Reference Work Entry

Encyclopedia of Cryptography and Security

pp 240-244

# Computational Diffie-Hellman Problem

• Igor ShparlinskiAffiliated withDepartment of Computing Faculty of Science, Macquarie University
<meta charset="utf-8"/><script type="text/x-mathjax-config"> MathJax.Hub.Config({ /* load MathML extension */ extensions: ["mml2jax.js"], /* configure input and output */ jax: ["input/TeX", "input/MathML", "output/HTML-CSS"], /* align display equations to the left */ displayAlign: "left", /* set zoom trigger */ menuSettings: { zoom: "Click" }, /* configuration of the tex2jax preprocessor: - set delimiters for inline and display equations - preview while equations are processed is "TeX" - allow \$to represent a literal dollar sign */ tex2jax: { inlineMath: [ ["\$","\$"] ], displayMath: [ ['$$','$$'], ["\$","\$"] ], preview: ["TeX"], processEscapes: true }, /* configuration of the mml2jax preprocessor: - preview while equations are processed is "MathML" */ mml2jax: { preview: ["MathML"] }, /* configuration of the HTML-CSS output processor: - available fonts are STIX and TeX - preferred font is STIX - use TeX as web-based font if none of the above is available on the user's computer - use TeX font for image fallback mode */ "HTML-CSS": { availableFonts: ["STIX","TeX"], preferredFont: ["STIX"], webFont: ["TeX"], imageFont: ["TeX"], styles: { '.MathJax_Display': { "margin": 0 } } }, TeX: { TagSide: "right", Macros: { upalpha: '{\\unicode{x03B1}}', upbeta: '{\\unicode{x03B2}}', upgamma: '{\\unicode{x03B3}}', updelta: '{\\unicode{x03B4}}', upepsilon: '{\\unicode{x03F5}}', upvarepsilon: '{\\unicode{x03B5}}', upzeta: '{\\unicode{x03B6}}', upeta: '{\\unicode{x03B7}}', uptheta: '{\\unicode{x03B8}}', upvartheta: '{\\unicode{x03D1}}', upiota: '{\\unicode{x03B9}}', upkappa: '{\\unicode{x03BA}}', uplambda: '{\\unicode{x03BB}}', upmu: '{\\unicode{x03BC}}', upnu: '{\\unicode{x03BD}}', upxi: '{\\unicode{x03BE}}', upomicron: '{\\unicode{x03BF}}', uppi: '{\\unicode{x03C0}}', uprho: '{\\unicode{x03C1}}', upsigma: '{\\unicode{x03C3}}', uptau: '{\\unicode{x03C4}}', upupsilon: '{\\unicode{x03C5}}', upphi: '{\\unicode{x03C6}}', upchi: '{\\unicode{x03C7}}', uppsi: '{\\unicode{x03C8}}', upomega: '{\\unicode{x03C9}}', Upalpha: '{\\unicode{x0391}}', Upbeta: '{\\unicode{x0392}}', Upgamma: '{\\unicode{x0393}}', Updelta: '{\\unicode{x0394}}', Upepsilon: '{\\unicode{x0395}}', Upzeta: '{\\unicode{x0396}}', Upeta: '{\\unicode{x0397}}', Uptheta: '{\\unicode{x0398}}', Upvartheta: '{\\unicode{x03F4}}', Upiota: '{\\unicode{x0399}}', Upkappa: '{\\unicode{x039A}}', Uplambda: '{\\unicode{x039B}}', Upmu: '{\\unicode{x039C}}', Upnu: '{\\unicode{x039D}}', Upxi: '{\\unicode{x039E}}', Upomicron: '{\\unicode{x039F}}', Uppi: '{\\unicode{x03A0}}', Uprho: '{\\unicode{x03A1}}', Upsigma: '{\\unicode{x03A3}}', Uptau: '{\\unicode{x03A4}}', Upupsilon: '{\\unicode{x03A5}}', Upphi: '{\\unicode{x03A6}}', Upchi: '{\\unicode{x03A7}}', Uppsi: '{\\unicode{x03A8}}', Upomega: '{\\unicode{x03A9}}', permille: '{\\unicode{x2030}}', hfill: '{\\enspace\\enspace}', enskip: '{\\enspace}', copyright: '{\\unicode{x00A9}}', dag: '{\\unicode{x2020}}', ddag: '{\\unicode{x2021}}', ointop: '{\\unicode{0x222E}}', P: '{\\unicode{0x00B6}}', textregistered: '{\\unicode{0x00AE}}', lhook: '{\\hookrightarrow}', rhook: '{\\hookleftarrow}', fancyscript: ['{\\scr #1}',1], varvec: ['\\pmb{#1}',1], parbox: ['\\mbox{#2}',2], AA: '{\\unicode{x00C5}}' } }, /* configuration of the math menu: - allow the user to select what font to use */ MathMenu: { showFontMenu: true } }); MathJax.Hub.Register.StartupHook("TeX Jax Ready", function () { MathJax.InputJax.TeX.prefilterHooks.Add(function (data) { data.math = data.math.replace(/\\kern *-\\nulldelimiterspace/g, ""); data.math = data.math.replace(/\\mapstochar/g, "\\mapsto"); data.math = data.math.replace(/{?\\ss}?/g, "ß"); data.math = data.math.replace(/\\"a|{\\"a}|\\"{a}/g, "ä"); data.math = data.math.replace(/\\"o|{\\"o}|\\"{o}/g, "ö"); data.math = data.math.replace(/\\"u|{\\"u}|\\"{u}/g, "ü"); data.math = data.math.replace(/\\parbox($\w$)?{(.*?)}{(.*?)\\\\ (.*?)}/g, "\\parbox{$2}{$3$4}"); data.math = data.math.replace(/\\user1/g, "\\mathcal"); data.math = data.math.replace(/\\user2/g, "\\pmb"); data.math = data.math.replace(/\\vspace\*?{[^}]+}/g, ""); data.math = data.math.replace(/\\text\\EUR/g, "€"); data.math = data.math.replace(/\\textrm/g, "\\mathrm"); data.math = data.math.replace(/\\bf{/g, "\\mathbf{"); data.math = data.math.replace(/\\text{\\small{([^}]+)}}/g, "\\scriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\footnotesize{([^}]+)}}/g, "\\scriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\scriptsize{([^}]+)}}/g, "\\scriptscriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\tiny{([^}]+)}}/g, "\\scriptscriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\mathop ([^{]+?)\\limits/g, "\\mathop{1}\\limits"); data.math = data.math.replace(/\big|Big|bigg|Bigg) *{([^}]+)}/g, "\\1 2"); data.math = data.math.replace(/\\textsc *{([^}]+)}/g, "{\\rm ~#~1~#~}"); data.math = data.math.replace(/\\upvarphi/g, "\\varphi"); var reg = new RegExp("~#~(.*?)~#~", "g"); var m; while (m = reg.exec(data.math)) { var result = ""; for (var index=0; index != m[1].length; index++) { var char = m[1].substr(index, 1); if (char.match(/[a-z]/)) { result += "{\\small "+ char.toUpperCase() + "}"; } else { result += char; } } data.math = data.math.replace(/~#~.*?~#~/, result); } }); } ); </script><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML-full.js"> </script></head><body><section id="Sec1_882" class="Section1 RenderAsSection1"><h2 class="Heading">Synonyms</h2><p class="Para"><span class="ExternalRef" id="XRef1_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_1382"><span class="RefSource">CDH</span></a></span>; <span class="ExternalRef" id="XRef2_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_1383"><span class="RefSource">DHP</span></a></span>; <span class="ExternalRef" id="XRef3_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_1384"><span class="RefSource">Diffie-Hellman problem</span></a></span></p></section></body></html><!--4.0--> <!DOCTYPE html SYSTEM "about:legacy-compat"> <html xmlns="http://www.w3.org/1999/xhtml" lang=""><head><title/><meta charset="utf-8"/><script type="text/x-mathjax-config"> MathJax.Hub.Config({ /* load MathML extension */ extensions: ["mml2jax.js"], /* configure input and output */ jax: ["input/TeX", "input/MathML", "output/HTML-CSS"], /* align display equations to the left */ displayAlign: "left", /* set zoom trigger */ menuSettings: { zoom: "Click" }, /* configuration of the tex2jax preprocessor: - set delimiters for inline and display equations - preview while equations are processed is "TeX" - allow \ to represent a literal dollar sign */ tex2jax: { inlineMath: [ ["\\(","\"] ], displayMath: [ ['$$','$$'], ["\$","\$"] ], preview: ["TeX"], processEscapes: true }, /* configuration of the mml2jax preprocessor: - preview while equations are processed is "MathML" */ mml2jax: { preview: ["MathML"] }, /* configuration of the HTML-CSS output processor: - available fonts are STIX and TeX - preferred font is STIX - use TeX as web-based font if none of the above is available on the user's computer - use TeX font for image fallback mode */ "HTML-CSS": { availableFonts: ["STIX","TeX"], preferredFont: ["STIX"], webFont: ["TeX"], imageFont: ["TeX"], styles: { '.MathJax_Display': { "margin": 0 } } }, TeX: { TagSide: "right", Macros: { upalpha: '{\\unicode{x03B1}}', upbeta: '{\\unicode{x03B2}}', upgamma: '{\\unicode{x03B3}}', updelta: '{\\unicode{x03B4}}', upepsilon: '{\\unicode{x03F5}}', upvarepsilon: '{\\unicode{x03B5}}', upzeta: '{\\unicode{x03B6}}', upeta: '{\\unicode{x03B7}}', uptheta: '{\\unicode{x03B8}}', upvartheta: '{\\unicode{x03D1}}', upiota: '{\\unicode{x03B9}}', upkappa: '{\\unicode{x03BA}}', uplambda: '{\\unicode{x03BB}}', upmu: '{\\unicode{x03BC}}', upnu: '{\\unicode{x03BD}}', upxi: '{\\unicode{x03BE}}', upomicron: '{\\unicode{x03BF}}', uppi: '{\\unicode{x03C0}}', uprho: '{\\unicode{x03C1}}', upsigma: '{\\unicode{x03C3}}', uptau: '{\\unicode{x03C4}}', upupsilon: '{\\unicode{x03C5}}', upphi: '{\\unicode{x03C6}}', upchi: '{\\unicode{x03C7}}', uppsi: '{\\unicode{x03C8}}', upomega: '{\\unicode{x03C9}}', Upalpha: '{\\unicode{x0391}}', Upbeta: '{\\unicode{x0392}}', Upgamma: '{\\unicode{x0393}}', Updelta: '{\\unicode{x0394}}', Upepsilon: '{\\unicode{x0395}}', Upzeta: '{\\unicode{x0396}}', Upeta: '{\\unicode{x0397}}', Uptheta: '{\\unicode{x0398}}', Upvartheta: '{\\unicode{x03F4}}', Upiota: '{\\unicode{x0399}}', Upkappa: '{\\unicode{x039A}}', Uplambda: '{\\unicode{x039B}}', Upmu: '{\\unicode{x039C}}', Upnu: '{\\unicode{x039D}}', Upxi: '{\\unicode{x039E}}', Upomicron: '{\\unicode{x039F}}', Uppi: '{\\unicode{x03A0}}', Uprho: '{\\unicode{x03A1}}', Upsigma: '{\\unicode{x03A3}}', Uptau: '{\\unicode{x03A4}}', Upupsilon: '{\\unicode{x03A5}}', Upphi: '{\\unicode{x03A6}}', Upchi: '{\\unicode{x03A7}}', Uppsi: '{\\unicode{x03A8}}', Upomega: '{\\unicode{x03A9}}', permille: '{\\unicode{x2030}}', hfill: '{\\enspace\\enspace}', enskip: '{\\enspace}', copyright: '{\\unicode{x00A9}}', dag: '{\\unicode{x2020}}', ddag: '{\\unicode{x2021}}', ointop: '{\\unicode{0x222E}}', P: '{\\unicode{0x00B6}}', textregistered: '{\\unicode{0x00AE}}', lhook: '{\\hookrightarrow}', rhook: '{\\hookleftarrow}', fancyscript: ['{\\scr #1}',1], varvec: ['\\pmb{#1}',1], parbox: ['\\mbox{#2}',2], AA: '{\\unicode{x00C5}}' } }, /* configuration of the math menu: - allow the user to select what font to use */ MathMenu: { showFontMenu: true } }); MathJax.Hub.Register.StartupHook("TeX Jax Ready", function () { MathJax.InputJax.TeX.prefilterHooks.Add(function (data) { data.math = data.math.replace(/\\kern *-\\nulldelimiterspace/g, ""); data.math = data.math.replace(/\\mapstochar/g, "\\mapsto"); data.math = data.math.replace(/{?\\ss}?/g, "ß"); data.math = data.math.replace(/\\"a|{\\"a}|\\"{a}/g, "ä"); data.math = data.math.replace(/\\"o|{\\"o}|\\"{o}/g, "ö"); data.math = data.math.replace(/\\"u|{\\"u}|\\"{u}/g, "ü"); data.math = data.math.replace(/\\parbox($\w$)?{(.*?)}{(.*?)\\\\ (.*?)}/g, "\\parbox{2}{$3$4}"); data.math = data.math.replace(/\\user1/g, "\\mathcal"); data.math = data.math.replace(/\\user2/g, "\\pmb"); data.math = data.math.replace(/\\vspace\*?{[^}]+}/g, ""); data.math = data.math.replace(/\\text\\EUR/g, "€"); data.math = data.math.replace(/\\textrm/g, "\\mathrm"); data.math = data.math.replace(/\\bf{/g, "\\mathbf{"); data.math = data.math.replace(/\\text{\\small{([^}]+)}}/g, "\\scriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\footnotesize{([^}]+)}}/g, "\\scriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\scriptsize{([^}]+)}}/g, "\\scriptscriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\tiny{([^}]+)}}/g, "\\scriptscriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\mathop ([^{]+?)\\limits/g, "\\mathop{1}\\limits"); data.math = data.math.replace(/\big|Big|bigg|Bigg) *{([^}]+)}/g, "\\1 2"); data.math = data.math.replace(/\\textsc *{([^}]+)}/g, "{\\rm ~#~1~#~}"); data.math = data.math.replace(/\\upvarphi/g, "\\varphi"); var reg = new RegExp("~#~(.*?)~#~", "g"); var m; while (m = reg.exec(data.math)) { var result = ""; for (var index=0; index != m[1].length; index++) { var char = m[1].substr(index, 1); if (char.match(/[a-z]/)) { result += "{\\small "+ char.toUpperCase() + "}"; } else { result += char; } } data.math = data.math.replace(/~#~.*?~#~/, result); } }); } ); </script><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML-full.js"> </script></head><body><section id="Sec2_882" class="Section1 RenderAsSection1"><h2 class="Heading">Related Concepts</h2><p class="Para"><span class="ExternalRef" id="XRef4_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_442"><span class="RefSource">Computational Complexity</span></a></span>; <span class="ExternalRef" id="XRef5_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_882"><span class="RefSource">Decisional Diffie-Hellman Problem</span></a></span>; <span class="ExternalRef" id="XRef6_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_75"><span class="RefSource">Diffie–Hellman Key Agreement</span></a></span>; <span class="ExternalRef" id="XRef7_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_445"><span class="RefSource">Discrete Logarithm Problem</span></a></span>; <span class="ExternalRef" id="XRef8_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_22"><span class="RefSource">Public Key Cryptography</span></a></span></p></section></body></html><!--4.0--> <!DOCTYPE html SYSTEM "about:legacy-compat"> <html xmlns="http://www.w3.org/1999/xhtml" lang=""><head><title/><meta charset="utf-8"/><script type="text/x-mathjax-config"> MathJax.Hub.Config({ /* load MathML extension */ extensions: ["mml2jax.js"], /* configure input and output */ jax: ["input/TeX", "input/MathML", "output/HTML-CSS"], /* align display equations to the left */ displayAlign: "left", /* set zoom trigger */ menuSettings: { zoom: "Click" }, /* configuration of the tex2jax preprocessor: - set delimiters for inline and display equations - preview while equations are processed is "TeX" - allow \ to represent a literal dollar sign */ tex2jax: { inlineMath: [ ["\\(","\"] ], displayMath: [ ['$$','$$'], ["\$","\$"] ], preview: ["TeX"], processEscapes: true }, /* configuration of the mml2jax preprocessor: - preview while equations are processed is "MathML" */ mml2jax: { preview: ["MathML"] }, /* configuration of the HTML-CSS output processor: - available fonts are STIX and TeX - preferred font is STIX - use TeX as web-based font if none of the above is available on the user's computer - use TeX font for image fallback mode */ "HTML-CSS": { availableFonts: ["STIX","TeX"], preferredFont: ["STIX"], webFont: ["TeX"], imageFont: ["TeX"], styles: { '.MathJax_Display': { "margin": 0 } } }, TeX: { TagSide: "right", Macros: { upalpha: '{\\unicode{x03B1}}', upbeta: '{\\unicode{x03B2}}', upgamma: '{\\unicode{x03B3}}', updelta: '{\\unicode{x03B4}}', upepsilon: '{\\unicode{x03F5}}', upvarepsilon: '{\\unicode{x03B5}}', upzeta: '{\\unicode{x03B6}}', upeta: '{\\unicode{x03B7}}', uptheta: '{\\unicode{x03B8}}', upvartheta: '{\\unicode{x03D1}}', upiota: '{\\unicode{x03B9}}', upkappa: '{\\unicode{x03BA}}', uplambda: '{\\unicode{x03BB}}', upmu: '{\\unicode{x03BC}}', upnu: '{\\unicode{x03BD}}', upxi: '{\\unicode{x03BE}}', upomicron: '{\\unicode{x03BF}}', uppi: '{\\unicode{x03C0}}', uprho: '{\\unicode{x03C1}}', upsigma: '{\\unicode{x03C3}}', uptau: '{\\unicode{x03C4}}', upupsilon: '{\\unicode{x03C5}}', upphi: '{\\unicode{x03C6}}', upchi: '{\\unicode{x03C7}}', uppsi: '{\\unicode{x03C8}}', upomega: '{\\unicode{x03C9}}', Upalpha: '{\\unicode{x0391}}', Upbeta: '{\\unicode{x0392}}', Upgamma: '{\\unicode{x0393}}', Updelta: '{\\unicode{x0394}}', Upepsilon: '{\\unicode{x0395}}', Upzeta: '{\\unicode{x0396}}', Upeta: '{\\unicode{x0397}}', Uptheta: '{\\unicode{x0398}}', Upvartheta: '{\\unicode{x03F4}}', Upiota: '{\\unicode{x0399}}', Upkappa: '{\\unicode{x039A}}', Uplambda: '{\\unicode{x039B}}', Upmu: '{\\unicode{x039C}}', Upnu: '{\\unicode{x039D}}', Upxi: '{\\unicode{x039E}}', Upomicron: '{\\unicode{x039F}}', Uppi: '{\\unicode{x03A0}}', Uprho: '{\\unicode{x03A1}}', Upsigma: '{\\unicode{x03A3}}', Uptau: '{\\unicode{x03A4}}', Upupsilon: '{\\unicode{x03A5}}', Upphi: '{\\unicode{x03A6}}', Upchi: '{\\unicode{x03A7}}', Uppsi: '{\\unicode{x03A8}}', Upomega: '{\\unicode{x03A9}}', permille: '{\\unicode{x2030}}', hfill: '{\\enspace\\enspace}', enskip: '{\\enspace}', copyright: '{\\unicode{x00A9}}', dag: '{\\unicode{x2020}}', ddag: '{\\unicode{x2021}}', ointop: '{\\unicode{0x222E}}', P: '{\\unicode{0x00B6}}', textregistered: '{\\unicode{0x00AE}}', lhook: '{\\hookrightarrow}', rhook: '{\\hookleftarrow}', fancyscript: ['{\\scr #1}',1], varvec: ['\\pmb{#1}',1], parbox: ['\\mbox{#2}',2], AA: '{\\unicode{x00C5}}' } }, /* configuration of the math menu: - allow the user to select what font to use */ MathMenu: { showFontMenu: true } }); MathJax.Hub.Register.StartupHook("TeX Jax Ready", function () { MathJax.InputJax.TeX.prefilterHooks.Add(function (data) { data.math = data.math.replace(/\\kern *-\\nulldelimiterspace/g, ""); data.math = data.math.replace(/\\mapstochar/g, "\\mapsto"); data.math = data.math.replace(/{?\\ss}?/g, "ß"); data.math = data.math.replace(/\\"a|{\\"a}|\\"{a}/g, "ä"); data.math = data.math.replace(/\\"o|{\\"o}|\\"{o}/g, "ö"); data.math = data.math.replace(/\\"u|{\\"u}|\\"{u}/g, "ü"); data.math = data.math.replace(/\\parbox($\w$)?{(.*?)}{(.*?)\\\\ (.*?)}/g, "\\parbox{2}{$3$4}"); data.math = data.math.replace(/\\user1/g, "\\mathcal"); data.math = data.math.replace(/\\user2/g, "\\pmb"); data.math = data.math.replace(/\\vspace\*?{[^}]+}/g, ""); data.math = data.math.replace(/\\text\\EUR/g, "€"); data.math = data.math.replace(/\\textrm/g, "\\mathrm"); data.math = data.math.replace(/\\bf{/g, "\\mathbf{"); data.math = data.math.replace(/\\text{\\small{([^}]+)}}/g, "\\scriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\footnotesize{([^}]+)}}/g, "\\scriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\scriptsize{([^}]+)}}/g, "\\scriptscriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\tiny{([^}]+)}}/g, "\\scriptscriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\mathop ([^{]+?)\\limits/g, "\\mathop{1}\\limits"); data.math = data.math.replace(/\$big|Big|bigg|Bigg) *{([^}]+)}/g, "\\1 2"); data.math = data.math.replace(/\\textsc *{([^}]+)}/g, "{\\rm ~#~1~#~}"); data.math = data.math.replace(/\\upvarphi/g, "\\varphi"); var reg = new RegExp("~#~(.*?)~#~", "g"); var m; while (m = reg.exec(data.math)) { var result = ""; for (var index=0; index != m[1].length; index++) { var char = m[1].substr(index, 1); if (char.match(/[a-z]/)) { result += "{\\small "+ char.toUpperCase() + "}"; } else { result += char; } } data.math = data.math.replace(/~#~.*?~#~/, result); } }); } ); </script><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML-full.js"> </script></head><body><section id="Sec3_882" class="Section1 RenderAsSection1"><h2 class="Heading">Definition</h2><p class="Para">Let <em class="EmphasisTypeItalic ">G</em> be a cyclic <span class="ExternalRef" id="XRef9_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_411"><span class="RefSource">group</span></a></span> with <span class="ExternalRef" id="XRef10_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_410"><span class="RefSource">generator</span></a></span><em class="EmphasisTypeItalic ">g</em> and let <span class="InlineEquation" id="IEq1_882">\({g}^{x},{g}^{y} \in G$</span>. The <em class="EmphasisTypeItalic ">computational Diffie-Hellman problem</em> is to compute <em class="EmphasisTypeItalic ">g</em><sup><em class="EmphasisTypeItalic ">xy</em></sup>.</p></section></body></html><!--4.0--> <!DOCTYPE html SYSTEM "about:legacy-compat"> <html xmlns="http://www.w3.org/1999/xhtml" lang=""><head><title/><meta charset="utf-8"/><script type="text/x-mathjax-config"> MathJax.Hub.Config({ /* load MathML extension */ extensions: ["mml2jax.js"], /* configure input and output */ jax: ["input/TeX", "input/MathML", "output/HTML-CSS"], /* align display equations to the left */ displayAlign: "left", /* set zoom trigger */ menuSettings: { zoom: "Click" }, /* configuration of the tex2jax preprocessor: - set delimiters for inline and display equations - preview while equations are processed is "TeX" - allow \ to represent a literal dollar sign */ tex2jax: { inlineMath: [ ["\$","\$"] ], displayMath: [ ['$$','$$'], ["\$","\$"] ], preview: ["TeX"], processEscapes: true }, /* configuration of the mml2jax preprocessor: - preview while equations are processed is "MathML" */ mml2jax: { preview: ["MathML"] }, /* configuration of the HTML-CSS output processor: - available fonts are STIX and TeX - preferred font is STIX - use TeX as web-based font if none of the above is available on the user's computer - use TeX font for image fallback mode */ "HTML-CSS": { availableFonts: ["STIX","TeX"], preferredFont: ["STIX"], webFont: ["TeX"], imageFont: ["TeX"], styles: { '.MathJax_Display': { "margin": 0 } } }, TeX: { TagSide: "right", Macros: { upalpha: '{\\unicode{x03B1}}', upbeta: '{\\unicode{x03B2}}', upgamma: '{\\unicode{x03B3}}', updelta: '{\\unicode{x03B4}}', upepsilon: '{\\unicode{x03F5}}', upvarepsilon: '{\\unicode{x03B5}}', upzeta: '{\\unicode{x03B6}}', upeta: '{\\unicode{x03B7}}', uptheta: '{\\unicode{x03B8}}', upvartheta: '{\\unicode{x03D1}}', upiota: '{\\unicode{x03B9}}', upkappa: '{\\unicode{x03BA}}', uplambda: '{\\unicode{x03BB}}', upmu: '{\\unicode{x03BC}}', upnu: '{\\unicode{x03BD}}', upxi: '{\\unicode{x03BE}}', upomicron: '{\\unicode{x03BF}}', uppi: '{\\unicode{x03C0}}', uprho: '{\\unicode{x03C1}}', upsigma: '{\\unicode{x03C3}}', uptau: '{\\unicode{x03C4}}', upupsilon: '{\\unicode{x03C5}}', upphi: '{\\unicode{x03C6}}', upchi: '{\\unicode{x03C7}}', uppsi: '{\\unicode{x03C8}}', upomega: '{\\unicode{x03C9}}', Upalpha: '{\\unicode{x0391}}', Upbeta: '{\\unicode{x0392}}', Upgamma: '{\\unicode{x0393}}', Updelta: '{\\unicode{x0394}}', Upepsilon: '{\\unicode{x0395}}', Upzeta: '{\\unicode{x0396}}', Upeta: '{\\unicode{x0397}}', Uptheta: '{\\unicode{x0398}}', Upvartheta: '{\\unicode{x03F4}}', Upiota: '{\\unicode{x0399}}', Upkappa: '{\\unicode{x039A}}', Uplambda: '{\\unicode{x039B}}', Upmu: '{\\unicode{x039C}}', Upnu: '{\\unicode{x039D}}', Upxi: '{\\unicode{x039E}}', Upomicron: '{\\unicode{x039F}}', Uppi: '{\\unicode{x03A0}}', Uprho: '{\\unicode{x03A1}}', Upsigma: '{\\unicode{x03A3}}', Uptau: '{\\unicode{x03A4}}', Upupsilon: '{\\unicode{x03A5}}', Upphi: '{\\unicode{x03A6}}', Upchi: '{\\unicode{x03A7}}', Uppsi: '{\\unicode{x03A8}}', Upomega: '{\\unicode{x03A9}}', permille: '{\\unicode{x2030}}', hfill: '{\\enspace\\enspace}', enskip: '{\\enspace}', copyright: '{\\unicode{x00A9}}', dag: '{\\unicode{x2020}}', ddag: '{\\unicode{x2021}}', ointop: '{\\unicode{0x222E}}', P: '{\\unicode{0x00B6}}', textregistered: '{\\unicode{0x00AE}}', lhook: '{\\hookrightarrow}', rhook: '{\\hookleftarrow}', fancyscript: ['{\\scr #1}',1], varvec: ['\\pmb{#1}',1], parbox: ['\\mbox{#2}',2], AA: '{\\unicode{x00C5}}' } }, /* configuration of the math menu: - allow the user to select what font to use */ MathMenu: { showFontMenu: true } }); MathJax.Hub.Register.StartupHook("TeX Jax Ready", function () { MathJax.InputJax.TeX.prefilterHooks.Add(function (data) { data.math = data.math.replace(/\\kern *-\\nulldelimiterspace/g, ""); data.math = data.math.replace(/\\mapstochar/g, "\\mapsto"); data.math = data.math.replace(/{?