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.hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the cryptosystem. For traditional Scottish and Irish social gathering, see <a href="/wiki/C%C3%A8ilidh" title="Cèilidh">Cèilidh</a>.</div> <p><b>CEILIDH</b> is a <a href="/wiki/Public_key" class="mw-redirect" title="Public key">public key</a> <a href="/wiki/Cryptosystem" title="Cryptosystem">cryptosystem</a> based on the <a href="/wiki/Discrete_logarithm_problem" class="mw-redirect" title="Discrete logarithm problem">discrete logarithm problem</a> in <a href="/wiki/Algebraic_torus" title="Algebraic torus">algebraic torus</a>. This idea was first introduced by <a href="/wiki/Alice_Silverberg" title="Alice Silverberg">Alice Silverberg</a> and <a href="/wiki/Karl_Rubin" title="Karl Rubin">Karl Rubin</a> in 2003; Silverberg named CEILIDH after her cat.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-maa_2-0" class="reference"><a href="#cite_note-maa-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The main advantage of the system is the reduced size of the keys for the same security over basic schemes.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Avoid_weasel_words" class="mw-redirect" title="Wikipedia:Avoid weasel words"><span title="The material near this tag possibly uses too vague attribution or weasel words. (April 2011)">which?</span></a></i>]</sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Algorithms">Algorithms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=CEILIDH&action=edit&section=1" title="Edit section: Algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Parameters">Parameters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=CEILIDH&action=edit&section=2" title="Edit section: Parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Let <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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" /></span> be a prime power.</li> <li>An <a href="/wiki/Integer" title="Integer">integer</a> <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 chosen such that : <ul><li>The torus <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 T_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9241493be76739f2400f258f32c24f9689161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" /></span> has an explicit rational parametrization.</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 \Phi _{n}(q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}(q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7392e3e77fc28809a82f5c09f224f1144750a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.775ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(q)}" /></span> is divisible by a big prime <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c3c96a91205fb1ae9d97b9e93b763b424bbac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \Phi _{n}}" /></span> is the <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^{th}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{th}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d54f58e8109a3d758d6712278b03f6aea6e696c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.167ex; height:2.676ex;" alt="{\displaystyle n^{th}}" /></span> <a href="/wiki/Cyclotomic_polynomial" title="Cyclotomic polynomial">Cyclotomic polynomial</a>.</li></ul></li> <li>Let <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=\phi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=\phi (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93fc32c5b9bd039a178a796d3b6e5dea89c6edd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.728ex; height:2.843ex;" alt="{\displaystyle m=\phi (n)}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" /></span> is the <a href="/wiki/Euler_function" title="Euler function">Euler function</a>.</li> <li>Let <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 \rho :T_{n}(\mathbb {F} _{q})\rightarrow {\mathbb {F} _{q}}^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>:</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho :T_{n}(\mathbb {F} _{q})\rightarrow {\mathbb {F} _{q}}^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b23039c68903b110a9c86af4d650d7860843d4db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.631ex; height:3.009ex;" alt="{\displaystyle \rho :T_{n}(\mathbb {F} _{q})\rightarrow {\mathbb {F} _{q}}^{m}}" /></span> a birational map and its inverse <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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" /></span>.