Toric forms of elliptic curves and their arithmetic

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• 作者：Wouter Castryck^a ; ¹ ; ^{wouter.castryck@gmail.com"" rel=""nofollow} ; ; Frederik Vercauteren^b ; ^{fvercaut@esat.kuleuven.be"" rel=""nofollow} ;
• 关键词：Elliptic curve ; Cryptography ; Arithmetic ; Newton polytope ; Toric geometry
• 刊名：Journal of Symbolic Computation
• 出版年：2011
• 出版时间：August 2011
• 年：2011
• 卷：46
• 期：8
• 页码：943-966
• 全文大小：399 K

文摘

We scan a large class of one-parameter families of elliptic curves for efficient arithmetic. The construction of the class is inspired by toric geometry, which provides a natural framework for the study of various forms of elliptic curves. The class both encompasses many prominent known forms and includes thousands of new forms. A powerful algorithm is described that automatically computes the most compact group operation formulas for any parameterized family of elliptic curves. The generality of this algorithm is further illustrated by computing uniform addition formulas and formulas for generalized Montgomery arithmetic.