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<title>Balanced Boolean function</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Balanced Boolean function</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Computer_science" title="Computer science">computer science</a>, a <b>balanced Boolean function</b> is a <a href="Boolean_function" title="Boolean function">Boolean function</a> whose output yields as many <b>0</b>s as <b>1</b>s over its <a href="Domain_of_a_function" title="Domain of a function">input set</a>. This means that for a uniformly random input string of bits, the probability of getting a <b>1</b> is 1/2.<sup id="cite_ref-bsw_1-0" class="reference"><a href="#cite_note-bsw-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<p>Examples of balanced Boolean functions are the <a href="Majority_function" title="Majority function">majority function</a>,<sup id="cite_ref-bsw_1-1" class="reference"><a href="#cite_note-bsw-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the "dictatorship function" that copies the first bit of its input to the output,<sup id="cite_ref-bsw_1-2" class="reference"><a href="#cite_note-bsw-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and the <a href="Parity_check" class="mw-redirect" title="Parity check">parity check</a> function that produces the <a href="Exclusive_or" title="Exclusive or">exclusive or</a> of the input bits.<sup id="cite_ref-ch_2-0" class="reference"><a href="#cite_note-ch-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>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 f}">
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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is a <a href="Bent_function" title="Bent function">bent function</a> on <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 n}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> bits, 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 \alpha }">
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<mi>α<!-- α --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
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</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> is any nonzero vector 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 n}">
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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> bits, then the function that maps <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}">
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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> to <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 f(x)\oplus f(x\oplus \alpha )}">
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<annotation encoding="application/x-tex">{\displaystyle f(x)\oplus f(x\oplus \alpha )}</annotation>
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</math></span><img src="./2016a011a1417290f7de7e60d91f6f14de590907.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.004ex; height:2.843ex;" alt="{\displaystyle f(x)\oplus f(x\oplus \alpha )}" loading="lazy"></span> is balanced. The bent functions are exactly the functions for which this is true, for all nonzero choices 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 \alpha }">
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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
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</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>.<sup id="cite_ref-szz_3-0" class="reference"><a href="#cite_note-szz-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p><p>The dictatorship function can be evaluated after examining only a single bit of the input, but that bit must always be examined. Benjamini, Schramm, and Wilson describe a more complex example based on <a href="Percolation_theory" title="Percolation theory">percolation theory</a> with the property that a randomized <a href="Las_Vegas_algorithm" title="Las Vegas algorithm">Las Vegas algorithm</a> can compute the function exactly while ensuring that the probability of reading any particular input bit is small, roughly inversely proportional to the square root of the number of bits.<sup id="cite_ref-bsw_1-3" class="reference"><a href="#cite_note-bsw-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Application">Application</h2></div>
<p>Balanced Boolean functions are used in <a href="Cryptography" title="Cryptography">cryptography</a>, where being balanced is one of "the most important criteria for cryptographically strong Boolean functions".<sup id="cite_ref-szz_3-1" class="reference"><a href="#cite_note-szz-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> If a function is not balanced, it will have a <a href="Statistical_bias" class="mw-redirect" title="Statistical bias">statistical bias</a>, making it subject to <a href="Cryptanalysis" title="Cryptanalysis">cryptanalysis</a> such as the <a href="Correlation_attack" title="Correlation attack">correlation attack</a>.
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-bsw-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-bsw_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-bsw_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-bsw_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-bsw_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFBenjaminiSchrammWilson2005" class="citation cs2"><a href="Itai_Benjamini" title="Itai Benjamini">Benjamini, Itai</a>; <a href="Oded_Schramm" title="Oded Schramm">Schramm, Oded</a>; Wilson, David Bruce (2005), "Balanced boolean functions that can be evaluated so that every input bit is unlikely to be read", in Gabow, Harold N.; Fagin, Ronald (eds.), <i>Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, May 22–24, 2005</i>, Association for Computing Machinery, pp. <span class="nowrap">244–</span>250, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.PR/0410282">math.PR/0410282</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1060590.1060627">10.1145/1060590.1060627</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-58113-960-8</bdi></cite></span>
</li>
<li id="cite_note-ch-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-ch_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFChakrabartyHayes1998" class="citation cs2">Chakrabarty, K.; Hayes, J.P. (1998), "Balanced Boolean functions", <i>IEE Proceedings - Computers and Digital Techniques</i>, <b>145</b> (1): 52, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1049%2Fip-cdt%3A19981769">10.1049/ip-cdt:19981769</a> (inactive 11 July 2025)</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{citation}}</code>: CS1 maint: DOI inactive as of July 2025 (link)</span></span>
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<li id="cite_note-szz-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-szz_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-szz_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFSeberryZhangZheng1993" class="citation cs2"><a href="Jennifer_Seberry" title="Jennifer Seberry">Seberry, Jennifer</a>; Zhang, Xian-Mo; Zheng, Yuliang (1993), "Nonlinearly balanced Boolean functions and their propagation characteristics", in Stinson, Douglas R. (ed.), <i>Advances in Cryptology – CRYPTO '93, 13th Annual International Cryptology Conference, Santa Barbara, California, USA, August 22–26, 1993, Proceedings</i>, Lecture Notes in Computer Science, vol. 773, Springer, pp. <span class="nowrap">49–</span>60, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-48329-2_5">10.1007/3-540-48329-2_5</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-57766-9</bdi></cite></span>
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