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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Diskrete Exponentialfunktion</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>diskrete Exponentialfunktion</b> (auch <b>modulare Exponentiation</b> oder <b>modulares Potenzieren</b>)
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{x}\ {\bmod {\ }}m}">
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<annotation encoding="application/x-tex">{\displaystyle b^{x}\ {\bmod {\ }}m}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05581775d06d74f155ba05ad4aaa500db96b16ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.053ex; height:2.343ex;" alt="{\displaystyle b^{x}\ {\bmod {\ }}m}" loading="lazy"></span></dd></dl>
<p>liefert den Rest bei Division von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{x}}">
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<annotation encoding="application/x-tex">{\displaystyle b^{x}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bb7406a338fb530330582bc63420d091897c709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.17ex; height:2.343ex;" alt="{\displaystyle b^{x}}" loading="lazy"></span> durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">
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<mi>m</mi>
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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span>. Die Umkehrung der diskreten Exponentialfunktion heißt <a href="Diskreter_Logarithmus" title="Diskreter Logarithmus">diskreter Logarithmus</a>.
</p><p>Die diskrete Exponentialfunktion ist auch für große <a href="Exponent_(Mathematik)" class="mw-redirect" title="Exponent (Mathematik)">Exponenten</a> <a href="Effizienz_(Informatik)" title="Effizienz (Informatik)">effizient</a> berechenbar. Für die Umkehrung, also die Berechnung des Exponenten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>, bei gegebener Basis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">
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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" 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}" loading="lazy"></span>, Modul <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">
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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> und gewünschtem Ergebnis, ist allerdings bis heute kein schneller Algorithmus bekannt. Die diskrete Exponentialfunktion wird daher als <a href="Einwegfunktion" title="Einwegfunktion">Einwegfunktion</a> in <a href="Asymmetrisches_Kryptosystem" title="Asymmetrisches Kryptosystem">asymmetrischen Kryptosystemen</a> verwendet.
</p><p>Zur effizienten Berechnung der diskreten Exponentialfunktion kann der <a href="Satz_von_Euler" title="Satz von Euler">Satz von Euler</a> und das <a href="Bin%C3%A4re_Exponentiation" title="Binäre Exponentiation">Square & Multiply</a>-Verfahren verwendet werden.
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<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://www-users.cse.umn.edu/~garrett/crypto/a01/FastPow.html">Applet für Fast Modular Exponentiation</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2023-04-13" href="https://de.wikipedia.org/wiki/?title=Diskrete_Exponentialfunktion&oldid=232782261">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.
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