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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algorithm for computing the greatest common divisor</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_GCD_algorithm_visualisation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Binary_GCD_algorithm_visualisation.svg/400px-Binary_GCD_algorithm_visualisation.svg.png" decoding="async" width="400" height="450" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Binary_GCD_algorithm_visualisation.svg/600px-Binary_GCD_algorithm_visualisation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Binary_GCD_algorithm_visualisation.svg/800px-Binary_GCD_algorithm_visualisation.svg.png 2x" data-file-width="512" data-file-height="576" /></a><figcaption>Visualisation of using the binary GCD algorithm to find the greatest common divisor (GCD) of 36 and 24. Thus, the GCD is 2<sup>2</sup> × 3 = 12.</figcaption></figure> <p>The <b>binary GCD algorithm</b>, also known as <b>Stein's algorithm</b> or the <b>binary Euclidean algorithm</b>,<sup id="cite_ref-brenta_1-0" class="reference"><a href="#cite_note-brenta-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-brentb_2-0" class="reference"><a href="#cite_note-brentb-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is an algorithm that computes the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> (GCD) of two nonnegative integers. Stein's algorithm uses simpler arithmetic operations than the conventional <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a>; it replaces division with <a href="/wiki/Arithmetic_shift" title="Arithmetic shift">arithmetic shifts</a>, comparisons, and subtraction. </p><p>Although the algorithm in its contemporary form was first published by the physicist and programmer Josef Stein in 1967,<sup id="cite_ref-Stein_3-0" class="reference"><a href="#cite_note-Stein-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> it was known by the 2nd century BCE, in ancient China.<sup id="cite_ref-Knuth98_4-0" class="reference"><a href="#cite_note-Knuth98-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binary_GCD_algorithm&action=edit&section=1" title="Edit section: Algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The algorithm finds the GCD of two nonnegative numbers <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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> by repeatedly applying these identities: </p> <ol><li><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 \gcd(u,0)=u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(u,0)=u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef1e69125a955e23d9ba9aa85bdd454bcdabc9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.251ex; height:2.843ex;" alt="{\displaystyle \gcd(u,0)=u}"></span>: everything divides zero, 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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></span> is the largest number that divides <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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></span>.</li> <li><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 \gcd(2u,2v)=2\cdot \gcd(u,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>u</mi> <mo>,</mo> <mn>2</mn> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(2u,2v)=2\cdot \gcd(u,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a734fcf1c48050bd330630cffbaa46c4336c303c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.841ex; height:2.843ex;" alt="{\displaystyle \gcd(2u,2v)=2\cdot \gcd(u,v)}"></span>: <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" 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 2}"></span> is a common divisor.</li> <li><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 \gcd(u,2v)=\gcd(u,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mn>2</mn> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(u,2v)=\gcd(u,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09eff6604f0b548557cdd617f3467eed562f0c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.837ex; height:2.843ex;" alt="{\displaystyle \gcd(u,2v)=\gcd(u,v)}"></span> if <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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></span> is odd: <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" 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 2}"></span> is then not a common divisor.</li> <li><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 \gcd(u,v)=\gcd(u,v-u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(u,v)=\gcd(u,v-u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf044d8845bce7ab3295a4d4553c6cd8e1da03b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.844ex; height:2.843ex;" alt="{\displaystyle \gcd(u,v)=\gcd(u,v-u)}"></span> if <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 u,v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66f4b32a0181923cc1337a5634f38241e5c697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.491ex; height:2.009ex;" alt="{\displaystyle u,v}"></span> odd 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 u\leq v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>≤</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\leq v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5730c3f2fd9ab2b37f23c0f5ae368e254aeb9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.556ex; height:2.176ex;" alt="{\displaystyle u\leq v}"></span>.</li></ol> <p>As GCD is commutative (<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 \gcd(u,v)=\gcd(v,u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(u,v)=\gcd(v,u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325549fb63d7097b6d8c3bc47453e3859107664a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.674ex; height:2.843ex;" alt="{\displaystyle \gcd(u,v)=\gcd(v,u)}"></span>), those identities still apply if the operands are swapped: <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 \gcd(0,v)=v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(0,v)=v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43886dd51925016026daa8d4264a92ca4a587a59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.847ex; height:2.843ex;" alt="{\displaystyle \gcd(0,v)=v}"></span>, <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 \gcd(2u,v)=\gcd(u,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(2u,v)=\gcd(u,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8d470ea6892e95651ce14b39a24eb5f06865e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.837ex; height:2.843ex;" alt="{\displaystyle \gcd(2u,v)=\gcd(u,v)}"></span> if <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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> is odd, etc. </p> <div class="mw-heading mw-heading2"><h2 