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id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Pseudorandomness</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p class="mw-empty-elt"> </p> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Appearing random but actually being generated by a deterministic, causal process</div> <p>A <b>pseudorandom</b> sequence of numbers is one that appears to be <a href="/wiki/Statistical_randomness" title="Statistical randomness">statistically random</a>, despite having been produced by a completely <a href="/wiki/Deterministic_system" title="Deterministic system">deterministic</a> and repeatable process.<sup id="cite_ref-RandomArticle_Phys.NYT2001_1-0" class="reference"><a href="#cite_note-RandomArticle_Phys.NYT2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Pseudorandom_number_generator" title="Pseudorandom number generator">Pseudorandom number generators</a> are often used in computer programming, as traditional sources of randomness available to humans (such as rolling dice) rely on physical processes not readily available to computer programs, although developments in <a href="/wiki/Hardware_random_number_generator" title="Hardware random number generator">hardware random number generator</a> technology have challenged this. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pseudorandomness&action=edit&section=1" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The generation of random numbers has many uses, such as for <a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">random sampling</a>, <a href="/wiki/Monte_Carlo_methods" class="mw-redirect" title="Monte Carlo methods">Monte Carlo methods</a>, <a href="/wiki/Board_game" title="Board game">board games</a>, or <a href="/wiki/Gambling" title="Gambling">gambling</a>. In <a href="/wiki/Physics" title="Physics">physics</a>, however, most processes, such as gravitational acceleration, are deterministic, meaning that they always produce the same outcome from the same starting point. Some notable exceptions are <a href="/wiki/Radioactive_decay" title="Radioactive decay">radioactive decay</a> and <a href="/wiki/Quantum_measurement" class="mw-redirect" title="Quantum measurement">quantum measurement</a>, which are both modeled as being truly random processes in the underlying physics. Since these processes are not practical sources of random numbers, pseudorandom numbers are used, which ideally have the unpredictability of a truly random sequence, despite being generated by a deterministic process.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In many applications, the deterministic process is a <a href="/wiki/Algorithm" title="Algorithm">computer algorithm</a> called a <a href="/wiki/Pseudorandom_number_generator" title="Pseudorandom number generator">pseudorandom number generator</a>, which must first be provided with a number called a <a href="/wiki/Random_seed" title="Random seed">random seed</a>. Since the same seed will yield the same sequence every time, it is important that the seed be well chosen and kept hidden, especially in <a href="/wiki/Computer_security" title="Computer security">security</a> applications, where the pattern's unpredictability is a critical feature.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In some cases where it is important for the sequence to be demonstrably unpredictable, physical sources of random numbers have been used, such as radioactive decay, atmospheric electromagnetic noise harvested from a radio tuned between stations, or intermixed timings of <a href="/wiki/Keystroke_dynamics" title="Keystroke dynamics">keystrokes</a>.<sup id="cite_ref-RandomArticle_Phys.NYT2001_1-1" class="reference"><a href="#cite_note-RandomArticle_Phys.NYT2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-RandomArticle.SS1998_4-0" class="reference"><a href="#cite_note-RandomArticle.SS1998-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The time investment needed to obtain these numbers leads to a compromise: using some of these physics readings as a seed for a pseudorandom number generator. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pseudorandomness&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Before modern computing, researchers requiring random numbers would either generate them through various means (<a href="/wiki/Dice" title="Dice">dice</a>, <a href="/wiki/Playing_cards" class="mw-redirect" title="Playing cards">cards</a>, <a href="/wiki/Roulette" title="Roulette">roulette wheels</a>,<sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> etc.) or use existing random number tables. </p><p>The first attempt to provide researchers with a ready supply of random digits was in 1927, when the Cambridge University Press published a table of 41,600 digits developed by <a href="/wiki/L.H.C._Tippett" class="mw-redirect" title="L.H.C. Tippett">L.H.C. Tippett</a>. In 1947, the <a href="/wiki/RAND_Corporation" title="RAND Corporation">RAND Corporation</a> generated numbers by the electronic simulation of a roulette wheel;<sup id="cite_ref-:0_5-1" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> the results were eventually published in 1955 as <i><a href="/wiki/A_Million_Random_Digits_with_100,000_Normal_Deviates" title="A Million Random Digits with 100,000 Normal Deviates">A Million Random Digits with 100,000 Normal Deviates</a></i>. </p> <div class="mw-heading mw-heading2"><h2 id="In_computational_complexity">In computational complexity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pseudorandomness&action=edit&section=3" title="Edit section: In computational complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>, a <a href="/wiki/Probability_distribution" title="Probability distribution">distribution</a> is <b>pseudorandom</b> against a class of adversaries if no adversary from the class can distinguish it from the uniform distribution with significant advantage.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This notion of pseudorandomness is studied in <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a> and has applications to <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>. </p><p>Formally, let <i>S</i> and <i>T</i> be finite sets and let <b>F</b> = {<i>f</i>: <i>S</i> → <i>T</i>} be a class of functions. A <a href="/wiki/Probability_distribution" title="Probability distribution">distribution</a> <b>D</b> over <i>S</i> is ε-<b>pseudorandom</b> against <b>F</b> if for every <i>f</i> in <b>F</b>, the <a href="/wiki/Total_variation_distance" class="mw-redirect" title="Total variation distance">statistical distance</a> between the distributions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b884e2d65b3356219702968b6751485fb8f38570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle f(X)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bb3d7bad157bf7c8dc994deaa8bad42db364998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.861ex; height:2.843ex;" alt="{\displaystyle f(Y)}"></span>, where <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> is sampled from <b>D</b> and <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> is sampled from the <a href="/wiki/Uniform_distribution_(discrete)" class="mw-redirect" title="Uniform distribution (discrete)">uniform distribution</a> on <i>S</i>, is at most ε. </p><p>In typical applications, the class <b>F</b> describes a model of computation with bounded resources and one is interested in designing distributions <b>D</b> with certain properties that are pseudorandom against <b>F</b>. The distribution <b>D</b> is often specified as the output of a <a href="/wiki/Pseudorandom_generator" title="Pseudorandom generator">pseudorandom generator</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pseudorandomness&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cryptographically_secure_pseudorandom_number_generator" title="Cryptographically secure pseudorandom number generator">Cryptographically secure pseudorandom number generator</a> – Type of functions designed for being unsolvable by root-finding algorithms</li> <li><a href="/wiki/List_of_random_number_generators" title="List of random number generators">List of random number generators</a></li> <li><a href="/wiki/Pseudorandom_binary_sequence" title="Pseudorandom binary sequence">Pseudorandom binary sequence</a> – Seemingly random, difficult to predict bit stream created by a deterministic algorithm</li> <li><a href="/wiki/Pseudorandom_ensemble" title="Pseudorandom ensemble">Pseudorandom ensemble</a></li> <li><a href="/wiki/Pseudorandom_number_generator" title="Pseudorandom number generator">Pseudorandom number generator</a> – Algorithm that generates an approximation of a random number sequence</li> <li><a href="/wiki/Low-discrepancy_sequence" title="Low-discrepancy sequence">Low-discrepancy sequence</a> – Type of mathematical sequence</li> <li><a href="/wiki/Random_number_generation" title="Random number generation">Random number generation</a> – Producing a sequence that cannot be predicted