ECM at work

Abstract

The performance of the elliptic curve method (ECM) for integer factorization plays an important role in the security assessment of RSA-based protocols as a cofactorization tool inside the number field sieve. The efficient arithmetic for Edwards curves found an application by speeding up ECM. We propose techniques based on generating and combining addition-subtracting chains to optimize Edwards ECM in terms of both performance and memory requirements. This makes our approach very suitable for memory-constrained devices such as graphics processing units (GPU). For commonly used ECM parameters we are able to lower the required memory up to a factor 55 compared to the state-of-the-art Edwards ECM approach. Our ECM implementation on a GTX 580 GPU sets a new throughput record, outperforming the best GPU, CPU and FPGA results reported in literature. © International Association for Cryptologic Research 2012.