Broken-Karatsuba multiplication and its application to Montgomery modular multiplication

Large number multiplication has always been an essential operation in cryptographic algorithms. In this paper, we propose Broken-Karatsuba multiplication by applying the non-least-positive form to represent large numbers and dig the parallelism hidden in conventional Karatsuba multiplication. Further, we modify Montgomery modular multiplication algorithm with Broken-Karatsuba multiplication to make it suitable for pipeline implementation with fewer hardware resources. Based on this modified algorithm, a 256-bit two-stage modular multiplier is constructed. There is no stall in the pipeline when performing consecutive modular multiplications and the delay of a modular multiplication is reduced significantly. Implemented on Virtex-6 FPGA platforms, our design outperforms most previous works in terms of modular multiplication latency and area-time product, which makes it suitable for server-side applications.