An integrated parallel GNFS algorithm for integer factorization based on Linbox Montgomery block Lanczos method over GF(2)

Integer factorization is known to be one of the most important and useful methods in number theory and arithmetic. It also has a very close relationship to some algorithms in cryptography such as RSA algorithm. The RSA cryptosystem is one of the most popular and attractive public-key cryptosystems in the world today. Its security is based on the difficulty of integer factorization. Solving a large and sparse linear system over GF(2) is one of the most time consuming steps in most modern integer factorization algorithms including the fastest one, GNFS algorithm. The Montgomery block Lanczos method from Linbox [13] is for solving large and sparse linear systems over finite fields and it can be integrated into the general number field sieve (GNFS) algorithm which is the best known algorithm for factoring large integers over 110 digits. This paper will present an improved Montgomery block Lanczos method integrated with parallel GNFS algorithm. The experimental results show that the improved Montgomery block Lanczos method has a better performance compared with the original method. It can find more solutions or dependencies than the original method with less time complexities. Implementation details and experimental results are provided in this paper as well.