better than by random chance</li> <li><a href="/wiki/Pseudorandom_noise" title="Pseudorandom noise">Pseudorandom noise</a> – Pseudo-random signal with characteristics similar to noise</li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pseudorandomness&action=edit&section=5" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Donald E. Knuth (1997) <i><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>, Volume 2: Seminumerical Algorithms (3rd edition)</i>. Addison-Wesley Professional, <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-89684-2" title="Special:BookSources/0-201-89684-2">0-201-89684-2</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldreich2008" class="citation book cs1">Goldreich, Oded (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EuguvA-w5OEC"><i>Computational Complexity: A Conceptual Perspective</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88473-0" title="Special:BookSources/978-0-521-88473-0"><bdi>978-0-521-88473-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Complexity%3A+A+Conceptual+Perspective&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-0-521-88473-0&rft.aulast=Goldreich&rft.aufirst=Oded&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEuguvA-w5OEC&rfr_id=info%3Asid%2Fen.wikipedia.org%3APseudorandomness" class="Z3988"></span> See especially Chapter 8: Pseudorandom generators, pp. 284–348, and Appendix C.2: Pseudorandomness, pp. 490–493.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVadhan2012" class="citation journal cs1">Vadhan, S. P. (2012). "Pseudorandomness". <i>Foundations and Trends in Theoretical Computer Science</i>. <b>7</b> (<span class="nowrap">1–</span>3): <span class="nowrap">1–</span>336. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1561%2F0400000010">10.1561/0400000010</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+and+Trends+in+Theoretical+Computer+Science&rft.atitle=Pseudorandomness&rft.volume=7&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E336&rft.date=2012&rft_id=info%3Adoi%2F10.1561%2F0400000010&rft.aulast=Vadhan&rft.aufirst=S.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APseudorandomness" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pseudorandomness&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.fourmilab.ch/hotbits">HotBits: Genuine random numbers, generated by radioactive decay</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20051023025710/http://www.merrymeet.com/jon/usingrandom.html">Using and Creating Cryptographic-Quality Random Numbers</a></li> <li>In RFC 4086, the use of pseudorandom number sequences in cryptography is discussed at length.<sup id="cite_ref-rfc4086_8-0" class="reference"><a href="#cite_note-rfc4086-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pseudorandomness&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-RandomArticle_Phys.NYT2001-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-RandomArticle_Phys.NYT2001_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-RandomArticle_Phys.NYT2001_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_Johnson2001" class="citation news cs1">George Johnson (June 12, 2001). <a rel="nofollow" class="external text" href="https://www.nytimes.com/2001/06/12/science/connoisseurs-of-chaos-offer-a-valuable-product-randomness.html">"Connoisseurs of Chaos Offer A Valuable Product: Randomness"</a>. <i><a href="/wiki/The_New_York_Times" title="The New York Times">The New York Times</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+York+Times&rft.atitle=Connoisseurs+of+Chaos+Offer+A+Valuable+Product%3A+Randomness&rft.date=2001-06-12&rft.au=George+Johnson&rft_id=https%3A%2F%2Fwww.nytimes.com%2F2001%2F06%2F12%2Fscience%2Fconnoisseurs-of-chaos-offer-a-valuable-product-randomness.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APseudorandomness" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS._P._Vadhan2012" class="citation book cs1">S. P. Vadhan (2012). <i>Pseudorandomness</i>. <q>pseudorandomness, the theory of efficiently generating objects that "look random" despite being constructed using little or no randomness</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pseudorandomness&rft.date=2012&rft.au=S.+P.