\\ss}?/g, "ß"); data.math = data.math.replace(/\\"a|{\\"a}|\\"{a}/g, "ä"); data.math = data.math.replace(/\\"o|{\\"o}|\\"{o}/g, "ö"); data.math = data.math.replace(/\\"u|{\\"u}|\\"{u}/g, "ü"); data.math = data.math.replace(/\\parbox($\w$)?{(.*?)}{(.*?)\\\\ (.*?)}/g, "\\parbox{$2}{$3 $4}"); data.math = data.math.replace(/\\user1/g, "\\mathcal"); data.math = data.math.replace(/\\user2/g, "\\pmb"); data.math = data.math.replace(/\\vspace\*?{[^}]+}/g, ""); data.math = data.math.replace(/\\text\\EUR/g, "€"); data.math = data.math.replace(/\\textrm/g, "\\mathrm"); data.math = data.math.replace(/\\bf{/g, "\\mathbf{"); data.math = data.math.replace(/\\text{\\small{([^}]+)}}/g, "\\scriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\footnotesize{([^}]+)}}/g, "\\scriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\scriptsize{([^}]+)}}/g, "\\scriptscriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\text{\\tiny{([^}]+)}}/g, "\\scriptscriptstyle{\\text{$1}}"); data.math = data.math.replace(/\\mathop ([^{]+?)\\limits/g, "\\mathop{$1}\\limits"); data.math = data.math.replace(/\$big|Big|bigg|Bigg) *{([^}]+)}/g, "\\1 2"); data.math = data.math.replace(/\\textsc *{([^}]+)}/g, "{\\rm ~#~1~#~}"); data.math = data.math.replace(/\\upvarphi/g, "\\varphi"); var reg = new RegExp("~#~(.*?)~#~", "g"); var m; while (m = reg.exec(data.math)) { var result = ""; for (var index=0; index != m[1].length; index++) { var char = m[1].substr(index, 1); if (char.match(/[a-z]/)) { result += "{\\small "+ char.toUpperCase() + "}"; } else { result += char; } } data.math = data.math.replace(/~#~.*?~#~/, result); } }); } ); </script><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML-full.js"> </script></head><body><section id="Sec4_882" class="Section1 RenderAsSection1"><h2 class="Heading">Background</h2><div class="Para">In their pioneering paper, Diffie and Hellman [<span class="CitationRef"><a href="#CR10_882">10</a></span>] proposed an elegant, reliable, and efficient way to establish a common key between two communicating parties. In the most general setting their idea can be described as follows (see <span class="ExternalRef" id="XRef11_882"><a href="http://dx.doi.org/10.1007/978-1-4419-5906-5_75"><span class="RefSource">Diffie-Hellman key agreement</span></a></span> for further discussion). Given a cyclic group<em class="EmphasisTypeItalic ">G</em> and agenerator <em class="EmphasisTypeItalic ">g</em>of <em class="EmphasisTypeItalic ">G</em>,two communicating parties Alice and Bob execute the following protocol:<div class="UnorderedList"><ul class="UnorderedListMarkBullet"><li><p class="Para">Alice selects secret <em class="EmphasisTypeItalic ">x</em>, Bob selects secret <em class="EmphasisTypeItalic ">y</em></p></li><li><p class="Para">Alice publishes <em class="EmphasisTypeItalic ">X</em> = <em class="EmphasisTypeItalic ">g</em><sup><em class="EmphasisTypeItalic ">x</em></sup>, Bob publishes <em class="EmphasisTypeItalic ">Y</em> = <em class="EmphasisTypeItalic ">g</em><sup><em class="EmphasisTypeItalic ">y</em></sup></p></li><li><p class="Para">Alice computes <em class="EmphasisTypeItalic ">K</em> = <em class="EmphasisTypeItalic ">Y</em><sup><em class="EmphasisTypeItalic ">x</em></sup>, Bob computes <em class="EmphasisTypeItalic ">K</em> = <em class="EmphasisTypeItalic ">X</em><sup><em class="EmphasisTypeItalic ">y</em></sup></p></li></ul></div></div><p class="Para">Therefore, at the end of the protocol the values <em class="EmphasisTypeItalic ">X</em> = <em class="EmphasisTypeItalic ">g</em><sup><em class="EmphasisTypeItalic ">x</em></sup> and <em class="EmphasisTypeItalic ">Y</em> = <em class="EmphasisTypeItalic ">g</em> ...</p></section></body></html><!--4.0--> </div> <div class="excerpt-message"></div> </div> <div id="kb-nav--aside" class="col-aside" role="complementary"> <div class="cover"> <div class="look-inside cover-image-animate"> <div id="reader-overlay"> <div id="reader-page-template"> <div id="page-%P"> <div class="page-number"> Page %P </div> <div class="pdf-page" id="page-img-container-%ID"> <img id="pdf-page-image-%ID" class="pdf-image" src="/static/1.738/images/pdf-preview/spacer.gif" alt="Loading..." width="%W" style="width:%Wpx;" data-loaded="false"/> </div> </div> </div> <div class="reader-position"> <div id="reader"> <div id="reader-buttons"> <a href="#close" id="pdf-close" class="btn btn-close right"> Close </a> <a href="#close" id="toggletext" class="btn" title="Toggle plain text"> Plain text </a> </div> <div id="viewer" class="show-pdf"> <div id="scroller"></div> </div> <div class="top-shadow"></div> <div class="bottom-shadow"></div> </div> </div> </div> <a class="webtrekk-track lookinside-href" href="/referenceworkentry/10.1007%2F978-1-4419-5906-5_882/lookinside/000.png" pageType="rd_springer_com.reference_work.entry_preview" parentContentType="ReferenceWork" contentType="ReferenceWorkEntry" doi="10.1007/978-1-4419-5906-5_882" viewType="Preview" publication="10.1007/978-1-4419-5906-5 | Encyclopedia of Cryptography and Security"> <img class="look-inside-cover" src="http://static-content.springer.com/cover/book/978-1-4419-5906-5.jpg" alt="Encyclopedia of Cryptography and Security" width="153" itemprop="image"/> <img class="look-inside-page" src="http://static-content.springer.com/lookinside/prt%3A978-1-4419-5906-5%2F3/000.png" alt="Encyclopedia of Cryptography and Security" width="153" itemprop="image"/> <span class="look-inside-badge"> Look <br/> Inside </span> </a> </div> </div> <div class="book-metrics SideBox hidden" data-url="http://bookmetrix-proxy.live.cf.public.springer.com" data-contenttype="chapter"> <h2 class="SideBox_title">Reference Work Entry Metrics</h2> <div class="book-metrics-items"></div> <div class="Sidebox_footer book-metrics-footer"> <a href="#"> <span>Provided by</span> <span><strong>Book</strong><span>metrix</span></span> </a> </div> </div> <div id="copublisher-container" data-url="http://copublishers.live.cf.public.springer.com/?id=978-1-4419-5906-5&id=978-1-4419-5905-8"></div> <div class="SideBox section-links"> </div> <div class="SideBox other-actions"> <h2 class="SideBox_title">Other actions</h2> <ul> <li> <a id="export-citation" href="/export-citation/referenceworkentry/10.1007/978-1-4419-5906-5_882"> Export citation </a> </li> <li> <a id="about-link" class="external" href="http://www.springer.com/978-1-4419-5905-8?wt_mc=ThirdParty.SpringerLink.3.EPR653.About_eBook" target="_blank" title="It opens in new window"> About this Reference Work </a> </li> <li> <a id="permissions-link" class="external" href="https://s100.copyright.com/AppDispatchServlet?publisherName=Springer&orderBeanReset=true&orderSource=SpringerLink&author=Igor+Shparlinski&contentID=10.1007%2F978-1-4419-5906-5_882&openAccess=false&endPage=244&publicationDate=2011&startPage=240&title=Computational+Diffie-Hellman+Problem&imprint=Springer+Science%2BBusiness+Media%2C+LLC&publication=eBook&authorAddress=Sydney%2C+Australia" target="_blank" title="It opens in new window">Reprints and