Sparse linear algebra in function field sieve

Abstract

Public key cryptography is based on mathematically hard problems namely integer factorization and Discrete Logarithm Problem (DLP). The Galoise fields GF(p) and GF(pn) are used in DLP based public key cryptography. The number field sieve and function field sieve are two popular methods to solve these hard problems. These methods involves two main phases such as pre computation and computation phases. The precomputation phase in turn consists of two steps such as relation collection and solving linear system of equations. The solving step is a big bottle neck in these methods. The popular methods used in this step are Lanczos and Wiedemann. The Lanczos is being used for long period of time in this step, recently Block Wiedemann is being used to solve in the field GF(2809) bits. Block Wiedemann started getting attention to solve DLP in higher field. In this paper the block Wiedemann method for solving DLP is investigated. The main motive of using Block Wiedemann method in higher field is analyzed. Also the performance of Block Wiedemann is critically analyzed on the sparsity of the matrix for solving DLP in one of the large sub group and the results are reported. This may insist to maintain the matrix as sparse as possible in the previous phases of linear algebra of Function Field Sieve to arrive computationally fast solution in the linear algebra phase.