High-Performance Integer Factoring with Reconfigurable Devices

Abstract

We present a novel FPGA-based implementation of the Elliptic Curve Method (ECM) for the factorization of medium-sized composite integers. More precisely, we demonstrate an ECM implementation capable to determine prime factors of up to 2,424 151-bit integers per second using a single Xilinx Virtex-4 SX35 FPGA. Using this implementation on a cluster like the COPACOBANA is beneficial for attacking cryptographic primitives like the well-known RSA cryptosystem with advanced methods such as the Number Field Sieve (NFS). To provide this vast number of integer factorizations per FPGA, we make use of the available DSP blocks on each Virtex-4 device to accelerate low-level arithmetic computations. This methodology allows the development of a time-area efficient design that runs 24 ECM cores in parallel, implementing both phase 1 and phase 2 of the ECM. Moreover, our design is fully scalable and supports composite integers in the range from 66 to 236 bits without any significant modifications to the hardware. Compared to the implementation by Gaj et al., who reported an ECM design for the same Virtex-4 platform, our improved architecture provides an advanced cost-performance ratio which is better by a factor of 37.