Reconfigurable hardware implementation of mesh routing in number field sieve factorization

Factorization of large numbers has been a constant source of interest in cryptanalysis. The fastest known algorithm for factoring large numbers is the number field sieve (NFS). The two most time consuming phases of NFS are sieving and matrix step. We propose an efficient way of implementing the matrix step in reconfigurable hardware. Our solution is based on the mesh-routing method proposed by Lenstra et al. We determine the practical size of a partial mesh that can fit in one FFGA device, Xilinx Virtex II XC2V6000. We further extrapolate the computation time for the case of a square systolic array of FFGAs for 512-bit and 1024-bit numbers' factorization. We demonstrate that for practical sizes of numbers used in cryptography, 1024 bits, the matrix step of factorization can be performed using 1024 Virtex II FFGAs in less than 40 days.