</li> <li>Choose <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 \alpha \in T_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in T_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82433b0045b85b5c6cb93d67326ec65352258d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.904ex; height:2.509ex;" alt="{\displaystyle \alpha \in T_{n}}" /></span> of order <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" /></span> and let <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 g=\rho (\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=\rho (\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a42dce5105b9e32f23b14f3ddacc682b3f17df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.713ex; height:2.843ex;" alt="{\displaystyle g=\rho (\alpha )}" /></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Key_agreement_scheme">Key agreement scheme</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=CEILIDH&action=edit&section=3" title="Edit section: Key agreement scheme"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This Scheme is based on the <a href="/wiki/Diffie-Hellman_key_agreement" class="mw-redirect" title="Diffie-Hellman key agreement">Diffie-Hellman key agreement</a>. </p> <ul><li>Alice chooses a random number <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\ {\pmod {\Phi _{n}(q)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ {\pmod {\Phi _{n}(q)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00744917b92d8a90db410af3f717549d6c76d8c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.27ex; height:2.843ex;" alt="{\displaystyle a\ {\pmod {\Phi _{n}(q)}}}" /></span>.</li> <li>She computes <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 P_{A}=\rho (\psi (g)^{a})\in \mathbb {F} _{q}^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{A}=\rho (\psi (g)^{a})\in \mathbb {F} _{q}^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5ed76b51436c3572a2900f69d2cc8b7a05ef75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.543ex; height:3.176ex;" alt="{\displaystyle P_{A}=\rho (\psi (g)^{a})\in \mathbb {F} _{q}^{m}}" /></span> and sends it to Bob.</li> <li>Bob chooses a random number <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\ {\pmod {\Phi _{n}(q)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\ {\pmod {\Phi _{n}(q)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96278dae13087951497ceef96694eb576f1baf8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.038ex; height:2.843ex;" alt="{\displaystyle b\ {\pmod {\Phi _{n}(q)}}}" /></span>.</li> <li>He computes <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 P_{B}=\rho (\psi (g)^{b})\in \mathbb {F} _{q}^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{B}=\rho (\psi (g)^{b})\in \mathbb {F} _{q}^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70384751b438b3cb90f20587ac1e238a593994e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.394ex; height:3.509ex;" alt="{\displaystyle P_{B}=\rho (\psi (g)^{b})\in \mathbb {F} _{q}^{m}}" /></span> and sends it to Alice.</li> <li>Alice computes <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 \rho (\psi (P_{B}))^{a})\in \mathbb {F} _{q}^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\psi (P_{B}))^{a})\in \mathbb {F} _{q}^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b05dd1b2ff2356b80515cb2547c4d912403cde1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.248ex; height:3.176ex;" alt="{\displaystyle \rho (\psi (P_{B}))^{a})\in \mathbb {F} _{q}^{m}}" /></span></li> <li>Bob computes <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 \rho (\psi (P_{A}))^{b})\in \mathbb {F} _{q}^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\psi (P_{A}))^{b})\in \mathbb {F} _{q}^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4a5be437065eb033e2d2bd78e1236ca239b1ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.069ex; height:3.509ex;" alt="{\displaystyle \rho (\psi (P_{A}))^{b})\in \mathbb {F} _{q}^{m}}" /></span></li></ul> <p><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 \psi \circ \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>∘</mo> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \circ \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0401674b2703fc6e87a3c85278b5c602528b69e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.91ex; height:2.676ex;" alt="{\displaystyle \psi \circ \rho }" /></span> is the identity, thus we have : <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 \rho (\psi (P_{B}))^{a})=\rho (\psi (P_{A}))^{b})=\rho (\psi (g)^{ab})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\psi (P_{B}))^{a})=\rho (\psi (P_{A}))^{b})=\rho (\psi (g)^{ab})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0563c5b1988b86ef3ebdc071c5d327bc80ecb16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.899ex; height:3.176ex;" alt="{\displaystyle \rho (\psi (P_{B}))^{a})=\rho (\psi (P_{A}))^{b})=\rho (\psi (g)^{ab})}" /></span> which is the shared secret of Alice and Bob. </p> <div class="mw-heading mw-heading3"><h3 id="Encryption_scheme">Encryption scheme</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=CEILIDH&action=edit&section=4" title="Edit section: Encryption scheme"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This scheme is based on the <a href="/wiki/ElGamal_encryption" title="ElGamal encryption">ElGamal encryption</a>. </p> <ul><li><b>Key Generation</b> <ul><li>Alice chooses a random number <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\ {\pmod {\Phi _{n}(q)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ {\pmod {\Phi _{n}(q)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00744917b92d8a90db410af3f717549d6c76d8c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.27ex; height:2.843ex;" alt="{\displaystyle a\ {\pmod {\Phi _{n}(q)}}}" /></span> as her private key.</li> <li>The resulting public key 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 P_{A}=\rho (\psi (g)^{a})\in \mathbb {F} _{q}^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{A}=\rho (\psi (g)^{a})\in \mathbb {F} _{q}^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5ed76b51436c3572a2900f69d2cc8b7a05ef75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.543ex; height:3.176ex;" alt="{\displaystyle P_{A}=\rho (\psi (g)^{a})\in \mathbb {F} _{q}^{m}}" /></span>.</li></ul></li> <li><b>Encryption</b> <ul><li>The message <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> is an element 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 \mathbb {F} _{q}^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b0101ce39a08427032a5d64e98501fe6f687fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.095ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{q}^{m}}" /></span>.</li> <li>Bob chooses a random integer <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> in the range <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 1\leq k\leq l-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>l</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq k\leq l-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc614a08f20c6414e78c1835680deca70dddfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.267ex; height:2.343ex;" alt="{\displaystyle 1\leq k\leq l-1}" /></span>.</li> <li>Bob computes <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 \gamma =\rho (\psi (g)^{k})\in \mathbb {F} _{q}^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\rho (\psi (g)^{k})\in \mathbb {F} _{q}^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe40a9fe686f1ae3bd3d8a70afcf003389c92814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.835ex; height:3.509ex;" alt="{\displaystyle \gamma =\rho (\psi (g)^{k})\in \mathbb {F} _{q}^{m}}" /></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 \delta =\rho (\psi (M)\psi (P_{A})^{k})\in \mathbb {F} _{q}^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta =\rho (\psi (M)\psi (P_{A})^{k})\in \mathbb {F} _{q}^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac30b04cdea32f83f973ad26f8b35144443ccc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.227ex; height:3.509ex;" alt="{\displaystyle \delta =\rho (\psi (M)\psi (P_{A})^{k})\in \mathbb {F} _{q}^{m}}" /></span>.