Impact of Optimization and Parallelism on Factorization Speed of SIQS

Dominik Breitenbacher, Ivan Homoliak, Jiri Jaros, Petr Hanacek

This paper examines optimization possibilities of Self-Initialization Quadratic Sieve (SIQS), which is enhanced version of Quadratic Sieve factorization method. SIQS is considered the second fastest factorization method at all and the fastest one for numbers shorter than 100 decimal digits, respectively. Although, SIQS is the fastest method up to 100 decimal digits, it cannot be effectively utilized to work in polynomial time. Therefore, it is desirable to look for options how to speed up the method as much as possible. Two feasible ways of achieving it are code optimization and parallelism. Both of them are utilized in this paper. The goal of this paper is to show how it is possible to take advantage of parallelism in SIQS as well as reach a large speed-up thanks to detailed source code analysis with optimization. Our implementation process consists of two phases. In the first phase, the complete serial algorithm is implemented in the simplest way which does not consider any requirements for execution speed. The solution from the first phase serves as the reference implementation for further experiments. An improvement of factorization speed is performed in the second phase of the SIQS implementation, where we use the method of iterative modifications in order to examine contribution of each proposed step. The final optimized version of the SIQS implementation has achieved over 200x speed-up.

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