id="Implementation">Implementation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binary_GCD_algorithm&action=edit&section=2" title="Edit section: Implementation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While the above description of the algorithm is mathematically correct, performant software implementations typically differ from it in a few notable ways: </p> <ul><li>eschewing <a href="/wiki/Trial_division" title="Trial division">trial division</a> by <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" 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 2}"></span> in favour of a single bitshift and the <a href="/wiki/Count_trailing_zeros" class="mw-redirect" title="Count trailing zeros">count trailing zeros</a> primitive; this is functionally equivalent to repeatedly applying identity 3, but much faster;</li> <li>expressing the algorithm <a href="/wiki/Iteration#Computing" title="Iteration">iteratively</a> rather than <a href="/wiki/Recursion_(computer_science)" title="Recursion (computer science)">recursively</a>: the resulting implementation can be laid out to avoid repeated work, invoking identity 2 at the start and maintaining as invariant that both numbers are odd upon entering the loop, which only needs to implement identities 3 and 4;</li> <li>making the loop's body <a href="/wiki/Branch_(computer_science)#Branch-free_code" title="Branch (computer science)">branch-free</a> except for its exit condition (<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 v=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3d414a23bf4ecfa36cdd039241efc60a5bd9e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.389ex; height:2.176ex;" alt="{\displaystyle v=0}"></span>): in the example below, the exchange of <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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> (ensuring <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 u\leq v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>≤</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\leq v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5730c3f2fd9ab2b37f23c0f5ae368e254aeb9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.556ex; height:2.176ex;" alt="{\displaystyle u\leq v}"></span>) compiles down to <a href="/wiki/Conditional_moves" class="mw-redirect" title="Conditional moves">conditional moves</a>;<sup id="cite_ref-rust_disassembly_5-0" class="reference"><a href="#cite_note-rust_disassembly-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> hard-to-predict branches can have a large, negative impact on performance.<sup id="cite_ref-intel_6-0" class="reference"><a href="#cite_note-intel-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lemire_7-0" class="reference"><a href="#cite_note-lemire-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul> <p>The following is an implementation of the algorithm in <a href="/wiki/Rust_(programming_language)" title="Rust (programming language)">Rust</a> exemplifying those differences, adapted from <a rel="nofollow" class="external text" href="https://github.com/uutils/coreutils/blob/1eabda91cf35ec45c78cb95c77d5463607daed65/src/uu/factor/src/numeric/gcd.rs"><i>uutils</i></a>: </p> <div class="mw-highlight mw-highlight-lang-rust mw-content-ltr" dir="ltr"><pre><span></span><span class="k">use</span><span class="w"> </span><span class="n">std</span>::<span class="n">cmp</span>::<span class="n">min</span><span class="p">;</span> <span class="k">use</span><span class="w"> </span><span class="n">std</span>::<span class="n">mem</span>::<span class="n">swap</span><span class="p">;</span> <span class="k">pub</span><span class="w"> </span><span class="k">fn</span> <span class="nf">gcd</span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">u</span>: <span class="kt">u64</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">v</span>: <span class="kt">u64</span><span class="p">)</span><span class="w"> </span>-> <span class="kt">u64</span> <span class="p">{</span> <span class="w"> </span><span class="c1">// Base cases: gcd(n, 0) = gcd(0, n) = n</span> <span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">v</span><span class="p">;</span> <span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">v</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">u</span><span class="p">;</span> <span class="w"> </span><span class="p">}</span> <span class="w"> </span><span class="c1">// Using identities 2 and 3:</span> <span class="w"> </span><span class="c1">// gcd(2ⁱ u, 2ʲ v) = 2ᵏ gcd(u, v) with u, v odd and k = min(i, j)</span> <span class="w"> </span><span class="c1">// 2ᵏ is the greatest power of two that divides both 2ⁱ u and 2ʲ v</span> <span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">u</span><span class="p">.</span><span class="n">trailing_zeros</span><span class="p">();</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="o">>>=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span> <span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">v</span><span class="p">.</span><span class="n">trailing_zeros</span><span class="p">();</span><span class="w"> </span><span class="n">v</span><span class="w"> </span><span class="o">>>=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span> <span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span> <span class="w"> </span><span class="k">loop</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="c1">// u and v are odd at the start of the loop</span> <span class="w"> </span><span class="fm">debug_assert!</span><span class="p">(</span><span class="n">u</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="s">"u = {} should be odd"</span><span class="p">,</span><span class="w"> </span><span class="n">u</span><span class="p">);</span> <span class="w"> </span><span class="fm">debug_assert!</span><span class="p">(</span><span class="n">v</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="s">"v = {} should be odd"</span><span class="p">,</span><span class="w"> </span><span class="n">v</span><span class="p">);</span> <span class="w"> </span><span class="c1">// Swap if necessary so u ≤ v</span> <span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">v</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="n">swap</span><span class="p">(</span><span class="o">&</span><span class="k">mut</span><span class="w"> </span><span class="n">u</span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="k">mut</span><span class="w"> </span><span class="n">v</span><span class="p">);</span> <span class="w"> </span><span class="p">}</span> <span class="w"> </span><span class="c1">// Identity 4: gcd(u, v) = gcd(u, v-u) as u ≤ v and u, v are both odd </span> <span class="w"> </span><span class="n">v</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">u</span><span class="p">;</span> <span class="w"> </span><span class="c1">// v is now even</span> <span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">v</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="c1">// Identity 1: gcd(u, 0) = u</span> <span class="w"> </span><span class="c1">// The shift by k is necessary to add back the 2ᵏ factor that was removed before the loop</span> <span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">k</span><span class="p">;</span> <span class="w"> </span><span class="p">}</span> <span class="w"> </span><span class="c1">// Identity 3: gcd(u, 2ʲ v) = gcd(u, v) as u is odd</span> <span class="w"> </span><span class="n">v</span><span class="w"> </span><span class="o">>>=</span><span class="w"> </span><span class="n">v</span><span class="p">.