</li> <li>Bob sends the ciphertext <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 (\gamma ,\delta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\gamma ,\delta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd58a43be55ebccc65b864c27b40f5f6c9f96c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.154ex; height:2.843ex;" alt="{\displaystyle (\gamma ,\delta )}" /></span> to Alice.</li></ul></li> <li><b>Decryption</b> <ul><li>Alice computes <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=\rho (\psi (\delta )\psi (\gamma )^{-a})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\rho (\psi (\delta )\psi (\gamma )^{-a})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea5cdcd76abf01c20f2ccf1ed0089f47e9708c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.888ex; height:3.009ex;" alt="{\displaystyle M=\rho (\psi (\delta )\psi (\gamma )^{-a})}" /></span>.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Security">Security</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=CEILIDH&action=edit&section=5" title="Edit section: Security"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The CEILIDH scheme is based on the ElGamal scheme and thus has similar security properties. </p><p>If the <a href="/wiki/Computational_Diffie-Hellman_assumption" class="mw-redirect" title="Computational Diffie-Hellman assumption">computational Diffie-Hellman assumption</a> holds the underlying cyclic group <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /></span>, then the encryption function is <a href="/wiki/One-way_function" title="One-way function">one-way</a>.<sup id="cite_ref-cryptutor_3-0" class="reference"><a href="#cite_note-cryptutor-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> If the <a href="/wiki/Decisional_Diffie-Hellman_assumption" class="mw-redirect" title="Decisional Diffie-Hellman assumption">decisional Diffie-Hellman assumption</a> (DDH) holds in <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /></span>, then CEILIDH achieves <a href="/wiki/Semantic_security" title="Semantic security">semantic security</a>.<sup id="cite_ref-cryptutor_3-1" class="reference"><a href="#cite_note-cryptutor-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Semantic security is not implied by the computational Diffie-Hellman assumption alone.<sup id="cite_ref-DHAES_4-0" class="reference"><a href="#cite_note-DHAES-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> See <a href="/wiki/Decisional_Diffie-Hellman_assumption" class="mw-redirect" title="Decisional Diffie-Hellman assumption">decisional Diffie-Hellman assumption</a> for a discussion of groups where the assumption is believed to hold. </p><p>CEILIDH encryption is unconditionally <a href="/wiki/Malleability_(cryptography)" title="Malleability (cryptography)">malleable</a>, and therefore is not secure under <a href="/wiki/Chosen_ciphertext_attack" class="mw-redirect" title="Chosen ciphertext attack">chosen ciphertext attack</a>. For example, given an encryption <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 (c_{1},c_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{1},c_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f16dff2d0a99e639b57fd5ebf52ef4c558f3a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.965ex; height:2.843ex;" alt="{\displaystyle (c_{1},c_{2})}" /></span> of some (possibly unknown) message <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" 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}" /></span>, one can easily construct a valid encryption <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 (c_{1},2c_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{1},2c_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4917b1e7f80702244aa4652241b37cc309261d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.128ex; height:2.843ex;" alt="{\displaystyle (c_{1},2c_{2})}" /></span> of the message <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 2m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e100f32f96dc84bf0591df4f5c5bd40d71189f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle 