Accelerating Iterative SpMV for the Discrete Logarithm Problem Using GPUs

Abstract

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.

Discrete Logarithm in GF(2809) with FFS

“…However, for brevity purposes, this presentation is in no way complete nor exhaustive. For more details, we refer the interested reader to the extensive literature on this topic, starting from the first theoretical articles [2,3,25] to the algorithmic advances and implementation reports which followed [18,14,15,7,11,10,16].…”

“…The implementation used Residue Number System (RNS) arithmetic to accelerate arithmetic over Z/ Z, since this representation system offers the opportunity to increase the parallelism between the computational units, and is well suited to the GPU execution framework. This approach is described in details in [16].…”

“…The development of the present work has led to several preprints and articles providing detailed insights on the improvements of the individual steps of FFS [7,10,11,16]. For brevity, this article does not repeat these details, and the interested reader is referred to these articles for more detail.…”

Generating Optimized Sparse Matrix Vector Product over Finite Fields

“…The difficulty is then transferred to the reduction modulo p that cannot be done in the RNS basis. However, one can use the explicit chinese remainder theorem [2,6] to provide a reduction that can use SIMD instructions. Furthermore, one can use matrix multiplication to perform modular reduction of a vector of RNS values and then better exploit data locality and SIMD.…”

“…Our main concern is weather the numerical formats are still satisfying when computing over finite fields and which arithmetic strategy is the most suited to the particularity of SpMV. This question has been already addressed in many papers, as in [3,6], but no general optimization approach has been designed. In this paper, we propose a general framework which incorporates most of the optimization techniques for SpMV over finite field.…”