</span><span class="n">trailing_zeros</span><span class="p">();</span> <span class="w"> </span><span class="p">}</span> <span class="p">}</span> </pre></div> <p><b>Note</b>: The implementation above accepts <i>unsigned</i> (non-negative) integers; given that <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 \gcd(u,v)=\gcd(\pm {}u,\pm {}v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>u</mi> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(u,v)=\gcd(\pm {}u,\pm {}v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd0c003fe1a8645c748a877d25d24470a4af7a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.291ex; height:2.843ex;" alt="{\displaystyle \gcd(u,v)=\gcd(\pm {}u,\pm {}v)}"></span>, the signed case can be handled as follows: </p> <div class="mw-highlight mw-highlight-lang-rust mw-content-ltr" dir="ltr"><pre><span></span><span class="sd">/// Computes the GCD of two signed 64-bit integers</span> <span class="sd">/// The result is unsigned and not always representable as i64: gcd(i64::MIN, i64::MIN) == 1 << 63</span> <span class="k">pub</span><span class="w"> </span><span class="k">fn</span> <span class="nf">signed_gcd</span><span class="p">(</span><span class="n">u</span>: <span class="kt">i64</span><span class="p">,</span><span class="w"> </span><span class="n">v</span>: <span class="kt">i64</span><span class="p">)</span><span class="w"> </span>-> <span class="kt">u64</span> <span class="p">{</span> <span class="w"> </span><span class="n">gcd</span><span class="p">(</span><span class="n">u</span><span class="p">.</span><span class="n">unsigned_abs</span><span class="p">(),</span><span class="w"> </span><span class="n">v</span><span class="p">.</span><span class="n">unsigned_abs</span><span class="p">())</span> <span class="p">}</span> </pre></div> <div class="mw-heading mw-heading2"><h2 id="Complexity">Complexity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binary_GCD_algorithm&action=edit&section=3" title="Edit section: Complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Big_O_notation" title="Big O notation">Asymptotically</a>, the algorithm requires <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 O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></span> steps, 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span> is the number of bits in the larger of the two numbers, as every two steps reduce at least one of the operands by at least a factor of <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" 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 2}"></span>. Each step involves only a few arithmetic operations (<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 O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span> with a small constant); when working with <a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">word-sized</a> numbers, each arithmetic operation translates to a single machine operation, so the number of machine operations is on the order of <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span>, i.e. <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 \log _{2}(\max(u,v))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}(\max(u,v))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4512d4527a1ae1df8b6402d46e25fd9db8ebdee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.462ex; height:2.843ex;" alt="{\displaystyle \log _{2}(\max(u,v))}"></span>. </p><p>For arbitrarily large numbers, the <a href="/wiki/Asymptotic_notation" class="mw-redirect" title="Asymptotic notation">asymptotic complexity</a> of this algorithm is <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 O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd9594a16cb898b8f2a2dff9227a385ec183392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{2})}"></span>,<sup id="cite_ref-gmplib_8-0" class="reference"><a href="#cite_note-gmplib-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> as each arithmetic operation (subtract and shift) involves a linear number of machine operations (one per word in the numbers' binary representation). If the numbers can be represented in the machine's memory, <i>i.e.</i> each number's <i>size</i> can be represented by a single machine word, this bound is reduced to: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\left({\frac {n^{2}}{\log _{2}n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O\left({\frac {n^{2}}{\log _{2}n}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed2935de32d92982c969098b2179b7c8a516da3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.226ex; height:6.343ex;" alt="{\displaystyle O\left({\frac {n^{2}}{\log _{2}n}}\right)}"></span> </p><p>This is the same as for the Euclidean algorithm, though a more precise analysis by Akhavi and Vallée proved that binary GCD uses about 60% fewer bit operations.<sup id="cite_ref-bit-complexity_9-0" class="reference"><a href="#cite_note-bit-complexity-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binary_GCD_algorithm&action=edit&section=4" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The binary GCD algorithm can be extended in several ways, either to output additional information, deal with <a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">arbitrarily large integers</a> more efficiently, or to compute GCDs in domains other than the integers. </p><p>The <i>extended binary GCD</i> algorithm, analogous to the <a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">extended Euclidean algorithm</a>, fits in the first kind of extension, as it provides the <a href="/wiki/B%C3%A9zout_coefficients" class="mw-redirect" title="Bézout coefficients">Bézout coefficients</a> in addition to the GCD: integers <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> such that <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 a\cdot {}u+b\cdot {}v=\gcd(u,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>u</mi> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>v</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot {}u+b\cdot {}v=\gcd(u,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0abdb2d9756b77a30737c62f6f7b54c40e8d5ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.77ex; height:2.843ex;" alt="{\displaystyle a\cdot {}u+b\cdot {}v=\gcd(u,v)}"></span>.