Factoring Large Numbers with the TWIRL Device
Abstract
The security of the RSA cryptosystem depends on the dif- ficulty of factoring large integers. The best current factoring algorithm is the Number Field Sieve (NFS), and its most difficult part is the siev- ing 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 previ- ously published designs (such as the optoelectronic TWINKLE and the mesh-based sieving). Based on a detailed analysis of all the critical com- ponents (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 $10M device. Coupled with recent results about the cost of the NFS matrix step, this raises some concerns about the security of this key size.
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