2m}" /></span>. </p> <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=CEILIDH&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFSilverberg2006" class="citation web cs1">Silverberg, Alice (November 2006). <a rel="nofollow" class="external text" href="https://www.maa.org/sites/default/files/pdf/pubs/nov06focus.pdf">"Alice in NUMB3Rland"</a> <span class="cs1-format">(PDF)</span>. <i>Focus</i>. <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a><span class="reference-accessdate">. Retrieved <span class="nowrap">12 July</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Focus&rft.atitle=Alice+in+NUMB3Rland&rft.date=2006-11&rft.aulast=Silverberg&rft.aufirst=Alice&rft_id=https%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2Fpdf%2Fpubs%2Fnov06focus.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACEILIDH" class="Z3988"></span></span> </li> <li id="cite_note-maa-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-maa_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKirsch2010" class="citation web cs1">Kirsch, Rachel (December 2010). <a rel="nofollow" class="external text" href="https://www.maa.org/meetings/calendar-events/cryptography-how-to-keep-a-secret">"Cryptography: How to Keep a Secret"</a>. <i>Mathematical Association of America</i><span class="reference-accessdate">. Retrieved <span class="nowrap">12 July</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mathematical+Association+of+America&rft.atitle=Cryptography%3A+How+to+Keep+a+Secret&rft.date=2010-12&rft.aulast=Kirsch&rft.aufirst=Rachel&rft_id=https%3A%2F%2Fwww.maa.org%2Fmeetings%2Fcalendar-events%2Fcryptography-how-to-keep-a-secret&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACEILIDH" class="Z3988"></span></span> </li> <li id="cite_note-cryptutor-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-cryptutor_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-cryptutor_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090421130001/http://crypto.cs.uiuc.edu/wiki/index.php/Elgamal_encryption_scheme">"El-gamal Encryption Scheme"</a>. <i>CRYPTUTOR</i>. Archived from <a rel="nofollow" class="external text" href="http://crypto.cs.uiuc.edu/wiki/index.php/Elgamal_encryption_scheme">the original</a> on 2009-04-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-04-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=CRYPTUTOR&rft.atitle=El-gamal+Encryption+Scheme&rft_id=http%3A%2F%2Fcrypto.cs.uiuc.edu%2Fwiki%2Findex.php%2FElgamal_encryption_scheme&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACEILIDH" class="Z3988"></span></span> </li> <li id="cite_note-DHAES-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-DHAES_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAbdallaBellareRogaway1998" class="citation web cs1">Abdalla, M.; Bellare, M.; Rogaway, P. (September 1998). <a rel="nofollow" class="external text" href="http://web.cs.ucdavis.edu/~rogaway/papers/dhies.pdf">"DHIES: An encryption scheme based on the Diffie-Hellman Problem (Appendix A)"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=DHIES%3A+An+encryption+scheme+based+on+the+Diffie-Hellman+Problem+%28Appendix+A%29&rft.date=1998-09&rft.aulast=Abdalla&rft.aufirst=M.&rft.au=Bellare%2C+M.&rft.au=Rogaway%2C+P.&rft_id=http%3A%2F%2Fweb.cs.ucdavis.edu%2F~rogaway%2Fpapers%2Fdhies.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACEILIDH" class="Z3988"></span></span> </li> </ol></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRubinSilverberg2003" class="citation book cs1">Rubin, K.; Silverberg, A. (2003). "Torus-Based Cryptography". In Boneh, D. (ed.). <i>Advances in Cryptology - CRYPTO 2003</i>. Lecture Notes in Computer Science. Vol. 2729. Springer, Berlin, Heidelberg. pp. <span class="nowrap">349–</span>365. <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%2F978-3-540-45146-4_21">10.1007/978-3-540-45146-4_21</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540406747" title="Special:BookSources/9783540406747"><bdi>9783540406747</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Torus-Based+Cryptography&rft.btitle=Advances+in+Cryptology+-+CRYPTO+2003&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E349-%3C%2Fspan%3E365&rft.pub=Springer%2C+Berlin%2C+Heidelberg&rft.date=2003&rft_id=info%3Adoi%2F10.1007%2F978-3-540-45146-4_21&rft.isbn=9783540406747&rft.aulast=Rubin&rft.aufirst=K.