<sup id="cite_ref-egcd-knuth_10-0" class="reference"><a href="#cite_note-egcd-knuth-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-egcd-applied-crypto_11-0" class="reference"><a href="#cite_note-egcd-applied-crypto-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-egcd-cohen_12-0" class="reference"><a href="#cite_note-egcd-cohen-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>In the case of large integers, the best asymptotic complexity is <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 O(M(n)\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(M(n)\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e57e81074d7e9445ed9756baeccc8406f25a8a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.369ex; height:2.843ex;" alt="{\displaystyle O(M(n)\log n)}"></span>, with <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 M(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c856803491e153263b91aacc660e26ef129d467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.646ex; height:2.843ex;" alt="{\displaystyle M(n)}"></span> the cost of <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span>-bit multiplication; this is near-linear and vastly smaller than the binary GCD algorithm's <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 O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd9594a16cb898b8f2a2dff9227a385ec183392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{2})}"></span>, though concrete implementations only outperform older algorithms for numbers larger than about 64 kilobits (<i>i.e.</i> greater than 8×10<sup>19265</sup>). This is achieved by extending the binary GCD algorithm using ideas from the <a href="/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm" title="Schönhage–Strassen algorithm">Schönhage–Strassen algorithm</a> for fast integer multiplication.<sup id="cite_ref-stehlé-zimmermann_13-0" class="reference"><a href="#cite_note-stehlé-zimmermann-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>The binary GCD algorithm has also been extended to domains other than natural numbers, such as <a href="/wiki/Gaussian_integers" class="mw-redirect" title="Gaussian integers">Gaussian integers</a>,<sup id="cite_ref-weilert_14-0" class="reference"><a href="#cite_note-weilert-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Eisenstein_integers" class="mw-redirect" title="Eisenstein integers">Eisenstein integers</a>,<sup id="cite_ref-eisenstein_15-0" class="reference"><a href="#cite_note-eisenstein-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> quadratic rings,<sup id="cite_ref-some-quadratic-rings_16-0" class="reference"><a href="#cite_note-some-quadratic-rings-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-UFD-quadratic-rings_17-0" class="reference"><a href="#cite_note-UFD-quadratic-rings-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Ring_of_integers" title="Ring of integers">integer rings</a> of <a href="/wiki/Number_fields" class="mw-redirect" title="Number fields">number fields</a>.<sup id="cite_ref-integer-rings_18-0" class="reference"><a href="#cite_note-integer-rings-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Historical_description">Historical description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binary_GCD_algorithm&action=edit&section=5" title="Edit section: Historical description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An algorithm for computing the GCD of two numbers was known in ancient China, under the <a href="/wiki/Han_dynasty" title="Han dynasty">Han dynasty</a>, as a method to reduce fractions: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>If possible halve it; otherwise, take the denominator and the numerator, subtract the lesser from the greater, and do that alternately to make them the same. Reduce by the same number.</p><div class="templatequotecite">— <cite>Fangtian – Land surveying, <i><a href="/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">The Nine Chapters on the Mathematical Art</a></i></cite></div></blockquote> <p>The phrase "if possible halve it" is ambiguous,<sup id="cite_ref-Knuth98_4-1" class="reference"><a href="#cite_note-Knuth98-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <ul><li>if this applies when <i>either</i> of the numbers become even, the algorithm is the binary GCD algorithm;</li> <li>if this only applies when <i>both</i> numbers are even, the algorithm is similar to the <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a>.</li></ul> <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=Binary_GCD_algorithm&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output 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href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a></li> <li><a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">Extended Euclidean algorithm</a></li> <li><a href="/wiki/Least_common_multiple" title="Least common multiple">Least common multiple</a></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=Binary_GCD_algorithm&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 mw-references-columns"><ol class="references"> <li id="cite_note-brenta-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-brenta_1-0">^</a></b></span> <span class="reference-text"><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><cite id="CITEREFBrent1999" class="citation conference cs1"><a href="/wiki/Richard_P._Brent" title="Richard P. Brent">Brent, Richard P.</a> (13–15 September 1999). <a rel="nofollow" class="external text" href="http://wwwmaths.anu.edu.au/~brent/pub/pub183.html"><i>Twenty years' analysis of the Binary Euclidean Algorithm</i></a>. 1999 Oxford-Microsoft Symposium in honour of Professor Sir Antony Hoare. Oxford.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Twenty+years%27+analysis+of+the+Binary+Euclidean+Algorithm&rft.place=Oxford&rft.date=1999-09-13%2F1999-09-15&rft.aulast=Brent&rft.aufirst=Richard+P.&rft_id=http%3A%2F%2Fwwwmaths.anu.edu.au%2F~brent%2Fpub%2Fpub183.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-brentb-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-brentb_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrent1999" class="citation techreport cs1"><a href="/wiki/Richard_P._Brent" title="Richard P. Brent">Brent, Richard P.</a> (November 1999). <a rel="nofollow" class="external text" href="https://www.cs.ox.ac.uk/techreports/oucl/tr-7-99.html"><i>Further analysis of the Binary Euclidean algorithm</i></a> (Technical report). Oxford University Computing Laboratory. <a href="/wiki/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/1303.2772">1303.2772</a></span>. PRG TR-7-99.