A Parallel GNFS Algorithm with the Improved Linbox Montgomery Block Lanczos Method for Integer Factorization

RSA algorithm is a very popular public key cryptosystem which has been widely used in industries. Its security relies on the difficulty of factoring large integers. The general number field sieve (GNFS) is so far the best known algorithm for factoring large integers over 110 digits. The Montgomery´s block Lanczos method from Linbox is for solving large and sparse linear systems over finite fields and it can be integrated into GNFS algorithm. This paper introduces an improved Montgomery block Lanczos method, based on the version developed in Linbox, integrated with our previously developed parallel GNFS algorithm. This method has a better performance comparing with the original one, can find more solutions or dependencies than the original one with less time complexities. Implementation details and experimental results will be provided as well in the paper as well.