&rft.au=Silverberg%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACEILIDH" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=CEILIDH&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.math.uci.edu/~asilverb/bibliography/ceilidh.pdf">Torus-Based Cryptography</a>: the paper introducing the concept (in PDF from Silverberg's university web page).</li></ul> <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 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title="Rabin cryptosystem">Rabin</a></li> <li><a href="/wiki/RSA_cryptosystem" title="RSA cryptosystem">RSA</a></li> <li><a href="/wiki/Okamoto%E2%80%93Uchiyama_cryptosystem" title="Okamoto–Uchiyama cryptosystem">Okamoto–Uchiyama</a></li> <li><a href="/wiki/Schmidt-Samoa_cryptosystem" title="Schmidt-Samoa cryptosystem">Schmidt–Samoa</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group wraplinks" 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" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boneh%E2%80%93Lynn%E2%80%93Shacham" class="mw-redirect" title="Boneh–Lynn–Shacham">BLS</a></li> <li><a href="/wiki/Cramer%E2%80%93Shoup_cryptosystem" title="Cramer–Shoup cryptosystem">Cramer–Shoup</a></li> <li><a href="/wiki/Diffie%E2%80%93Hellman_key_exchange" title="Diffie–Hellman key exchange">DH</a></li> <li><a href="/wiki/Digital_Signature_Algorithm" title="Digital Signature Algorithm">DSA</a></li> <li><a href="/wiki/Elliptic-curve_Diffie%E2%80%93Hellman" title="Elliptic-curve Diffie–Hellman">ECDH</a> <ul><li><a href="/wiki/Curve25519" title="Curve25519">X25519</a></li> <li><a href="/wiki/Curve448" title="Curve448">X448</a></li></ul></li> <li><a href="/wiki/Elliptic_Curve_Digital_Signature_Algorithm" title="Elliptic Curve Digital Signature Algorithm">ECDSA</a></li> <li><a href="/wiki/EdDSA" title="EdDSA">EdDSA</a> <ul><li><a href="/wiki/EdDSA#Ed25519" title="EdDSA">Ed25519</a></li> <li><a href="/wiki/EdDSA#Ed448" title="EdDSA">Ed448</a></li></ul></li> <li><a href="/wiki/ECMQV" class="mw-redirect" title="ECMQV">ECMQV</a></li> <li><a href="/wiki/Encrypted_key_exchange" title="Encrypted key exchange">EKE</a></li> <li><a href="/wiki/ElGamal_encryption" title="ElGamal encryption">ElGamal</a> <ul><li><a href="/wiki/ElGamal_signature_scheme" title="ElGamal signature scheme">signature scheme</a></li></ul></li> <li><a href="/wiki/MQV" title="MQV">MQV</a></li> <li><a href="/wiki/Schnorr_signature" title="Schnorr signature">Schnorr</a></li> <li><a href="/wiki/SPEKE" title="SPEKE">SPEKE</a></li> <li><a href="/wiki/Secure_Remote_Password_protocol" title="Secure Remote Password protocol">SRP</a></li> <li><a href="/wiki/Station-to-Station_protocol" title="Station-to-Station protocol">STS</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group wraplinks" style="width:1%"><a href="/wiki/Lattice-based_cryptography" title="Lattice-based cryptography">Lattice/SVP/CVP</a>/<wbr /><a href="/wiki/Learning_with_errors" title="Learning with errors">LWE</a>/<wbr /><a href="/wiki/Short_integer_solution_problem" title="Short integer solution problem">SIS</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BLISS_signature_scheme" title="BLISS signature scheme">BLISS</a></li> <li><a href="/wiki/Kyber" title="Kyber">Kyber</a></li> <li><a href="/wiki/NewHope" title="NewHope">NewHope</a></li> <li><a href="/wiki/NTRUEncrypt" title="NTRUEncrypt">NTRUEncrypt</a></li> <li><a href="/wiki/NTRUSign" title="NTRUSign">NTRUSign</a></li> <li><a href="/wiki/RLWE-KEX" class="mw-redirect" title="RLWE-KEX">RLWE-KEX</a></li> <li><a href="/wiki/RLWE-SIG" class="mw-redirect" title="RLWE-SIG">RLWE-SIG</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group wraplinks" style="width:1%">Others</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_Eraser" title="Algebraic Eraser">AE</a></li> <li><a class="mw-selflink selflink">CEILIDH</a></li> <li><a href="/wiki/Efficient_Probabilistic_Public-Key_Encryption_Scheme" title="Efficient Probabilistic Public-Key Encryption Scheme">EPOC</a></li> <li><a href="/wiki/Hidden_Field_Equations" title="Hidden Field Equations">HFE</a></li> <li><a