazy"></span> Bob does the following:

</p>

<ol><li>derives the shared secret: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=P_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a070a1b20035f67f8e6e3d904cf8ce3e1d53e5d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.262ex; height:2.509ex;" alt="{\displaystyle S=P_{x}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(P_{x},P_{y})=k_{B}R}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P=(P_{x},P_{y})=k_{B}R}</annotation>

</semantics>

</math></span><img src="./52d79f9868e9b15c635a3e5843c1226b310af170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.447ex; height:3.009ex;" alt="{\displaystyle P=(P_{x},P_{y})=k_{B}R}" loading="lazy"></span> (it is the same as the one Alice derived because <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=k_{B}R=k_{B}rG=rk_{B}G=rK_{B}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P=k_{B}R=k_{B}rG=rk_{B}G=rK_{B}}</annotation>

</semantics>

</math></span><img src="./cd3e722dc1295c3c1dfc784f8e0585774cff44cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.228ex; height:2.509ex;" alt="{\displaystyle P=k_{B}R=k_{B}rG=rk_{B}G=rK_{B}}" loading="lazy"></span>), or outputs <i>failed</i> if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=O}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4ef3bf35a01a10039ac05aa2ce7602ad22350ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.617ex; height:2.176ex;" alt="{\displaystyle P=O}" loading="lazy"></span></li>

<li>derives keys the same way as Alice did: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{E}\|k_{M}={\textrm {KDF}}(S\|S_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{E}\|k_{M}={\textrm {KDF}}(S\|S_{1})}</annotation>

</semantics>

</math></span><img src="./008282fa2694eb52a8bc1a2a4c68501d254f967b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.182ex; height:2.843ex;" alt="{\displaystyle k_{E}\|k_{M}={\textrm {KDF}}(S\|S_{1})}" loading="lazy"></span></li>

<li>uses <a href="Message_authentication_code" title="Message authentication code">MAC</a> to check the tag and outputs <i>failed</i> if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\neq {\textrm {MAC}}(k_{M};c\|S_{2})}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\neq {\textrm {MAC}}(k_{M};c\|S_{2})}</annotation>

</semantics>

</math></span><img src="./ea4b7eaeccb60a792547cd471a63ec0be2d262be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.529ex; height:2.843ex;" alt="{\displaystyle d\neq {\textrm {MAC}}(k_{M};c\|S_{2})}" loading="lazy"></span></li>

<li>uses symmetric encryption scheme to decrypt the message <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=E^{-1}(k_{E};c)}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=E^{-1}(k_{E};c)}</annotation>

</semantics>

</math></span><img src="./ebb071fdae59d6c9465efec58c27783b628cafc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.815ex; height:3.176ex;" alt="{\displaystyle m=E^{-1}(k_{E};c)}" loading="lazy"></span></li></ol>

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

<ul><li><a href="SECG" title="SECG">SECG</a>, <a rel="nofollow" class="external text" href="http://www.secg.org/sec1-v2.pdf">Standards for efficient cryptography, SEC 1: Elliptic Curve Cryptography</a>, Version 2.0, May 21, 2009.</li>

<li>Gayoso Martínez, Hernández Encinas, Sánchez Ávila: <i><a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/255970113_A_Survey_of_the_Elliptic_Curve_Integrated_Encryption_Scheme">A Survey of the Elliptic Curve Integrated Encryption Scheme</a></i>, Journal of Computer Science and Engineering, 2, 2 (2010), 7–13.</li>

<li>Ladar Levison: <a rel="nofollow" class="external text" href="http://article.gmane.org/gmane.comp.encryption.openssl.devel/17997/">Code for using ECIES to protect data (ECC + AES + SHA)</a>, openssl-devel mailing list, August 6, 2010.</li>

<li><a href="IEEE_P1363#Traditional_public-key_cryptography_(IEEE_Std_1363-2000_and_1363a-2004)" title="IEEE P1363">IEEE 1363a</a> (non-public standard) specifies DLIES and ECIES</li>