Permissions</a> </li> <li> <a id="papers-link" class="external webtrekk-track" href="http://redirect.papersapp.com/redirect?url=http://link.springer.com/referenceworkentry/10.1007%2F978-1-4419-5906-5_882" target="_blank" title="It opens in new window" gaCategory="Export" publication="10.1007/978-1-4419-5906-5_882 | Computational Diffie-Hellman Problem" viewType="Add to Papers">Add to Papers</a> </li> </ul> </div> <div class="SideBox other-actions share"> <h2 class="SideBox_title">Share</h2> <a id="facebook" href="javascript:void(0)" title="Share this content on Facebook">Share this content on Facebook</a> <a id="twitter" href="javascript:void(0)" title="Share this content on Twitter">Share this content on Twitter</a> <a id="linkedin" href="javascript:void(0)" title="Share this content on LinkedIn">Share this content on LinkedIn</a> </div> </div> </div> <div id="kb-nav--aside" class="document-aside" role="complementary"> <input id="relatedItemsSource" type="hidden" value="none"/> <div id="abstract-esm" class="expander expander-empty"> <div class="expander-title"> <div class="heading"> <h2><button>Supplementary Material (0)</button></h2> </div> </div> </div> <div id="abstract-references" class="expander"> <div class="expander-title"> <div class="heading"> <h2><button>References (40)</button></h2> </div> </div> <div class="expander-content"> <div class="expander-content-inner"> <div class="formatted"> <!DOCTYPE html SYSTEM "about:legacy-compat"> <div xmlns="http://www.w3.org/1999/xhtml" xmlns:meta="http://www.springer.com/app/meta" xmlns:epub="http://www.idpf.org/2007/ops" class="FulltextWrapper"><aside class="Bibliography" id="Bib1_882"><h3 class="Heading">Recommended Reading</h3><ol class="BibliographyWrapper"><li class="Citation"><div class="CitationNumber">1.</div><div class="CitationContent" id="CR1_882">Biham E, Boneh D, Reingold O (1999) Breaking generalized Diffie-Hellman modulo a composite is no easier than factoring. Inform Proc Letts 70:83–87<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?1003.94521"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1695966"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1016/S0020-0190(99)00047-2"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">2.</div><div class="CitationContent" id="CR2_882">Blake I, Seroussi G, Smart NP (1999) Elliptic curves in cryptography. In: London mathematical society, Lecture notes series, vol 265. Cambridge University Press, Cambridge</div></li><li class="Citation"><div class="CitationNumber">3.</div><div class="CitationContent" id="CR3_882">Bleichenbacher D, Bosma W, Lenstra AK (1995) Some remarks on Lucas-based cryptosystems. In: Coppersmith D (ed) Advances in cryptology – CRYPTO’95. Lecture notes in computer science, vol 963. Springer, Berlin, pp 386–396</div></li><li class="Citation"><div class="CitationNumber">4.</div><div class="CitationContent" id="CR4_882">Boneh D, Franklin M (2001) Identity-based encryption from the Weil pairing. In: Kilian J (ed) Advances in cryptology – CRYPTO 2001. Lecture notes in computer science, vol 2139. Springer, Berlin, pp 213–229<span class="Occurrences"><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/3-540-44647-8_13"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">5.</div><div class="CitationContent" id="CR5_882">Boneh D, Lipton R (1996) Algorithms for black-box fields and their applications to cryptography. In: Koblitz N (ed) Advances in cryptology – CRYPTO’96. Lecture notes in computer science, vol 1109. Springer, Berlin, pp 283–297</div></li><li class="Citation"><div class="CitationNumber">6.</div><div class="CitationContent" id="CR6_882">Boneh D, Venkatesan R (1996) Hardness of computing the most significant bits of secret keys in Diffie-Hellman and related schemes. In: Koblitz N (ed) Advances in cryptology – CRYPTO’96. Lecture notes in computer science, vol 1109. Springer, Berlin, pp 129–142</div></li><li class="Citation"><div class="CitationNumber">7.</div><div class="CitationContent" id="CR7_882">Boneh D, Venkatesan R (1997) Rounding in lattices and its cryptographic applications. In: Proceedings of 8th Annual ACM-SIAM symposium on discrete algorithms. ACM, New York, pp 675–681</div></li><li class="Citation"><div class="CitationNumber">8.</div><div class="CitationContent" id="CR8_882">Cherepnev MA (1996) On the connection between the discrete logarithms and the Diffie-Hellman problem. Diskretnaja Matem (in Russian) 6:341–349<span class="Occurrences"><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1422345"><span><span>MathSciNet</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">9.</div><div class="CitationContent" id="CR9_882">Coppersmith D, Shparlinski IE (2000) On polynomial approximation of the discrete logarithm and the Diffie-Hellman mapping. J Crypto 13:339–360<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?1038.94007"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1768482"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/s001450010002"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">10.</div><div class="CitationContent" id="CR10_882">Diffie W, Hellman ME (1976) New directions in cryptography. IEEE Trans Inform Theory 22:109–112<span class="Occurrences"><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=437208"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1109/TIT.1976.1055638"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">11.</div><div class="CitationContent" id="CR11_882">El Mahassni E, Shparlinski IE (2001) Polynomial representations of the Diffie-Hellman mapping. Bull Aust Math Soc 63:467–473<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?0974.11040"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1834947"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1017/S0004972700019547"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">12.</div><div class="CitationContent" id="CR12_882">Enge A (1999) Elliptic curves and their applications to cryptography. Kluwer, Dordrecht<span class="Occurrences"><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/978-1-4615-5207-9"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">13.</div><div class="CitationContent" id="CR13_882">Galbraith SD (2001) Supersingular curves in cryptography. In: Boyd C (ed) Advances in cryptology – ASIACRYPT 2001. Lecture notes in computer science, vol 2248. Springer, Berlin, pp 495–513<span class="Occurrences"><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/3-540-45682-1_29"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">14.</div><div class="CitationContent" id="CR14_882">Gaudry P, Hess F, Smart NP (2002) Constructive and destructive facets of Weil descent on elliptic curves. J Crypto 15:19–46<span class="Occurrences"><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1880933"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/s00145-001-0011-x"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">15.</div><div class="CitationContent" id="CR15_882">Gonzalez Vasco MI, Shparlinski IE (2001) On the security of Diffie-Hellman bits. In: Proceedings of workshop on cryptography and computational number theory, Singapore, Birkh<span class="InlineEquation" id="IEq36_882">\(\ddot{\text{ a}}$</span>user, pp 257–268</div></li><li class="Citation"><div class="CitationNumber">16.