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Further+analysis+of+the+Binary+Euclidean+algorithm&rft.pub=Oxford+University+Computing+Laboratory&rft.date=1999-11&rft_id=info%3Aarxiv%2F1303.2772&rft.aulast=Brent&rft.aufirst=Richard+P.&rft_id=https%3A%2F%2Fwww.cs.ox.ac.uk%2Ftechreports%2Foucl%2Ftr-7-99.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-Stein-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stein_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStein1967" class="citation cs2">Stein, J. (February 1967), "Computational problems associated with Racah algebra", <i>Journal of Computational Physics</i>, <b>1</b> (3): 397–405, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967JCoPh...1..397S">1967JCoPh...1..397S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2867%2990047-2">10.1016/0021-9991(67)90047-2</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-9991">0021-9991</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Physics&rft.atitle=Computational+problems+associated+with+Racah+algebra&rft.volume=1&rft.issue=3&rft.pages=397-405&rft.date=1967-02&rft.issn=0021-9991&rft_id=info%3Adoi%2F10.1016%2F0021-9991%2867%2990047-2&rft_id=info%3Abibcode%2F1967JCoPh...1..397S&rft.aulast=Stein&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-Knuth98-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Knuth98_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Knuth98_4-1"><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="CITEREFKnuth1998" class="citation cs2"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1998), <i>Seminumerical Algorithms</i>, <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>, vol. 2 (3rd ed.), Addison-Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-89684-8" title="Special:BookSources/978-0-201-89684-8"><bdi>978-0-201-89684-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Seminumerical+Algorithms&rft.series=The+Art+of+Computer+Programming&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=1998&rft.isbn=978-0-201-89684-8&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-rust_disassembly-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-rust_disassembly_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGodbolt" class="citation web cs1">Godbolt, Matt. <a rel="nofollow" class="external text" href="https://rust.godbolt.org/z/56jva3KPn">"Compiler Explorer"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">4 February</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Compiler+Explorer&rft.aulast=Godbolt&rft.aufirst=Matt&rft_id=https%3A%2F%2Frust.godbolt.org%2Fz%2F56jva3KPn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-intel-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-intel_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKapoor2009" class="citation web cs1">Kapoor, Rajiv (21 February 2009). <a rel="nofollow" class="external text" href="https://software.intel.com/content/www/us/en/develop/articles/avoiding-the-cost-of-branch-misprediction.html">"Avoiding the Cost of Branch Misprediction"</a>. <i>Intel Developer Zone</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Intel+Developer+Zone&rft.atitle=Avoiding+the+Cost+of+Branch+Misprediction&rft.date=2009-02-21&rft.aulast=Kapoor&rft.aufirst=Rajiv&rft_id=https%3A%2F%2Fsoftware.intel.com%2Fcontent%2Fwww%2Fus%2Fen%2Fdevelop%2Farticles%2Favoiding-the-cost-of-branch-misprediction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-lemire-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-lemire_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLemire2019" class="citation web cs1">Lemire, Daniel (15 October 2019). <a rel="nofollow" class="external text" href="https://lemire.me/blog/2019/10/15/mispredicted-branches-can-multiply-your-running-times/">"Mispredicted branches can multiply your running times"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mispredicted+branches+can+multiply+your+running+times&rft.date=2019-10-15&rft.aulast=Lemire&rft.aufirst=Daniel&rft_id=https%3A%2F%2Flemire.me%2Fblog%2F2019%2F10%2F15%2Fmispredicted-branches-can-multiply-your-running-times%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-gmplib-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-gmplib_8-0">^</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="http://gmplib.org/manual/Binary-GCD.html">"GNU MP 6.1.2: Binary GCD"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=GNU+MP+6.1.2%3A+Binary+GCD&rft_id=http%3A%2F%2Fgmplib.org%2Fmanual%2FBinary-GCD.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-bit-complexity-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-bit-complexity_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAkhaviVallée2000" class="citation cs2">Akhavi, Ali; Vallée, Brigitte (2000), <a rel="nofollow" class="external text" href="https://vallee.users.greyc.fr/Publications/icalp8-2000.ps">"Average Bit-Complexity of Euclidean Algorithms"</a>, <i>Proceedings ICALP'00, Lecture Notes Computer Science 1853</i>: 373–387, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.7616">10.1.1.42.7616</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+ICALP%2700%2C+Lecture+Notes+Computer+Science+1853&rft.atitle=Average+Bit-Complexity+of+Euclidean+Algorithms&rft.pages=373-387&rft.date=2000&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.42.7616%23id-name%3DCiteSeerX&rft.aulast=Akhavi&rft.aufirst=Ali&rft.au=Vall%C3%A9e%2C+Brigitte&rft_id=https%3A%2F%2Fvallee.users.greyc.fr%2FPublications%2Ficalp8-2000.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-egcd-knuth-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-egcd-knuth_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKnuth1998">Knuth 1998</a>, p. 646, answer to exercise 39 of section 4.5.2</span> </li> <li id="cite_note-egcd-applied-crypto-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-egcd-applied-crypto_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMenezesvan_OorschotVanstone1996" class="citation book cs1">Menezes, Alfred J.; van Oorschot, Paul C.; Vanstone, Scott A. (October 1996). <a rel="nofollow" class="external text" href="http://cacr.uwaterloo.ca/hac/about/chap14.pdf#page=17">"§14.4 Greatest Common Divisor Algorithms"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="http://cacr.uwaterloo.ca/hac/"><i>Handbook of Applied Cryptography</i></a>. CRC Press. pp. 606–610. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8493-8523-7" title="Special:BookSources/0-8493-8523-7"><bdi>0-8493-8523-7</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">9 September</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A714.4+Greatest+Common+Divisor+Algorithms&rft.btitle=Handbook+of+Applied+Cryptography&rft.pages=606-610&rft.pub=CRC+Press&rft.date=1996-10&rft.isbn=0-8493-8523-7&rft.aulast=Menezes&rft.aufirst=Alfred+J.