<li>ANSI X9.63 (non-public standard)</li>

<li>ISO/IEC 18033-2 (non-public standard)</li>

<li>Victor Shoup, <a rel="nofollow" class="external text" href="http://www.shoup.net/papers/iso-2_1.pdf">A proposal for an ISO standard for public key encryption</a>, Version 2.1, December 20, 2001.</li>

<li>Abdalla, Michel and Bellare, Mihir and Rogaway, Phillip: <a rel="nofollow" class="external text" href="http://web.cs.ucdavis.edu/~rogaway/papers/dhies.pdf">DHIES: An Encryption Scheme Based on the Diffie–Hellman Problem</a>, IACR Cryptology ePrint Archive, 1999.</li></ul>

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</style></div><div role="navigation" class="navbox" aria-label="Navbox0" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Public-key_cryptography64" style="font-size:114%;margin:0 4em"><a href="Public-key_cryptography" title="Public-key cryptography">Public-key cryptography</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algorithms</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group wraplinks" style="width:1%"><a href="Integer_factorization" title="Integer factorization">Integer factorization</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Benaloh_cryptosystem" title="Benaloh cryptosystem">Benaloh</a></li>

<li><a href="Blum%E2%80%93Goldwasser_cryptosystem" title="Blum–Goldwasser cryptosystem">Blum–Goldwasser</a></li>

<li><a href="Cayley%E2%80%93Purser_algorithm" title="Cayley–Purser algorithm">Cayley–Purser</a></li>

<li><a href="Damg%C3%A5rd%E2%80%93Jurik_cryptosystem" title="Damgård–Jurik cryptosystem">Damgård–Jurik</a></li>

<li><a href="Goldwasser%E2%80%93Micali_cryptosystem" title="Goldwasser–Micali cryptosystem">Goldwasser–Micali</a></li>

<li><a href="Naccache%E2%80%93Stern_cryptosystem" title="Naccache–Stern cryptosystem">Naccache–Stern</a></li>

<li><a href="Paillier_cryptosystem" title="Paillier cryptosystem">Paillier</a></li>

<li><a href="Rabin_signature" class="mw-redirect" title="Rabin signature">Rabin</a></li>

<li><a href="Okamoto%E2%80%93Uchiyama_cryptosystem" title="Okamoto–Uchiyama cryptosystem">Okamoto–Uchiyama</a></li>

<li><a href="Schmidt-Samoa_cryptosystem" title="Schmidt-Samoa cryptosystem">Schmidt–Samoa</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group wraplinks" style="width:1%"><a href="Discrete_logarithm" title="Discrete logarithm">Discrete logarithm</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Cramer%E2%80%93Shoup_cryptosystem" title="Cramer–Shoup cryptosystem">Cramer–Shoup</a></li>

<li><a href="Elliptic_Curve_Digital_Signature_Algorithm" title="Elliptic Curve Digital Signature Algorithm">ECDSA</a></li>

<li><a href="EdDSA" title="EdDSA">EdDSA</a>

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

<li><a href="ElGamal_encryption" title="ElGamal encryption">ElGamal</a>

<ul><li><a href="ElGamal_signature_scheme" title="ElGamal signature scheme">signature scheme</a></li></ul></li>

<li><a href="Schnorr_signature" title="Schnorr signature">Schnorr</a></li>

<li><a href="SPEKE" title="SPEKE">SPEKE</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group wraplinks" style="width:1%"><a href="Lattice-based_cryptography" title="Lattice-based cryptography">Lattice/SVP/CVP</a>/<wbr><a href="Learning_with_errors" title="Learning with errors">LWE</a>/<wbr><a href="Short_integer_solution_problem" title="Short integer solution problem">SIS</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="BLISS_signature_scheme" title="BLISS signature scheme">BLISS</a></li>

<li><a href="Kyber" title="Kyber">Kyber</a></li>

<li><a href="NewHope" title="NewHope">NewHope</a></li>

<li><a href="NTRUEncrypt" title="NTRUEncrypt">NTRUEncrypt</a></li>

<li><a href="NTRUSign" title="NTRUSign">NTRUSign</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group wraplinks" style="width:1%">Others</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="CEILIDH" title="CEILIDH">CEILIDH</a></li>