</div><div class="CitationContent" id="CR16_882">Joux A (2000) A one round protocol for tripartite Diffie-Hellman. In: Bosma W (ed) Proceedings of ANTS-IV. Lecture notes in computer science, vol 1838. Springer, Berlin, pp 385–393</div></li><li class="Citation"><div class="CitationNumber">17.</div><div class="CitationContent" id="CR17_882">Joux A (2002) The Weil and Tate pairings as building blocks for public key cryptosystems. In: Kohel D, Fieker C (eds) Proceedings of ANTS V. Lecture notes in computer science, vol 2369. Springer, Berlin, pp 20–32</div></li><li class="Citation"><div class="CitationNumber">18.</div><div class="CitationContent" id="CR18_882">Koblitz N (1987) Elliptic curve cryptosystems. Math Comp 48:203–209<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$0622.94015"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=866109"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1090/S0025-5718-1987-0866109-5"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">19.</div><div class="CitationContent" id="CR19_882">Koblitz N (2002) Good and bad uses of elliptic curves in cryptography. Moscow Math J 2:693–715<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$1063.11051"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1986087"><span><span>MathSciNet</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">20.</div><div class="CitationContent" id="CR20_882">Koblitz N, Menezes A (2010) Intractable problems in cryptography. In: Proceedings of 9th International Conference Finite Fields and Their Applications, Contemporary Math., vol 518, pp 279–300</div></li><li class="Citation"><div class="CitationNumber">21.</div><div class="CitationContent" id="CR21_882">Koblitz N, Menezes A, Shparlinski IE (2011) Discrete logarithms, Diffie-Hellman, and reductions to appear in Vietnam Journal of Mathematics</div></li><li class="Citation"><div class="CitationNumber">22.</div><div class="CitationContent" id="CR22_882">Lenstra AK, Verheul ER (2000) The XTR public key system. In: Bellare M (ed) Advances in cryptology – CRYPTO 2000. Lecture notes in computer science, vol 1880. Springer, Berlin, pp 1–19<span class="Occurrences"><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/3-540-44598-6_1"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">23.</div><div class="CitationContent" id="CR23_882">Lenstra AK, Verheul ER (2000) Key improvements to XTR. In: Okamoto T (ed) Advances in cryptography – ASIACRYPT 2000. Lecture notes in computer science, vol 1976. Springer, Berlin, pp 220–233<span class="Occurrences"><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/3-540-44448-3_17"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">24.</div><div class="CitationContent" id="CR24_882">Lenstra AK, Verheul ER (2001) Fast irreducibility and subgroup membership testing in XTR. In: Kim K (ed) PKC 2001. Lecture notes in computer science, vol 1992. Springer, Berlin, pp 73–86</div></li><li class="Citation"><div class="CitationNumber">25.</div><div class="CitationContent" id="CR25_882">Li W-CW, Näslund M, Shparlinski IE (2002) The hidden number problem with the trace and bit security of XTR and LUC. In: Yung M (ed) Advances in cryptology – CRYPTO 2002. Lecture notes in computer science, vol 2442. Springer, Berlin, pp 433–448<span class="Occurrences"><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/3-540-45708-9_28"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">26.</div><div class="CitationContent" id="CR26_882">Maurer UM, Wolf S (1999) The relationship between breaking the Diffie-Hellman protocol and computing discrete logarithms. SIAM J Comput 28:1689–1721<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$1053.94014"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1694168"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1137/S0097539796302749"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">27.</div><div class="CitationContent" id="CR27_882">Maurer UM, Wolf S (2000) The Diffie-Hellman protocol. Designs, Codes and Cryptogr 19:147–171<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$0983.94037"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1759615"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1023/A%3A1008302122286"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">28.</div><div class="CitationContent" id="CR28_882">Meidl W, Winterhof A (2002) A polynomial representation of the Diffie-Hellman mapping. Appl Algebra in Engin Commun Comput 13:313–318<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$1013.94012"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1953197"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/s00200-002-0104-2"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">29.</div><div class="CitationContent" id="CR29_882">Menezes AJ, Koblitz N, Vanstone SA (2000) The state of elliptic curve cryptography. Designs, Codes and Cryptogr 19: 173–193<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$0996.94032"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1759616"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1023/A%3A1008354106356"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">30.</div><div class="CitationContent" id="CR30_882">Menezes AJ, van Oorschot PC, Vanstone SA (1996) Handbook of applied cryptography. CRC Press, Boca Raton<span class="Occurrences"><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1201/9781439821916"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">31.</div><div class="CitationContent" id="CR31_882">Miller VC (1986) Use of elliptic curves in cryptography. In: Williams HC (ed) Advances in cryptology — CRYPTO’85. Lecture notes in computer science, vol 218. Springer, Berlin, pp 417–426</div></li><li class="Citation"><div class="CitationNumber">32.</div><div class="CitationContent" id="CR32_882">Pomerance C (1987) Fast, rigorous factorization and discrete logarithm algorithms. Discrete Algorithms and Complexity. Academic Press, New York, pp 119–143</div></li><li class="Citation"><div class="CitationNumber">33.</div><div class="CitationContent" id="CR33_882">Rubin K, Silverberg A (2002) Supersingular abelian varieties in cryptology. In: Yung M (ed) Advances in cryptology – CRYPTO 2002. Lecture notes in computer science, vol 2442. Springer, Berlin, pp 336–353<span class="Occurrences"><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/3-540-45708-9_22"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">34.</div><div class="CitationContent" id="CR34_882">Schirokauer O (1993) Discrete logarithms and local units. Philos Trans R Soc Lond Ser A 345:409–423<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$0795.11063"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1253502"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1098/rsta.1993.0139"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">35.</div><div class="CitationContent" id="CR35_882">Schirokauer O, Weber D, Denny T (1996) Discrete logarithms: the effectiveness of the index calculus method. In: Cohen H (ed) Proceedings of ANTS-II. Lecture notes in computer science, vol 1122. Springer, Berlin, pp 337–362</div></li><li class="Citation"><div class="CitationNumber">36.</div><div class="CitationContent" id="CR36_882">Shoup V (1997) Lower bounds for discrete logarithms and related problems. In: Fumy W (ed) Advances in cryptology – EUROCRYPT’97. Lecture notes in computer science, vol 1233. Springer, Berlin, pp 256–266</div></li><li class="Citation"><div class="CitationNumber">37.