Flexible and scalable FPGA-oriented design of multipliers for large binary polynomials

With the recent advances in quantum computing, code-based cryptography is foreseen to be one of the few mathematical solutions to design quantum resistant public-key cryptosystems. The binary polynomial multiplication dominates the computational time of the primitives in such cryptosystems, thus the design of efficient multipliers is crucial to optimize the performance of post-quantum public-key cryptographic solutions. This manuscript presents a flexible template architecture for the hardware implementation of large binary polynomial multipliers. The architecture combines the iterative application of the Karatsuba algorithm, to minimize the number of required partial products, with the Comba algorithm, used to optimize the schedule of their computations. In particular, the proposed multiplier architecture supports operands in the order of dozens of thousands of bits, and it offers a wide range of performance-resources trade-offs that is made independent from the size of the input operands. To demonstrate the effectiveness of our solution, we employed the nine configurations of the LEDAcrypt public-key cryptosystem as representative use cases for large-degree binary polynomial multiplications. For each configuration we showed that our template architecture can deliver a performance-optimized multiplier implementation for each FPGA of the Xilinx Artix-7 mid-range family. The experimental validation performed by implementing our multiplier for all the LEDAcrypt configurations on the Artix-7 12 and 200 FPGAs, i.e., the smallest and the largest devices of the Artix-7 family, demonstrated an average performance gain of 3.6x and 33.3x with respect to an optimized software implementation employing the gf2x C library.