基于三类代数曲线上的双线性对实现

摘要

In pairings,the most common is the Tate pairing,Eta pairing and Ate pairing are the variations of the Tate pairing. For the elliptic curve y 2 + y = x 3 + x + b where b ∈ F 2 , the implementation algorithms of Tate pairing and Eta pairing in four cases are given by discussing respectively.Meanwhile,the implementation algorithms of Tate pairing and Ate pairing based on the hyperelliptic curve y 2 + y =x 5 +ax +b where a,b ∈ F 2 and the implementation algorithms of Ate pairing based on the hyperelliptic curve y 2 =x p -ax -b where a ,b ∈F p are proposed.It provides a reference for the research and application of the pairing-based cryptosystem.%双线性对中,最常见的是 Tate 对,Eta 对和 Ate 对是 Tate 对的变种。针对椭圆曲线 y 2+y =x 3+x+b,其中 b∈F 2,通过分类讨论,给出了4种情况下的 Tate 对和 Eta 对实现算法。同时,还提出了基于超椭圆曲线 y 2+y =x 5+ax+b(其中 a,b∈F 2)上的 Tate 对和 Ate 对实现算法,以及基于超椭圆曲线 y 2=x p -ax-b(其中 a,b∈F p )上的 Ate 对实现算法。为基于双线性对密码体制的研究和应用提供了一定的参考。