On The Optimization Of The Public Key Cryptographic Algorithm SM2 Based On Elliptic Curves

Elliptic curve cryptography(ECC)is now one of the most important methods for publickey encryption and signature schemes in the real world,With the same required security,it can be satisfied with a smaller key size.The smaller key size also brings many advantages such as fast execution,small storage capacity,and low transmission channel bandwidth requirements.It is very suitable for applications that currently have limited storage space and bandwidth but urgently need fast implementation.In this situation,the State Cryptography Management Office of China released a set of public key cryptography algorithm SM2 based on elliptic curves,which was standardized in ISO in 2017 to strengthen my country’s network security.In this work,we aim at this point and further optimize the SM2 elliptic curve cryptographic algorithm in order to obtain a more efficient implementation of the SM2 cryptosystem.In order to improve the efficiency of modular operations,the base domain characteristics of some standard elliptic curves in cryptography are often generalized Mersenne primes,such as NIST curves P-256 and SM2.In the actual underlying operations,such special prime numbers make the most influential modular multiplication operations very efficient.In order to promote the fast modular multiplication operation on the SM2 prime field,in order to obtain a modular reduction algorithm close to NIST P-256,we propose two optimization methods.In the first result,the approach of Solinas and Brown et al.is refined for the case of SM2 primes,reducing the number of required intermediate variables(of vectors of words)to 13.In the second result,a new processing technique was adopted to eliminate variables by appropriate transformation of the operation formula for further optimization,resulting in a method requiring only 11 intermediate variables(of vectors of words).This part of the work makes the modular multiplication operation on the SM2 prime field more efficient,and provides a valuable reference for the subsequent software and hardware optimization of SM2.The efficiency of ECC implementation mainly depends on the implementation of scalar multiplication,that is,for a given point P on the curve,the calculation of point kP=P+…+P(k times)is performed.In order to improve the execution efficiency of this operation,there have been a lot of related work on it,among which the multi-base chain method has received extensive attention.In order to improve the performance of calculating scalar multiplication by the multi-base chain method,we propose an improvement on the existing best 5P calculation method and propose a faster 5P calculation formula.Compared with the previous 5P formula,our two optimization results have a certain degree of efficiency improvement,and can make a good trade-off under different platforms.Based on this work,we have designed a more efficient multi-base chain method for improving the efficiency of computing scalar multiplications on elliptic curves.