+Vadhan&rfr_id=info%3Asid%2Fen.wikipedia.org%3APseudorandomness" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_Ward2015" class="citation news cs1">Mark Ward (August 9, 2015). <a rel="nofollow" class="external text" href="https://www.bbc.com/news/technology-33839925">"Web's random numbers are too weak, researchers warn"</a>. <i>BBC</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BBC&rft.atitle=Web%27s+random+numbers+are+too+weak%2C+researchers+warn&rft.date=2015-08-09&rft.au=Mark+Ward&rft_id=https%3A%2F%2Fwww.bbc.com%2Fnews%2Ftechnology-33839925&rfr_id=info%3Asid%2Fen.wikipedia.org%3APseudorandomness" class="Z3988"></span></span> </li> <li id="cite_note-RandomArticle.SS1998-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-RandomArticle.SS1998_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJonathan_Knudson1998" class="citation magazine cs1">Jonathan Knudson (January 1998). "Javatalk: Horseshoes, hand grenades and random numbers". <i>Sun Server</i>. pp. <span class="nowrap">16–</span>17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sun+Server&rft.atitle=Javatalk%3A+Horseshoes%2C+hand+grenades+and+random+numbers&rft.pages=%3Cspan+class%3D%22nowrap%22%3E16-%3C%2Fspan%3E17&rft.date=1998-01&rft.au=Jonathan+Knudson&rfr_id=info%3Asid%2Fen.wikipedia.org%3APseudorandomness" class="Z3988"></span></span> </li> <li id="cite_note-:0-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.rand.org/pubs/monograph_reports/MR1418/index2.html">"A Million Random Digits"</a>. RAND Corporation. January 2001<span class="reference-accessdate">. Retrieved <span class="nowrap">March 30,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Million+Random+Digits&rft.pub=RAND+Corporation&rft.date=2001-01&rft_id=http%3A%2F%2Fwww.rand.org%2Fpubs%2Fmonograph_reports%2FMR1418%2Findex2.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APseudorandomness" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Oded Goldreich. Computational Complexity: A Conceptual Perspective. Cambridge University Press. 2008.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://people.seas.harvard.edu/~salil/pseudorandomness/pseudorandomness-Aug12.pdf">"Pseudorandomness"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Pseudorandomness&rft_id=https%3A%2F%2Fpeople.seas.harvard.edu%2F~salil%2Fpseudorandomness%2Fpseudorandomness-Aug12.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APseudorandomness" class="Z3988"></span></span> </li> <li id="cite_note-rfc4086-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-rfc4086_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD._Eastlake,_3rdJ._SchillerS._Crocker2005" class="citation cs1">D. Eastlake, 3rd; J. Schiller; S. Crocker (June 2005). <a rel="nofollow" class="external text" href="https://datatracker.ietf.org/doc/html/rfc4086"><i>Randomness Requirements for Security</i></a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.17487%2FRFC4086">10.17487/RFC4086</a></span>. BCP 106. <a href="/wiki/Request_for_Comments" title="Request for Comments">RFC</a> <a rel="nofollow" class="external text" href="https://datatracker.ietf.org/doc/html/rfc4086">4086</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Randomness+Requirements+for+Security&rft.date=2005-06&rft_id=info%3Adoi%2F10.17487%2FRFC4086&rft.au=D.+Eastlake%2C+3rd&rft.au=J.+Schiller&rft.au=S.+Crocker&rft_id=https%3A%2F%2Fdatatracker.ietf.org%2Fdoc%2Fhtml%2Frfc4086&rfr_id=info%3Asid%2Fen.wikipedia.org%3APseudorandomness" class="Z3988"></span> <i>Best Current Practice 106.</i> Obsoletes <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222">RFC <a rel="nofollow" class="external text" href="https://datatracker.ietf.org/doc/html/rfc1750">1750</a>. </span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Pseudorandomness&oldid=1268323558">https://en.wikipedia.org/w/index.php?title=Pseudorandomness&oldid=1268323558</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Pseudorandomness" title="Category:Pseudorandomness">Pseudorandomness</a></li><li><a href="/wiki/Category:Theoretical_computer_science" title="Category:Theoretical computer science">Theoretical computer science</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Use_American_English_from_March_2019" title="Category:Use American English from March 2019">Use American English from March 2019</a></li><li><a href="/wiki/Category:All_Wikipedia_articles_written_in_American_English" title="Category:All Wikipedia articles written in American English">All Wikipedia articles written in American English</a></li><li><a href="/wiki/Category:Use_mdy_dates_from_July_2020" title="Category:Use mdy dates from July 2020">Use mdy dates from July 2020</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 9 January 2025, at 05:44<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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