</div><div class="CitationContent" id="CR37_882">Shparlinski IE (2003) Cryptographic applications of analytic number theory. Birkhäuser, Basel<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$1036.94001"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1007/978-3-0348-8037-4"><span><span>CrossRef</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">38.</div><div class="CitationContent" id="CR38_882">Smith PJ, Skinner CT (1995) A public-key cryptosystem and a digital signature system based on the Lucas function analogue to discrete logarithms. In: Pieprzyk J, Naini RS (eds) Advances in cryptography – ASIACRYPT’94. Lecture notes in computer science, vol 917. Springer, Berlin, pp 357–364</div></li><li class="Citation"><div class="CitationNumber">39.</div><div class="CitationContent" id="CR39_882">Stinson DR (1995) Cryptography: theory and practice. CRC Press, Boca Raton<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$0855.94001"><span><span>MATH</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">40.</div><div class="CitationContent" id="CR40_882">Winterhof A (2001) A note on the interpolation of the Diffie-Hellman mapping. Bull Aust Math Soc 64:475–477<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$0997.11108"><span><span>MATH</span></span></a></span><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1878898"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceDOI"><a href="http://dx.doi.org/10.1017/S0004972700019936"><span><span>CrossRef</span></span></a></span></span></div></li></ol></aside></div> </div> </div> </div> </div> <div id="abstract-about" class="expander expander-open"> <div class="expander-title"> <div class="heading"> <button> <h2>About this Reference Work Entry</h2> </button> </div> </div> <div class="expander-content"> <div class="expander-content-inner"> <div class="summary"> <dl> <dt>Title</dt> <dd id="abstract-about-title">Computational Diffie-Hellman Problem</dd> <dt>Reference Work Title</dt> <dd id="abstract-about-publication"> <a href="/book/10.1007/978-1-4419-5906-5">Encyclopedia of Cryptography and Security</a> </dd> <dt id="dt-abstract-about-book-chapter-page-ranges">Pages</dt> <dd id="abstract-about-book-chapter-page-ranges">pp 240-244</dd> <dt id="dt-abstract-about-book-chapter-copyright-year">Copyright</dt> <dd id="abstract-about-book-chapter-copyright-year">2011</dd> <dt>DOI</dt> <dd id="abstract-about-book-chapter-doi" class="doi">10.1007/978-1-4419-5906-5_882</dd> <dt id="dt-abstract-about-book-print-isbn">Print ISBN</dt> <dd id="abstract-about-book-print-isbn">978-1-4419-5905-8</dd> <dt id="dt-abstract-about-book-online-isbn">Online ISBN</dt> <dd id="abstract-about-book-online-isbn">978-1-4419-5906-5</dd> <dt id="dt-abstract-about-publisher">Publisher</dt> <dd id="abstract-about-publisher">Springer US</dd> <dt id="dt-abstract-about-book-copyright-holder">Copyright Holder</dt> <dd id="abstract-about-book-copyright-holder">Springer Science+Business Media, LLC</dd> <dt>Additional Links</dt> <dd id="abstract-about-additional-links"> <ul> <li> <a class="external" href="http://www.springer.com/978-1-4419-5905-8?wt_mc=ThirdParty.SpringerLink.3.EPR653.About_eBook" target="_blank" title="It opens in new window">About this Reference Work</a> </li> </ul> </dd> </dl> <dl> <dt>Topics</dt> <dd itemprop="genre"> <ul class="abstract-about-subject"> <li> <a href="/search?facet-subject=%22Data+Structures%2C+Cryptology+and+Information+Theory%22">Data Structures, Cryptology and Information Theory</a> </li> <li> <a href="/search?facet-subject=%22Data+Encryption%22">Data Encryption</a> </li> <li> <a href="/search?facet-subject=%22Coding+and+Information+Theory%22">Coding and Information Theory</a> </li> </ul> </dd> <dt>Industry Sectors</dt> <dd itemprop="genre"> <ul class="abstract-about-industrysectors"> <li> <a title="/industry/electronics" href="/industry/electronics">Electronics</a> </li> <li> <a title="/industry/telecom" href="/industry/telecom">Telecommunications</a> </li> <li> <a title="/industry/it" href="/industry/it">IT & Software</a> </li> </ul> </dd> <dt>eBook Packages</dt> <dd itemprop="genre"> <ul class="abstract-about-ebook-packages"> <li> <a href="/search?package=11645">eBook Package english Computer Science</a> </li> <li> <a href="/search?package=90011652">eBook Package english full Collection</a> </li> </ul> </dd> </dl> <dl> <dt>Editors</dt> <dd> <ul class="editors"> <li itemprop="editor" itemscope="itemscope" itemtype="http://schema.org/Person"> <a class="person" href="/search?facet-creator=%22Henk+C.+A.+van+Tilborg%22" itemprop="name">Henk C. A. van Tilborg</a> <sup title="Department of Mathematics and Computing Science, Eindhoven University of Technology">(376)</sup> </li> <li itemprop="editor" itemscope="itemscope" itemtype="http://schema.org/Person"> <a class="person" href="/search?facet-creator=%22Sushil+Jajodia%22" itemprop="name">Sushil Jajodia</a> <sup title="Center for Secure Information Systems, George Mason University">(377)</sup> </li> </ul> </dd> <dt>Editor Affiliations</dt> <dd> <ul class="editor-affiliations"> <li> <span class="position">376.</span> <span class="affiliation"> Department of Mathematics and Computing Science, Eindhoven University of Technology </span> </li> <li> <span class="position">377.</span> <span class="affiliation"> Center for Secure Information Systems, George Mason University </span> </li> </ul> </dd> <dt>Authors</dt> <dd> <ul class="authors"> <li itemprop="author" itemscope="itemscope" itemtype="http://schema.org/Person"> <a class="person" href="/search?facet-creator=%22Igor+Shparlinski%22" itemprop="name">Igor Shparlinski</a> <sup title="Department of Computing Faculty of Science, Macquarie University">(1)</sup> </li> </ul> </dd> <dt>Author Affiliations</dt> <dd> <ul class="author-affiliations"> <li> <span class="position">1.</span> <span class="affiliation"> Department of Computing Faculty of Science, Macquarie University, Sydney, Australia </span> </li> </ul> </dd> </dl> </div> </div> </div> </div> </div> </div> <div id="look-inside-interrupt" class="look-inside-interrupt"> <h3>Continue reading...</h3> <div class="col-1"> <p>To view the rest of this content please follow the download PDF link above.</p> </div> </div> </div> <div id="web-trekk-abstract" doi="10.1007/978-1-4419-5906-5_882" parentContentType="ReferenceWork" contentType="ReferenceWorkEntry" viewType="Abstract" publication="10.1007/978-1-4419-5906-5 | Encyclopedia of Cryptography and Security" unsiloed="" embedFulltext="false" openChoice="N"></div> <input id="contentType" type="hidden" value="ReferenceWorkEntry"/> <div id="footer"> <hr/> <div id="footer-verticals" class="section" role="navigation"> <span class="strapline">Over 9 million scientific documents at your fingertips</span> <div class="flyout"> <button class="pillow-btn open-disciplines"> Browse by Discipline <span class="caret"></span> </button> <ol class="disciplines"> <li> <a href="/search?facet-discipline="Architecture+%26+Design"" title="follow this link to go to Architecture & Design">Architecture & Design</a> </li> <li> <a href="/search?facet-discipline="Astronomy"" title="follow this link to go to Astronomy">Astronomy</a> </li> <li> <a href="/search?facet-discipline="Biomedical+Sciences"" title="follow this link to go to Biomedical Sciences">Biomedical Sciences</a> </li> <li> <a