href="/wiki/Integrated_Encryption_Scheme" title="Integrated Encryption Scheme">IES</a></li> <li><a href="/wiki/Lamport_signature" title="Lamport signature">Lamport</a></li> <li><a href="/wiki/McEliece_cryptosystem" title="McEliece cryptosystem">McEliece</a></li> <li><a href="/wiki/Merkle%E2%80%93Hellman_knapsack_cryptosystem" title="Merkle–Hellman knapsack cryptosystem">Merkle–Hellman</a></li> <li><span class="wraplinks"><a href="/wiki/Naccache%E2%80%93Stern_knapsack_cryptosystem" title="Naccache–Stern knapsack cryptosystem">Naccache–Stern knapsack cryptosystem</a></span></li> <li><a href="/wiki/Three-pass_protocol" title="Three-pass protocol">Three-pass protocol</a></li> <li><a href="/wiki/XTR" title="XTR">XTR</a></li> <li><a href="/wiki/SQIsign" title="SQIsign">SQIsign</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</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/Discrete_logarithm#Cryptography" title="Discrete logarithm">Discrete logarithm cryptography</a></li> <li><a href="/wiki/Elliptic-curve_cryptography" title="Elliptic-curve cryptography">Elliptic-curve cryptography</a></li> <li><a href="/wiki/Hash-based_cryptography" title="Hash-based cryptography">Hash-based cryptography</a></li> <li><a href="/wiki/Non-commutative_cryptography" title="Non-commutative cryptography">Non-commutative cryptography</a></li> <li><a href="/wiki/RSA_problem" title="RSA problem">RSA problem</a></li> <li><a href="/wiki/Trapdoor_function" title="Trapdoor function">Trapdoor function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Standardization</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/CRYPTREC" title="CRYPTREC">CRYPTREC</a></li> <li><a href="/wiki/IEEE_P1363" title="IEEE P1363">IEEE P1363</a></li> <li><a href="/wiki/NESSIE" title="NESSIE">NESSIE</a></li> <li><a href="/wiki/NSA_Suite_B_Cryptography" title="NSA Suite B Cryptography">NSA Suite B</a></li> <li><a href="/wiki/Commercial_National_Security_Algorithm_Suite" title="Commercial National Security Algorithm Suite">CNSA</a></li> <li><a href="/wiki/NIST_Post-Quantum_Cryptography_Standardization" title="NIST Post-Quantum Cryptography Standardization">Post-Quantum Cryptography</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Topics</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/Digital_signature" title="Digital signature">Digital signature</a></li> <li><a href="/wiki/Optimal_asymmetric_encryption_padding" title="Optimal asymmetric encryption padding">OAEP</a></li> <li><a href="/wiki/Public_key_fingerprint" title="Public key fingerprint">Fingerprint</a></li> <li><a href="/wiki/Public_key_infrastructure" title="Public key infrastructure">PKI</a></li> <li><a href="/wiki/Web_of_trust" title="Web of trust">Web of trust</a></li> <li><a href="/wiki/Key_size" title="Key size">Key size</a></li> <li><a href="/wiki/Identity-based_cryptography" title="Identity-based cryptography">Identity-based cryptography</a></li> <li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li> <li><a href="/wiki/OpenPGP_card" title="OpenPGP card">OpenPGP card</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Cryptography_navbox" title="Template:Cryptography navbox"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Cryptography_navbox" title="Template talk:Cryptography navbox"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Cryptography_navbox" title="Special:EditPage/Template:Cryptography navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Cryptography149" style="font-size:114%;margin:0 4em"><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_cryptography" title="History of cryptography">History of cryptography</a></li> <li><a href="/wiki/Outline_of_cryptography" title="Outline of cryptography">Outline of cryptography</a></li> <li><a href="/wiki/Classical_cipher" title="Classical cipher">Classical cipher</a></li> <li><a href="/wiki/Cryptographic_protocol" title="Cryptographic protocol">Cryptographic protocol</a> <ul><li><a href="/wiki/Authentication_protocol" title="Authentication protocol">Authentication protocol</a></li></ul></li> <li><a href="/wiki/Cryptographic_primitive" title="Cryptographic primitive">Cryptographic primitive</a></li> <li><a