<li><a href="Lamport_signature" title="Lamport signature">Lamport</a></li>

<li><a href="McEliece_cryptosystem" title="McEliece cryptosystem">McEliece</a></li>

<li><a href="Merkle%E2%80%93Hellman_knapsack_cryptosystem" title="Merkle–Hellman knapsack cryptosystem">Merkle–Hellman</a></li>

<li><span class="wraplinks"><a href="Naccache%E2%80%93Stern_knapsack_cryptosystem" title="Naccache–Stern knapsack cryptosystem">Naccache–Stern knapsack cryptosystem</a></span></li>

<li><a href="Three-pass_protocol" title="Three-pass protocol">Three-pass protocol</a></li>

<li><a href="SQIsign" title="SQIsign">SQIsign</a></li>

<li><a href="SPHINCS%2B" title="SPHINCS+">SPHINCS<sup>+</sup></a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Discrete_logarithm#Cryptography" title="Discrete logarithm">Discrete logarithm cryptography</a></li>

<li><a href="Elliptic-curve_cryptography" title="Elliptic-curve cryptography">Elliptic-curve cryptography</a></li>

<li><a href="Hash-based_cryptography" title="Hash-based cryptography">Hash-based cryptography</a></li>

<li><a href="Non-commutative_cryptography" title="Non-commutative cryptography">Non-commutative cryptography</a></li>

<li><a href="RSA_problem" title="RSA problem">RSA problem</a></li>

<li><a href="Trapdoor_function" title="Trapdoor function">Trapdoor function</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Standardization</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="CRYPTREC" title="CRYPTREC">CRYPTREC</a></li>

<li><a href="NESSIE" title="NESSIE">NESSIE</a></li>

<li><a href="NSA_Suite_B_Cryptography" title="NSA Suite B Cryptography">NSA Suite B</a></li>

<li><a href="NIST_Post-Quantum_Cryptography_Standardization" title="NIST Post-Quantum Cryptography Standardization">Post-Quantum Cryptography</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Digital_signature" title="Digital signature">Digital signature</a></li>

<li><a href="Public_key_fingerprint" title="Public key fingerprint">Fingerprint</a></li>

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

<li><a href="Identity-based_cryptography" title="Identity-based cryptography">Identity-based cryptography</a></li>

<li><a href="Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li>

<li><a href="OpenPGP_card" title="OpenPGP card">OpenPGP card</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Cryptography149" style="font-size:114%;margin:0 4em"><a href="Cryptography" title="Cryptography">Cryptography</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="History_of_cryptography" title="History of cryptography">History of cryptography</a></li>

<li><a href="Outline_of_cryptography" title="Outline of cryptography">Outline of cryptography</a></li>

<li><a href="Classical_cipher" title="Classical cipher">Classical cipher</a></li>

<li><a href="Cryptographic_protocol" title="Cryptographic protocol">Cryptographic protocol</a>

<ul><li><a href="Authentication_protocol" title="Authentication protocol">Authentication protocol</a></li></ul></li>

<li><a href="Cryptographic_primitive" title="Cryptographic primitive">Cryptographic primitive</a></li>

<li><a href="Cryptanalysis" title="Cryptanalysis">Cryptanalysis</a></li>

<li><a href="Cryptocurrency" title="Cryptocurrency">Cryptocurrency</a></li>

<li><a href="Cryptosystem" title="Cryptosystem">Cryptosystem</a></li>

<li><a href="Cryptographic_nonce" title="Cryptographic nonce">Cryptographic nonce</a></li>

<li><a href="Cryptovirology" title="Cryptovirology">Cryptovirology</a></li>

<li><a href="Hash_function" title="Hash function">Hash function</a>

<ul><li><a href="Cryptographic_hash_function" title="Cryptographic hash function">Cryptographic hash function</a></li>

<li><a href="Key_derivation_function" title="Key derivation function">Key derivation function</a></li>

<li><a href="Secure_Hash_Algorithms" title="Secure Hash Algorithms">Secure Hash Algorithms</a></li></ul></li>

