{"id":736684,"date":"2021-03-29T08:59:14","date_gmt":"2021-03-29T15:59:14","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=736684"},"modified":"2021-03-29T08:59:14","modified_gmt":"2021-03-29T15:59:14","slug":"a-simple-and-compact-algorithm-for-sidh-with-arbitrary-degree-isogenies","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/a-simple-and-compact-algorithm-for-sidh-with-arbitrary-degree-isogenies\/","title":{"rendered":"A Simple and Compact Algorithm for SIDH with Arbitrary Degree Isogenies"},"content":{"rendered":"<p id=\"Par1\" class=\"Para\">We derive a new formula for computing arbitrary odd-degree isogenies between elliptic curves in Montgomery form. The formula lends itself to a simple and compact algorithm that can efficiently compute any low odd-degree isogenies inside the supersingular isogeny Diffie-Hellman (SIDH) key exchange protocol. Our implementation of this algorithm shows that, beyond the commonly used 3-isogenies, there is a moderate degradation in relative performance of\u00a0<span id=\"IEq1\" class=\"InlineEquation\"><span class=\"MathJax_Preview\">(2d+1)<\/span><\/span>-isogenies as\u00a0<em class=\"EmphasisTypeItalic \">d<\/em>\u00a0grows, but that larger values of\u00a0<em class=\"EmphasisTypeItalic \">d<\/em>\u00a0can now be used in practical SIDH implementations.<\/p>\n<p id=\"Par2\" class=\"Para\">We further show that the proposed algorithm can be used to both compute isogenies of curves and evaluate isogenies at points, unifying the two main types of functions needed for isogeny-based public-key cryptography. Together, these results open the door for practical SIDH on a much wider class of curves, and allow for simplified SIDH implementations that only need to call one general-purpose function inside the fundamental computation of the large degree secret isogenies.<\/p>\n<p id=\"Par3\" class=\"Para\">As an additional contribution, we also give new explicit formulas for 3- and 4-isogenies, and show that these give immediate speedups when substituted into pre-existing SIDH libraries.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We derive a new formula for computing arbitrary odd-degree isogenies between elliptic curves in Montgomery form. The formula lends itself to a simple and compact algorithm that can efficiently compute any low odd-degree isogenies inside the supersingular isogeny Diffie-Hellman (SIDH) key exchange protocol. Our implementation of this algorithm shows that, beyond the commonly used 3-isogenies, [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Springer-Verlag","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"303","msr_page_range_end":"329","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"ASIACRYPT 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