&rft.au=van+Oorschot%2C+Paul+C.&rft.au=Vanstone%2C+Scott+A.&rft_id=http%3A%2F%2Fcacr.uwaterloo.ca%2Fhac%2Fabout%2Fchap14.pdf%23page%3D17&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-egcd-cohen-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-egcd-cohen_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen1993" class="citation book cs1"><a href="/wiki/Henri_Cohen_(number_theorist)" title="Henri Cohen (number theorist)">Cohen, Henri</a> (1993). "Chapter 1 : Fundamental Number-Theoretic Algorithms". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hXGr-9l1DXcC"><i>A Course In Computational Algebraic Number Theory</i></a>. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 138. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 17–18. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-55640-0" title="Special:BookSources/0-387-55640-0"><bdi>0-387-55640-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+1+%3A+Fundamental+Number-Theoretic+Algorithms&rft.btitle=A+Course+In+Computational+Algebraic+Number+Theory&rft.series=Graduate+Texts+in+Mathematics&rft.pages=17-18&rft.pub=Springer-Verlag&rft.date=1993&rft.isbn=0-387-55640-0&rft.aulast=Cohen&rft.aufirst=Henri&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhXGr-9l1DXcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-stehlé-zimmermann-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-stehlé-zimmermann_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStehléZimmermann2004" class="citation cs2">Stehlé, Damien; <a href="/wiki/Paul_Zimmermann_(mathematician)" title="Paul Zimmermann (mathematician)">Zimmermann, Paul</a> (2004), <a rel="nofollow" class="external text" href="http://hal.archives-ouvertes.fr/docs/00/07/15/33/PDF/RR-5050.pdf">"A binary recursive gcd algorithm"</a> <span class="cs1-format">(PDF)</span>, <i>Algorithmic number theory</i>, Lecture Notes in Comput. Sci., vol. 3076, Springer, Berlin, pp. 411–425, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.107.8612">10.1.1.107.8612</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-24847-7_31">10.1007/978-3-540-24847-7_31</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-22156-2" title="Special:BookSources/978-3-540-22156-2"><bdi>978-3-540-22156-2</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2138011">2138011</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3119374">3119374</a>, INRIA Research Report RR-5050</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+binary+recursive+gcd+algorithm&rft.btitle=Algorithmic+number+theory&rft.series=Lecture+Notes+in+Comput.+Sci.&rft.pages=411-425&rft.pub=Springer%2C+Berlin&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3119374%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-540-24847-7_31&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.107.8612%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2138011%23id-name%3DMR&rft.isbn=978-3-540-22156-2&rft.aulast=Stehl%C3%A9&rft.aufirst=Damien&rft.au=Zimmermann%2C+Paul&rft_id=http%3A%2F%2Fhal.archives-ouvertes.fr%2Fdocs%2F00%2F07%2F15%2F33%2FPDF%2FRR-5050.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span>.</span> </li> <li id="cite_note-weilert-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-weilert_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeilert2000" class="citation journal cs1">Weilert, André (July 2000). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjsco.2000.0422">"(1+i)-ary GCD Computation in Z[i] as an Analogue to the Binary GCD Algorithm"</a>. <i>Journal of Symbolic Computation</i>. <b>30</b> (5): 605–617. <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/10.1006%2Fjsco.2000.0422">10.1006/jsco.2000.0422</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Computation&rft.atitle=%281%2Bi%29-ary+GCD+Computation+in+Z%5Bi%5D+as+an+Analogue+to+the+Binary+GCD+Algorithm&rft.volume=30&rft.issue=5&rft.pages=605-617&rft.date=2000-07&rft_id=info%3Adoi%2F10.1006%2Fjsco.2000.0422&rft.aulast=Weilert&rft.aufirst=Andr%C3%A9&rft_id=https%3A%2F%2Fdoi.org%2F10.1006%252Fjsco.2000.0422&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-eisenstein-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-eisenstein_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDamgårdFrandsen2003" class="citation conference cs1">Damgård, Ivan Bjerre; Frandsen, Gudmund Skovbjerg (12–15 August 2003). <i>Efficient Algorithms for GCD and Cubic Residuosity in the Ring of Eisenstein Integers</i>. 14th International Symposium on the Fundamentals of Computation Theory. <a href="/wiki/Malm%C3%B6" title="Malmö">Malmö</a>, Sweden. pp. 109–117. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-45077-1_11">10.1007/978-3-540-45077-1_11</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Efficient+Algorithms+for+GCD+and+Cubic+Residuosity+in+the+Ring+of+Eisenstein+Integers&rft.place=Malm%C3%B6%2C+Sweden&rft.pages=109-117&rft.date=2003-08-12%2F2003-08-15&rft_id=info%3Adoi%2F10.1007%2F978-3-540-45077-1_11&rft.aulast=Damg%C3%A5rd&rft.aufirst=Ivan+Bjerre&rft.au=Frandsen%2C+Gudmund+Skovbjerg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-some-quadratic-rings-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-some-quadratic-rings_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAgarwalFrandsen2004" class="citation conference cs1">Agarwal, Saurabh; Frandsen, Gudmund Skovbjerg (13–18 June 2004). <i>Binary GCD Like Algorithms for Some Complex Quadratic Rings</i>. Algorithmic Number Theory Symposium. <a href="/wiki/Burlington,_VT" class="mw-redirect" title="Burlington, VT">Burlington, VT</a>, USA. pp. 57–71. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-24847-7_4">10.1007/978-3-540-24847-7_4</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Binary+GCD+Like+Algorithms+for+Some+Complex+Quadratic+Rings&rft.place=Burlington%2C+VT%2C+USA&rft.pages=57-71&rft.date=2004-06-13%2F2004-06-18&rft_id=info%3Adoi%2F10.1007%2F978-3-540-24847-7_4&rft.aulast=Agarwal&rft.aufirst=Saurabh&rft.au=Frandsen%2C+Gudmund+Skovbjerg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-UFD-quadratic-rings-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-UFD-quadratic-rings_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAgarwalFrandsen2006" class="citation conference cs1">Agarwal, Saurabh; Frandsen, Gudmund Skovbjerg (20–24 March 2006). <i>A New GCD Algorithm for Quadratic Number Rings with Unique Factorization</i>. 7th Latin American Symposium on Theoretical Informatics. Valdivia, Chile. pp. 30–42. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F11682462_8">10.1007/11682462_8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=A+New+GCD+Algorithm+for+Quadratic+Number+Rings+with+Unique+Factorization&rft.place=Valdivia%2C+Chile&rft.pages=30-42&rft.date=2006-03-20%2F2006-03-24&rft_id=info%3Adoi%2F10.1007%2F11682462_8&rft.aulast=Agarwal&rft.aufirst=Saurabh&rft.au=Frandsen%2C+Gudmund+Skovbjerg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-integer-rings-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-integer-rings_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWikström2005" class="citation conference cs1">Wikström, Douglas (11–15 July 2005). <i>On the l-Ary GCD-Algorithm in Rings of Integers</i>. Automata, Languages and Programming, 32nd International Colloquium. Lisbon, Portugal. pp. 1189–1201. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F11523468_96">10.1007/11523468_96</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=On+the+l-Ary+GCD-Algorithm+in+Rings+of+Integers&rft.place=Lisbon%2C+Portugal&rft.pages=1189-1201&rft.date=2005-07-11%2F2005-07-15&rft_id=info%3Adoi%2F10.1007%2F11523468_96&rft.aulast=Wikstr%C3%B6m&rft.aufirst=Douglas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></span> </li> </ol></div> <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=Binary_GCD_algorithm&action=edit&section=8" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1998). "§4.5 Rational arithmetic". <i>Seminumerical Algorithms</i>. <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>. Vol. 2 (3rd ed.). Addison-Wesley. pp. 330–417. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-89684-8" title="Special:BookSources/978-0-201-89684-8"><bdi>978-0-201-89684-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A74.5+Rational+arithmetic&rft.btitle=Seminumerical+Algorithms&rft.series=The+Art+of+Computer+Programming&rft.pages=330-417&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=1998&rft.isbn=978-0-201-89684-8&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></li></ul> <p>Covers the extended binary GCD, and a probabilistic analysis of the algorithm. </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen1993" class="citation book cs1"><a href="/wiki/Henri_Cohen_(number_theorist)" title="Henri Cohen (number theorist)">Cohen, Henri</a> (1993). "Chapter 1 : Fundamental Number-Theoretic Algorithms". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hXGr-9l1DXcC"><i>A Course In Computational Algebraic Number Theory</i></a>. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 138. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 12–24. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-55640-0" title="Special:BookSources/0-387-55640-0"><bdi>0-387-55640-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+1+%3A+Fundamental+Number-Theoretic+Algorithms&rft.btitle=A+Course+In+Computational+Algebraic+Number+Theory&rft.series=Graduate+Texts+in+Mathematics&rft.pages=12-24&rft.pub=Springer-Verlag&rft.date=1993&rft.isbn=0-387-55640-0&rft.aulast=Cohen&rft.aufirst=Henri&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhXGr-9l1DXcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></li></ul> <p>Covers a variety of topics, including the extended binary GCD algorithm which outputs <a href="/wiki/B%C3%A9zout_coefficients" class="mw-redirect" title="Bézout coefficients">Bézout coefficients</a>, efficient handling of multi-precision integers using a variant of <a href="/wiki/Lehmer%27s_GCD_algorithm" title="Lehmer's GCD algorithm">Lehmer's GCD algorithm</a>, and the relationship between GCD and <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">continued fraction expansions</a> of real numbers. </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVallée1998" class="citation journal cs1"><a href="/wiki/Brigitte_Vall%C3%A9e" title="Brigitte Vallée">Vallée, Brigitte</a> (September–October 1998). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110513012901/http://users.info.unicaen.fr/~brigitte/Publications/bin-gcd.ps">"Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators"</a>. <i>Algorithmica</i>. <b>22</b> (4): 660–685. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FPL00009246">10.1007/PL00009246</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:27441335">27441335</a>. Archived from <a rel="nofollow" class="external text" href="https://users.info.unicaen.fr/~brigitte/Publications/bin-gcd.ps">the original</a> <span class="cs1-format">(PS)</span> on 13 May 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Algorithmica&rft.atitle=Dynamics+of+the+Binary+Euclidean+Algorithm%3A+Functional+Analysis+and+Operators&rft.volume=22&rft.issue=4&rft.pages=660-685&rft.date=1998-09%2F1998-10&rft_id=info%3Adoi%2F10.1007%2FPL00009246&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A27441335%23id-name%3DS2CID&rft.aulast=Vall%C3%A9e&rft.aufirst=Brigitte&rft_id=https%3A%2F%2Fusers.info.unicaen.fr%2F~brigitte%2FPublications%2Fbin-gcd.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+GCD+algorithm" class="Z3988"></span></li></ul> <p>An analysis of the algorithm in the average case, through the lens of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>: the algorithms' main parameters are cast as a <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a>, and their average value is related to the <a href="/wiki/Invariant_measure" title="Invariant measure">invariant measure</a> of the system's <a href="/wiki/Transfer_operator" title="Transfer operator">transfer operator</a>. </p> <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=Binary_GCD_algorithm&action=edit&section=9" 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="https://xlinux.nist.gov/dads/HTML/binaryGCD.html">NIST Dictionary of Algorithms and Data Structures: binary GCD algorithm</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/blue/binary.shtml">Cut-the-Knot: Binary Euclid's Algorithm</a> at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://wwwmaths.anu.edu.au/~brent/pub/pub037.html"><i>Analysis of the Binary Euclidean Algorithm</i></a> (1976), a paper by <a href="/wiki/Richard_Brent_(scientist)" class="mw-redirect" title="Richard Brent (scientist)">Richard P. Brent</a>, including a variant using left shifts</li></ul> <p><br /> </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · 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theory">Number-theoretic</a> <a href="/wiki/Algorithm" title="Algorithm">algorithms</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primality_test" title="Primality test">Primality tests</a></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="/wiki/AKS_primality_test" title="AKS primality test">AKS</a></li> <li><a href="/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test" title="Adleman–Pomerance–Rumely primality test">APR</a></li> <li><a href="/wiki/Baillie%E2%80%93PSW_primality_test" title="Baillie–PSW primality test">Baillie–PSW</a></li> <li><a href="/wiki/Elliptic_curve_primality" title="Elliptic curve primality">Elliptic curve</a></li> <li><a href="/wiki/Pocklington_primality_test" title="Pocklington primality test">Pocklington</a></li> <li><a