CAIRN 3: An FPGA Implementation of the Sieving Step with the Lattice Sieving

The hardness of the integer factorization problem assures the security of some public-key cryptosystems including RSA, and the number field sieve method (NFS), the most efficient algorithm for factoring large integers currently, is a threat for such cryptosystems. Recently, Izu et al. developed a dedicated sieving device “CAIRN 2” with Xilinx’s FPGA which is designed to handle up to 768-bit integers. However, since CAIRN 2 uses the line sieving, it is not optimized from the viewpoint of the efficiency. In this paper, we report some results of an FPGA-based sieving hardware “CAIRN 3” with the lattice sieving. In the experimental sieving for a 768-bit integer (RSA768), CAIRN 3 is about 38 times faster than CAIRN 2. It is estimated that the full sieving for RSA768 requires about 270 years with single CAIRN 3.