href="/wiki/Cryptanalysis" title="Cryptanalysis">Cryptanalysis</a></li> <li><a href="/wiki/Cryptocurrency" title="Cryptocurrency">Cryptocurrency</a></li> <li><a href="/wiki/Cryptosystem" title="Cryptosystem">Cryptosystem</a></li> <li><a href="/wiki/Cryptographic_nonce" title="Cryptographic nonce">Cryptographic nonce</a></li> <li><a href="/wiki/Cryptovirology" title="Cryptovirology">Cryptovirology</a></li> <li><a href="/wiki/Hash_function" title="Hash function">Hash function</a> <ul><li><a href="/wiki/Cryptographic_hash_function" title="Cryptographic hash function">Cryptographic hash function</a></li> <li><a href="/wiki/Key_derivation_function" title="Key derivation function">Key derivation function</a></li> <li><a href="/wiki/Secure_Hash_Algorithms" title="Secure Hash Algorithms">Secure Hash Algorithms</a></li></ul></li> <li><a href="/wiki/Digital_signature" title="Digital signature">Digital signature</a></li> <li><a href="/wiki/Kleptography" title="Kleptography">Kleptography</a></li> <li><a href="/wiki/Key_(cryptography)" title="Key (cryptography)">Key (cryptography)</a></li> <li><a href="/wiki/Key_exchange" title="Key exchange">Key exchange</a></li> <li><a href="/wiki/Key_generator" title="Key generator">Key generator</a></li> <li><a href="/wiki/Key_schedule" title="Key schedule">Key schedule</a></li> <li><a href="/wiki/Key_stretching" title="Key stretching">Key stretching</a></li> <li><a href="/wiki/Keygen" title="Keygen">Keygen</a></li> <li><a href="/wiki/Template:Cryptography_machines" title="Template:Cryptography machines">Machines</a></li> <li><a href="/wiki/Cryptojacking_malware" class="mw-redirect" title="Cryptojacking malware">Cryptojacking malware</a></li> <li><a href="/wiki/Ransomware" title="Ransomware">Ransomware</a></li> <li><a href="/wiki/Random_number_generation" title="Random number generation">Random number generation</a> <ul><li><a href="/wiki/Cryptographically_secure_pseudorandom_number_generator" title="Cryptographically secure pseudorandom number generator">Cryptographically secure pseudorandom number generator</a> (CSPRNG)</li></ul></li> <li><a href="/wiki/Pseudorandom_noise" title="Pseudorandom noise">Pseudorandom noise</a> (PRN)</li> <li><a href="/wiki/Secure_channel" title="Secure channel">Secure channel</a></li> <li><a href="/wiki/Insecure_channel" class="mw-redirect" title="Insecure channel">Insecure channel</a></li> <li><a href="/wiki/Subliminal_channel" title="Subliminal channel">Subliminal channel</a></li> <li><a href="/wiki/Encryption" title="Encryption">Encryption</a></li> <li><a href="/wiki/Decryption" class="mw-redirect" title="Decryption">Decryption</a></li> <li><a href="/wiki/End-to-end_encryption" title="End-to-end encryption">End-to-end encryption</a></li> <li><a href="/wiki/Harvest_now,_decrypt_later" title="Harvest now, decrypt later">Harvest now, decrypt later</a></li> <li><a href="/wiki/Information-theoretic_security" title="Information-theoretic security">Information-theoretic security</a></li> <li><a href="/wiki/Plaintext" title="Plaintext">Plaintext</a></li> <li><a href="/wiki/Codetext" class="mw-redirect" title="Codetext">Codetext</a></li> <li><a href="/wiki/Ciphertext" title="Ciphertext">Ciphertext</a></li> <li><a href="/wiki/Shared_secret" title="Shared secret">Shared secret</a></li> <li><a href="/wiki/Trapdoor_function" title="Trapdoor function">Trapdoor function</a></li> <li><a href="/wiki/Trusted_timestamping" title="Trusted timestamping">Trusted timestamping</a></li> <li><a href="/wiki/Key-based_routing" title="Key-based routing">Key-based routing</a></li> <li><a href="/wiki/Onion_routing" title="Onion routing">Onion routing</a></li> <li><a href="/wiki/Garlic_routing" title="Garlic routing">Garlic routing</a></li> <li><a href="/wiki/Kademlia" title="Kademlia">Kademlia</a></li> <li><a href="/wiki/Mix_network" title="Mix network">Mix network</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cryptographic_hash_function" title="Cryptographic hash function">Cryptographic hash function</a></li> <li><a href="/wiki/Block_cipher" title="Block cipher">Block cipher</a></li> <li><a href="/wiki/Stream_cipher" title="Stream cipher">Stream cipher</a></li> <li><a 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