href="/wiki/Fermat_primality_test" title="Fermat primality test">Fermat</a></li> <li><a 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href="/wiki/Continued_fraction_factorization" title="Continued fraction factorization">Continued fraction (CFRAC)</a></li> <li><a href="/wiki/Dixon%27s_factorization_method" title="Dixon's factorization method">Dixon's</a></li> <li><a href="/wiki/Lenstra_elliptic-curve_factorization" title="Lenstra elliptic-curve factorization">Lenstra elliptic curve (ECM)</a></li> <li><a href="/wiki/Euler%27s_factorization_method" title="Euler's factorization method">Euler's</a></li> <li><a href="/wiki/Pollard%27s_rho_algorithm" title="Pollard's rho algorithm">Pollard's rho</a></li> <li><a href="/wiki/Pollard%27s_p_%E2%88%92_1_algorithm" title="Pollard's p − 1 algorithm"><i>p</i> − 1</a></li> <li><a href="/wiki/Williams%27s_p_%2B_1_algorithm" title="Williams's p + 1 algorithm"><i>p</i> + 1</a></li> <li><a href="/wiki/Quadratic_sieve" title="Quadratic sieve">Quadratic sieve (QS)</a></li> <li><a href="/wiki/General_number_field_sieve" title="General number field sieve">General number field sieve (GNFS)</a></li> <li><i><a href="/wiki/Special_number_field_sieve" title="Special number field sieve">Special number field sieve (SNFS)</a></i></li> <li><a href="/wiki/Rational_sieve" title="Rational sieve">Rational sieve</a></li> <li><a href="/wiki/Fermat%27s_factorization_method" title="Fermat's factorization method">Fermat's</a></li> <li><a href="/wiki/Shanks%27s_square_forms_factorization" title="Shanks's square forms factorization">Shanks's square forms</a></li> <li><a href="/wiki/Trial_division" title="Trial division">Trial division</a></li> <li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multiplication_algorithm" title="Multiplication algorithm">Multiplication</a></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="/wiki/Ancient_Egyptian_multiplication" title="Ancient Egyptian multiplication">Ancient Egyptian</a></li> <li><a href="/wiki/Long_multiplication" class="mw-redirect" title="Long multiplication">Long</a></li> <li><a href="/wiki/Karatsuba_algorithm" title="Karatsuba algorithm">Karatsuba</a></li> <li><a href="/wiki/Toom%E2%80%93Cook_multiplication" title="Toom–Cook multiplication">Toom–Cook</a></li> <li><a href="/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm" title="Schönhage–Strassen algorithm">Schönhage–Strassen</a></li> <li><a href="/wiki/F%C3%BCrer%27s_algorithm" class="mw-redirect" title="Fürer's algorithm">Fürer's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean</a> <a href="/wiki/Division_algorithm" title="Division algorithm">division</a></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="/wiki/Binary_division" class="mw-redirect" title="Binary division">Binary</a></li> <li><a href="/wiki/Chunking_(division)" title="Chunking (division)">Chunking</a></li> <li><a href="/wiki/Fourier_division" title="Fourier division">Fourier</a></li> <li><a href="/wiki/Goldschmidt_division" class="mw-redirect" title="Goldschmidt division">Goldschmidt</a></li> <li><a href="/wiki/Newton%E2%80%93Raphson_division" class="mw-redirect" title="Newton–Raphson division">Newton-Raphson</a></li> <li><a href="/wiki/Long_division" title="Long division">Long</a></li> <li><a href="/wiki/Short_division" title="Short division">Short</a></li> <li><a href="/wiki/SRT_division" class="mw-redirect" title="SRT division">SRT</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_logarithm" title="Discrete logarithm">Discrete logarithm</a></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="/wiki/Baby-step_giant-step" title="Baby-step giant-step">Baby-step giant-step</a></li> <li><a href="/wiki/Pollard%27s_rho_algorithm_for_logarithms" title="Pollard's rho algorithm for logarithms">Pollard rho</a></li> <li><a href="/wiki/Pollard%27s_kangaroo_algorithm" title="Pollard's kangaroo algorithm">Pollard kangaroo</a></li> <li><a href="/wiki/Pohlig%E2%80%93Hellman_algorithm" title="Pohlig–Hellman algorithm">Pohlig–Hellman</a></li> <li><a href="/wiki/Index_calculus_algorithm" title="Index calculus algorithm">Index calculus</a></li> <li><a href="/wiki/Function_field_sieve" title="Function field sieve">Function field sieve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">Greatest common divisor</a></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 class="mw-selflink selflink">Binary</a></li> <li><a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean</a></li> <li><a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">Extended Euclidean</a></li> <li><a href="/wiki/Lehmer%27s_GCD_algorithm" title="Lehmer's GCD algorithm">Lehmer's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadratic_residue" title="Quadratic residue">Modular square root</a></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="/wiki/Cipolla%27s_algorithm" title="Cipolla's algorithm">Cipolla</a></li> <li><a href="/wiki/Pocklington%27s_algorithm" title="Pocklington's algorithm">Pocklington's</a></li> <li><a href="/wiki/Tonelli%E2%80%93Shanks_algorithm" title="Tonelli–Shanks algorithm">Tonelli–Shanks</a></li> <li><a href="/wiki/Berlekamp%E2%80%93Rabin_algorithm" title="Berlekamp–Rabin algorithm">Berlekamp</a></li> <li><a href="/wiki/Kunerth%27s_algorithm" title="Kunerth's algorithm">Kunerth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other algorithms</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="/wiki/Chakravala_method" title="Chakravala method">Chakravala</a></li> <li><a href="/wiki/Cornacchia%27s_algorithm" title="Cornacchia's algorithm">Cornacchia</a></li> <li><a href="/wiki/Exponentiation_by_squaring" title="Exponentiation by squaring">Exponentiation by squaring</a></li> <li><a href="/wiki/Integer_square_root" title="Integer square root">Integer square root</a></li> <li><a href="/wiki/Integer_relation_algorithm" title="Integer relation algorithm">Integer relation</a> (<a href="/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm" title="Lenstra–Lenstra–Lovász lattice basis reduction algorithm">LLL</a>; <a href="/wiki/Korkine%E2%80%93Zolotarev_lattice_basis_reduction_algorithm" title="Korkine–Zolotarev lattice basis reduction algorithm">KZ</a>)</li> <li><a href="/wiki/Modular_exponentiation" title="Modular exponentiation">Modular exponentiation</a></li> <li><a href="/wiki/Montgomery_reduction" class="mw-redirect" title="Montgomery reduction">Montgomery reduction</a></li> <li><a href="/wiki/Schoof%27s_algorithm" title="Schoof's algorithm">Schoof</a></li> <li><a href="/wiki/Trachtenberg_system" title="Trachtenberg system">Trachtenberg system</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><i>Italics</i> indicate that algorithm is for numbers of special forms</li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" 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