<li><a href="Digital_signature" title="Digital signature">Digital signature</a></li>

<li><a href="Kleptography" title="Kleptography">Kleptography</a></li>

<li><a href="Key_(cryptography)" title="Key (cryptography)">Key (cryptography)</a></li>

<li><a href="Key_exchange" title="Key exchange">Key exchange</a></li>

<li><a href="Key_generator" title="Key generator">Key generator</a></li>

<li><a href="Key_schedule" title="Key schedule">Key schedule</a></li>

<li><a href="Key_stretching" title="Key stretching">Key stretching</a></li>

<li><a href="Keygen" title="Keygen">Keygen</a></li>

<li>Machines</li>

<li><a href="Cryptojacking_malware" class="mw-redirect" title="Cryptojacking malware">Cryptojacking malware</a></li>

<li><a href="Ransomware" title="Ransomware">Ransomware</a></li>

<li><a href="Random_number_generation" title="Random number generation">Random number generation</a>

<ul><li><a href="Cryptographically_secure_pseudorandom_number_generator" title="Cryptographically secure pseudorandom number generator">Cryptographically secure pseudorandom number generator</a> (CSPRNG)</li></ul></li>

<li><a href="Pseudorandom_noise" title="Pseudorandom noise">Pseudorandom noise</a> (PRN)</li>

<li><a href="Secure_channel" title="Secure channel">Secure channel</a></li>

<li><a href="Insecure_channel" class="mw-redirect" title="Insecure channel">Insecure channel</a></li>

<li><a href="Subliminal_channel" title="Subliminal channel">Subliminal channel</a></li>

<li><a href="Encryption" title="Encryption">Encryption</a></li>

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

<li><a href="End-to-end_encryption" title="End-to-end encryption">End-to-end encryption</a></li>

<li><a href="Harvest_now%2C_decrypt_later" title="Harvest now, decrypt later">Harvest now, decrypt later</a></li>

<li><a href="Information-theoretic_security" title="Information-theoretic security">Information-theoretic security</a></li>

<li><a href="Plaintext" title="Plaintext">Plaintext</a></li>

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

<li><a href="Ciphertext" title="Ciphertext">Ciphertext</a></li>

<li><a href="Shared_secret" title="Shared secret">Shared secret</a></li>

<li><a href="Trapdoor_function" title="Trapdoor function">Trapdoor function</a></li>

<li><a href="Trusted_timestamping" title="Trusted timestamping">Trusted timestamping</a></li>

<li><a href="Key-based_routing" title="Key-based routing">Key-based routing</a></li>

<li><a href="Onion_routing" title="Onion routing">Onion routing</a></li>

<li><a href="Garlic_routing" title="Garlic routing">Garlic routing</a></li>

<li><a href="Kademlia" title="Kademlia">Kademlia</a></li>

<li><a href="Mix_network" title="Mix network">Mix network</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cryptographic_hash_function" title="Cryptographic hash function">Cryptographic hash function</a></li>

<li><a href="Block_cipher" title="Block cipher">Block cipher</a></li>

<li><a href="Stream_cipher" title="Stream cipher">Stream cipher</a></li>

<li><a href="Symmetric-key_algorithm" title="Symmetric-key algorithm">Symmetric-key algorithm</a></li>

<li><a href="Authenticated_encryption" title="Authenticated encryption">Authenticated encryption</a></li>

<li><a href="Public-key_cryptography" title="Public-key cryptography">Public-key cryptography</a></li>

<li><a href="Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li>

<li><a href="Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li>

<li><a href="Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li>

<li><a href="Message_authentication_code" title="Message authentication code">Message authentication code</a></li>

<li><a href="Cryptographically_secure_pseudorandom_number_generator" title="Cryptographically secure pseudorandom number generator">Random numbers</a></li>

<li><a href="Steganography" title="Steganography">Steganography</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

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This article is issued from <a class="external text" title="Last edited on 2024-11-28" href="https://en.wikipedia.org/wiki/?title=Integrated_Encryption_Scheme&oldid=1260068330">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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