href="/search?facet-discipline="Business+%26+Management"" title="follow this link to go to Business & Management">Business & Management</a> </li> <li> <a href="/search?facet-discipline="Chemistry"" title="follow this link to go to Chemistry">Chemistry</a> </li> <li> <a href="/search?facet-discipline="Computer+Science"" title="follow this link to go to Computer Science">Computer Science</a> </li> <li> <a href="/search?facet-discipline="Earth+Sciences+%26+Geography"" title="follow this link to go to Earth Sciences & Geography">Earth Sciences & Geography</a> </li> <li> <a href="/search?facet-discipline="Economics"" title="follow this link to go to Economics">Economics</a> </li> <li> <a href="/search?facet-discipline="Education+%26+Language"" title="follow this link to go to Education & Language">Education & Language</a> </li> <li> <a href="/search?facet-discipline="Energy"" title="follow this link to go to Energy">Energy</a> </li> <li> <a href="/search?facet-discipline="Engineering"" title="follow this link to go to Engineering">Engineering</a> </li> <li> <a href="/search?facet-discipline="Environmental+Sciences"" title="follow this link to go to Environmental Sciences">Environmental Sciences</a> </li> <li> <a href="/search?facet-discipline="Food+Science+%26+Nutrition"" title="follow this link to go to Food Science & Nutrition">Food Science & Nutrition</a> </li> <li> <a href="/search?facet-discipline="Law"" title="follow this link to go to Law">Law</a> </li> <li> <a href="/search?facet-discipline="Life+Sciences"" title="follow this link to go to Life Sciences">Life Sciences</a> </li> <li> <a href="/search?facet-discipline="Materials"" title="follow this link to go to Materials">Materials</a> </li> <li> <a href="/search?facet-discipline="Mathematics"" title="follow this link to go to Mathematics">Mathematics</a> </li> <li> <a href="/search?facet-discipline="Medicine"" title="follow this link to go to Medicine">Medicine</a> </li> <li> <a href="/search?facet-discipline="Philosophy"" title="follow this link to go to Philosophy">Philosophy</a> </li> <li> <a href="/search?facet-discipline="Physics"" title="follow this link to go to Physics">Physics</a> </li> <li> <a href="/search?facet-discipline="Psychology"" title="follow this link to go to Psychology">Psychology</a> </li> <li> <a href="/search?facet-discipline="Public+Health"" title="follow this link to go to Public Health">Public Health</a> </li> <li> <a href="/search?facet-discipline="Social+Sciences"" title="follow this link to go to Social Sciences">Social Sciences</a> </li> <li> <a href="/search?facet-discipline="Statistics"" title="follow this link to go to Statistics">Statistics</a> </li> </ol> </div> </div> <div id="footer-nav" class="section"> <div id="footer-nav-misc"> <div id="footer-our-content" class="block" role="navigation"> <h2>Our Content</h2> <ul> <li> <a title="View Journals" href="/search?facet-content-type=%22Journal%22">Journals</a> </li> <li> <a title="View Books" href="/search?facet-content-type=%22Book%22">Books</a> </li> <li> <a title="View Book Series" href="/search?facet-content-type=%22BookSeries%22">Book Series</a> </li> <li> <a title="View Protocols" href="/search?facet-content-type=%22Protocol%22">Protocols</a> </li> <li> <a title="View Reference Works" href="/search?facet-content-type=%22ReferenceWork%22">Reference Works</a> </li> </ul> </div> <div id="footer-other-sites" class="block" role="navigation"> <h2>Other Sites</h2> <ul> <li> <a title="Visit Springer.com" href="http://www.springer.com/">Springer.com</a> </li> <li> <a title="Visit Springer Protocols" href="http://www.springerprotocols.com/">SpringerProtocols</a> </li> <li> <a title="Visit Springer Materials" href="http://materials.springer.com/">SpringerMaterials</a> </li> </ul> </div> <div class="block" role="navigation"> <h2>Help & Contacts</h2> <ul> <li> <a class="contact-us-link" title="Contact us" href="/contactus">Contact Us</a> </li> <li> <a class="impressum-link" title="View Impressum" href="/impressum">Impressum</a> </li> </ul> </div> </div> </div> <div id="footer-legal" class="section" role="contentinfo"> <div id="mobile-nav"> <span class="pillow-btn open-legal" title="Show Legal Information">Legal</span> </div> <div id="legal" role="contentinfo"> <span id="footer-copyright">© Springer International Publishing AG, Part of Springer Science+Business Media</span> <a id="footer-privacy" title="View Terms and Conditions" href="/termsandconditions">Privacy Policy, Disclaimer, General Terms & Conditions</a> <div id="diagnostic-info"> <span id="diagnostic-login-status">Not logged in</span> <span class="diagnostic-business-partners">Unaffiliated</span> <span id="diagnostic-ip">54.163.7.185</span> </div> </div> </div> <div id="footer-branding" class="section"> <div class="block"> <span id="branding-logo">Springer for Research & Development</span> </div> </div> <div id="google-analytics-account" style="display: none">UA-26408784-1</div> </div> <div id="doubleclick-ad" class="banner-advert"> <script type="text/javascript">if ( window.outerWidth || document.body.clientWidth > 1100 ) { googletag.cmd.push(function() { googletag.display('doubleclick-ad'); }); };</script> </div> </div> <noscript> <div id="jsnotice" class="prompt-bar"> <p> JavaScript is currently disabled<span>, this site works much better if you enable JavaScript in your browser.</span> </p> </div> </noscript> <link rel="stylesheet" media="print" href="/static/1.738/css/print.css"/> <script src="/static/js/webtrekk/webtrekk_v3.js"></script> <script type="text/javascript"> var webtrekkProperties = { trackDomain : "springergmbh01.webtrekk.net", trackId : "935649882378213", pageType : "rd_springer_com.reference_work.entry_abstract" }; </script> <noscript> <div id="webtrekk"> <img src="http://springergmbh01.webtrekk.net/935649882378213/wt?p=315,rd_springer_com.reference_work.entry_abstract" height="1" width="1" alt=""/> </div> </noscript> <script type="text/javascript"> (function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){ (i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o), m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m) })(window,document,'script','//www.google-analytics.com/analytics.js','ga'); ga('create', 'UA-26408784-1', 'auto'); ga('create', 'UA-26408784-4', 'auto', {'name': 'unsiloTracker'}); ga('set', 'dimension2', 'N'); ga('require', 'linkid', 'linkid.js'); ga('send', 'pageview'); </script> <script src="/static/js/lib/jquery.min.js"></script> <script src="/static/js/lib/jquery-ui.min.js"></script> <script src="/static/js/lib/jquery-migrate-1.2.1.min.js"></script> <script src="/static/js/lib/crossmark.js"></script> <script type="text/javascript" src="http://rng-snp-003.com/js/36708.js"></script> <noscript> <img src="http://rng-snp-003.com/36708.png" style="display:none;"/> </noscript> <script type="text/javascript"> var vgWortProperties = { vgWortDomain: 'springer.met.vgwort.de' }; </script> <script src="/static/1.738/js/all.js"></script> <script src="/static/1.738/js/internal/mathJaxConfig.js"></script> <script src="http://rd.springer.com/mathjax/v2.5/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script src="/static/1.738/js/internal/mathJaxHooks.js"></script> <script type="text/javascript">var _kiq = _kiq || [];</script> <script type="text/javascript" src="//s3.amazonaws.com/ki.js/47412/9CC.js" async="true"></script> </body> </html>