Accelerating Iterative SpMV for the Discrete Logarithm Problem Using GPUs

Conference paper

DOI: 10.1007/978-3-319-16277-5_2

Volume 9061 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Jeljeli H. (2015) Accelerating Iterative SpMV for the Discrete Logarithm Problem Using GPUs. In: Koç Ç., Mesnager S., Savaş E. (eds) Arithmetic of Finite Fields. WAIFI 2014. Lecture Notes in Computer Science, vol 9061. Springer, Cham

Abstract

In the context of cryptanalysis, computing discrete logarithms in large cyclic groups using index-calculus-based methods, such as the number field sieve or the function field sieve, requires solving large sparse systems of linear equations modulo the group order. Most of the fast algorithms used to solve such systems — e.g., the conjugate gradient or the Lanczos and Wiedemann algorithms — iterate a product of the corresponding sparse matrix with a vector (SpMV). This central operation can be accelerated on GPUs using specific computing models and addressing patterns, which increase the arithmetic intensity while reducing irregular memory accesses. In this work, we investigate the implementation of SpMV kernels on NVIDIA GPUs, for several representations of the sparse matrix in memory. We explore the use of Residue Number System (RNS) arithmetic to accelerate modular operations. We target linear systems arising when attacking the discrete logarithm problem on groups of size 100 to 1000 bits, which includes the relevant range for current cryptanalytic computations. The proposed SpMV implementation contributed to solving the discrete logarithm problem in GF(\(2^{619}\)) and GF(\(2^{809}\)) using the FFS algorithm.

Keywords

Discrete logarithm problem Sparse-matrix–vector product Modular arithmetic Residue number system GPUs 

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.CARAMEL project-teamLORIA, INRIA/CNRS/Université de LorraineVandœuvre-lés-Nancy CedexFrance