Factoring Large Numbers with the TWIRL Device

The security of the RSA cryptosystem depends on the difficulty of factoring large integers. The best current factoring algorithm is the Number Field Sieve (NFS), and its most difficult part is the sieving step. In 1999 a large distributed computation involving hundreds of workstations working for many months managed to factor a 512-bit RSA key, but 1024-bit keys were believed to be safe for the next 15-20 years. In this paper we describe a new hardware implementation of the NFS sieving step (based on standard 0.13μm, 1GHz silicon VLSI technology) which is 3-4 orders of magnitude more cost effective than the best previously published designs (such as the optoelectronic TWINKLE and the mesh-based sieving). Based on a detailed analysis of all the critical components (but without an actual implementation), we believe that the NFS sieving step for 512-bit RSA keys can be completed in less than ten minutes by a $10K device. For 1024-bit RSA keys, analysis of the NFS parameters (backed by experimental data where possible) suggests that sieving step can be completed in less than a year by a $10 M device. Coupled with recent results about the cost of the NFS matrix step, this raises some concerns about the security of this key size.