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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Distribution ensemble</span></span>
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<p>In <a href="Cryptography" title="Cryptography">cryptography</a>, a <b>distribution ensemble</b> or <b>probability ensemble</b> is a family of distributions or <a href="Random_variable" title="Random variable">random variables</a> <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=\{X_{i}\}_{i\in I}}">
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<mi>X</mi>
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</math></span><img src="./74714c2bcfee91e435aeadb9b26b8311b34b4b8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.852ex; height:2.843ex;" alt="{\displaystyle X=\{X_{i}\}_{i\in I}}" 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 I}">
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<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>
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</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span> is a random variable, or <a href="Probability_distribution" title="Probability distribution">probability distribution</a>. Often <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 I=\mathbb {N} }">
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</math></span><img src="./f00dfa40411dcce619a2d831188111085817a695.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.948ex; height:2.176ex;" alt="{\displaystyle I=\mathbb {N} }" loading="lazy"></span> and it is required that each <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_{n}}">
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</p><p>For example, a <b>uniform ensemble</b> <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 U=\{U_{n}\}_{n\in \mathbb {N} }}">
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<annotation encoding="application/x-tex">{\displaystyle U=\{U_{n}\}_{n\in \mathbb {N} }}</annotation>
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</math></span><img src="./4436654afae2e78006cb69bde69c738fbbf3d710.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.513ex; height:2.843ex;" alt="{\displaystyle U=\{U_{n}\}_{n\in \mathbb {N} }}" loading="lazy"></span> is a distribution ensemble where each <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 U_{n}}">
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<annotation encoding="application/x-tex">{\displaystyle U_{n}}</annotation>
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</math></span><img src="./67b641ce10cb5e865397c4efe647cbffed98a024.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.806ex; height:2.509ex;" alt="{\displaystyle U_{n}}" loading="lazy"></span> is <a href="Uniform_distribution_(discrete)" class="mw-redirect" title="Uniform distribution (discrete)">uniformly distributed</a> over strings of length&nbsp;<i>n</i>. In fact, many applications of probability ensembles implicitly assume that the probability spaces for the random variables all coincide in this way, so every probability ensemble is also a <a href="Stochastic_process#Definitions" title="Stochastic process">stochastic process</a>.
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Provable_security" title="Provable security">Provable security</a></li>
<li><a href="Statistically_close" class="mw-redirect" title="Statistically close">Statistically close</a></li>
<li><a href="Pseudorandom_ensemble" title="Pseudorandom ensemble">Pseudorandom ensemble</a></li>
<li><a href="Computational_indistinguishability" title="Computational indistinguishability">Computational indistinguishability</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li>Goldreich, Oded (2001). <i>Foundations of Cryptography: Volume 1, Basic Tools</i>. 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>&nbsp;<bdi>0-521-79172-3</bdi>. Fragments available at the <a rel="nofollow" class="external text" href="http://www.wisdom.weizmann.ac.il/~oded/frag.html">author's web site</a>.</li></ul>
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