The Choice Of Elliptic Curve And The Study Of Fast Calculation Based On Bilinear Pairing

Elliptic Curve Cryptography(ECC) has become hot point in Public Key Cryptosystems realm.As for applications one important issue is how to improve the efficiency while maintaining the security.The heart of the matters in Elliptic Curve Cryptography are parameters of elliptic curve,point multiplication and computation of bilinear pairings.This work focuses on fast algorithms for efficient point multiplication and pairing computation,chooses of pairing-friendly elliptic curve suitable for bilinear pairings.we studied these two aspects:1.Construct more suitable pairing-friendly elliptic curve for bilinear pairings,measured mainly by the parameters embedding degree k and choice of parameterρ.In this dissertation,we extend embedding degree;also reduce restrictions of r(x).The improved algorithm is suitable for all the bilinear pairings,while it is more superior to Ate pairing that the frobenius trace of elliptic curve reach the lower bound,thus effective application is in pairing-based cryptosystems schemes.2.Point multiplication directly determines the speed of Elliptic Curve Cryptosystem.So we improved bilinear pairings speed by modifing point multiply algorithm in elliptic curve.We studied operations over the finite field with which characteristic was prime.First of all,multiplication calculation was time-consuming more than square.In order to reduce multiplication times,we recombined quadratic inverse based on the existing algorithm and used "double-add"skill.The modified algorithm enhanced by 2.9%.Last,the optimized point multiplication algorithm has been directly applied to Miller algorithm combining representation of NAF,the double-add operation times had reduced.The improved Miller algorithm comparing with Miller algorithm enhanced nearly 47%, increased about 4.03% comparing with the algorithm in literature [55] . So we effectively improved the bilinear pairings.