SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Security parameter</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Security parameter</span></span>

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style>

<p>In <a href="Cryptography" title="Cryptography">cryptography</a>, a <b>security parameter</b> is a way of measuring of how "hard" it is for an <a href="Adversary_(cryptography)" title="Adversary (cryptography)">adversary</a> to break a cryptographic scheme. There are two main types of security parameter: <i>computational</i> and <i>statistical</i>, often denoted by <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 \kappa }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" 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 \lambda }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span>, respectively. Roughly speaking, the computational security parameter is a measure for the input size of the <a href="Computational_problem" title="Computational problem">computational problem</a> on which the cryptographic scheme is based, which determines its computational complexity, whereas the statistical security parameter is a measure of the probability with which an <a href="Adversary_(cryptography)" title="Adversary (cryptography)">adversary</a> can break the scheme (whatever that means for the protocol).

</p><p>Security parameters are usually expressed in <a href="Unary_numeral_system" title="Unary numeral system">unary representation</a> - i.e. <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 \kappa }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"></span> is expressed as a string 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 \kappa }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" 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 1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span>s, <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 \kappa =1\cdots 1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9ad62538e3b0ea4b6f574e4ca146ed75c1642033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.26ex; height:2.176ex;" alt="{\displaystyle \kappa =1\cdots 1}" loading="lazy"></span>, conventionally written as <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 1^{\kappa }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./48e057afebfa9fd906a29ee7d613797b312bd4a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.342ex; height:2.343ex;" alt="{\displaystyle 1^{\kappa }}" loading="lazy"></span> - so that the <a href="Time_complexity" title="Time complexity">time complexity</a> of the cryptographic algorithm is <a href="Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial</a> in the size of the input.

</p>

<div class="mw-heading mw-heading2"><h2 id="Computational_security">Computational security</h2></div>

<p>The security of cryptographic primitives relies on the hardness of some <a href="NP_(complexity)" title="NP (complexity)"> hard problems</a>. One sets the computational security parameter <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 \kappa }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"></span> such that <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 O(2^{\kappa })}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(2^{\kappa })}</annotation>

</semantics>

</math></span><img src="./5c999ace13c21d3f35c81074e8e4f7679577e9cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.924ex; height:2.843ex;" alt="{\displaystyle O(2^{\kappa })}" loading="lazy"></span> computation is considered <a href="Computational_complexity_theory#intractability" title="Computational complexity theory">intractable</a>.

</p>

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

<ul><li>If the security of a scheme depends on the secrecy of a key for a <a href="Pseudorandom_function" class="mw-redirect" title="Pseudorandom function">pseudorandom function</a> (PRF), then we may specify that the PRF key should be sampled from the space <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 \{0,1\}^{\kappa }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{0,1\}^{\kappa }}</annotation>

</semantics>

</math></span><img src="./1be4a297c14f7050e59eb1e101c3adaecd3376db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.863ex; height:2.843ex;" alt="{\displaystyle \{0,1\}^{\kappa }}" loading="lazy"></span> so that a <a href="Brute-force_search" title="Brute-force search">brute-force search</a> requires <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 O(2^{\kappa })}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(2^{\kappa })}</annotation>

</semantics>

<li>In the <a href="RSA_(cryptosystem)" class="mw-redirect" title="RSA (cryptosystem)">RSA cryptosystem</a>, the security parameter <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 \kappa }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"></span> denotes the length in bits of the modulus <i>n</i>; the positive integer <i>n</i> must therefore be a number in the set {0, ..., 2<sup><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 \kappa }">

</mstyle>

</mrow>

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

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Statistical_security">Statistical security</h2></div>

<p>Security in cryptography often relies on the fact that <a href="Total_variation_distance" class="mw-redirect" title="Total variation distance">statistical distance</a> between

</p>

<ul><li>a distribution predicated on a secret, and</li>

<li>a <i>simulated</i> distribution produced by an entity that does not know the secret</li></ul>

<p>is small. We formalise this using the statistical security parameter by saying that the distributions are <i>statistically close</i> if the statistical distance between distributions can be expressed as a <a href="Negligible_function" title="Negligible function">negligible function</a> in the security parameter. One sets the statistical security parameter <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 \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"></span> such that <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 O(2^{-\sigma })}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(2^{-\sigma })}</annotation>

</semantics>

</math></span><img src="./7301f82bab46430297d25144b5bee89957af8ddc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.196ex; height:3.009ex;" alt="{\displaystyle O(2^{-\sigma })}" loading="lazy"></span> is considered a "small enough" chance of the adversary winning.

</p><p>Consider the following two broad categories of attack of adversaries on a given cryptographic scheme: attacks in which the adversary tries to learn secret information, and attacks in which the adversary tries to convince an honest party to accept a false statement as true (or vice versa). In the first case, for example a <a href="Public-key_encryption" class="mw-redirect" title="Public-key encryption">public-key encryption scheme</a>, an adversary may be able to obtain a large amount of information from which he can attempt to learn secret information, e.g. by examining the distribution of ciphertexts for a fixed plaintext encrypted under different randomness. In the second case, it may be that the adversary must guess a challenge or a secret and can do so with some fixed probability; in this we can talk about distributions by considering the algorithm for sampling the challenge in the protocol. In <i>both</i> cases, we can talk about the chance of the adversary "winning" in a loose sense, and can parameterise the statistical security by requiring the distributions to be statistically close in the first case or defining a challenge space dependent on the statistical security parameter in the second case.

</p>

<div class="mw-heading mw-heading3"><h3 id="Examples_2">Examples</h3></div>

<ul><li>In <a href="Encryption" title="Encryption">encryption schemes</a>, one aspect of security is (at a high level) that anything that can be learnt about a plaintext given a <a href="Ciphertext" title="Ciphertext">ciphertext</a> can also be learnt from a randomly-sampled string (of the same length as ciphertexts) that is independent of the plaintext. Formally, one would need to show that a uniform distribution over a set of strings of fixed length is statistically close to a uniform distribution over the space of all possible ciphertexts.</li>

<li>In <a href="Zero_knowledge_proof" class="mw-redirect" title="Zero knowledge proof">zero knowledge</a> protocols, we can further subdivide the statistical security parameters into <i>zero knowledge</i> and <i>soundness</i> statistical security parameters. The former parameterises what the transcript leaks about the secret knowledge, and the latter parameterises the chance with which a dishonest prover can convince an honest verifier that he knows a secret even if he doesn't.</li>

<li>In <a href="Universal_composability" title="Universal composability">universal composability</a>, the security of a protocol relies on the statistical indistinguishability of distributions of a real-world and an ideal-world execution. Interestingly, for a <a href="Computationally_bounded_adversary" title="Computationally bounded adversary">computationally unbounded</a> environment it is not sufficient for distributions to be statistically indistinguishable since the environment can run the experiment enough times to observe which distribution is being produced (real or ideal); however, any standalone adversary against the protocol will only win with negligible probability in the statistical security parameter since it only engages in the protocol once.</li></ul>

<ul><li><a href="Security_level" title="Security level">Security level</a></li>

<li><a href="Negligible_function_(cryptography)" class="mw-redirect" title="Negligible function (cryptography)">Negligible function</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-01" href="https://en.wikipedia.org/wiki/?title=Security_parameter&oldid=1298209607">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.

</div>

</div></div>

</div>

</main>

</div>

</body></html>