DARM: A Low-Complexity and Fast Modular Multiplier for Lattice-Based Cryptography

The lattice-based cryptography (LBC) has been widely used recently in many compute-intensive applications, such as the post-quantum cryptography (PQC) and privacy-preserving deep learning, where the main task for such applications is to improve the computational efficiency. The modular multiplication operations, mainly involved in the number theoretic transform (NTT), comprise a large proportion of the whole computations required by an LBC. This paper presents a novel "decompose-and-reduce" modular multiplication algorithm (DARM), considering primes with the form of q = 2
^2N
−δ and δ < 2
^N−2
. The inherent structure of the modulus is exploited and the intermediates’ data widths are reduced. Moreover, a low-complexity and fast multiplier is elaborately devised based on DARM. To further validate the performance of our multiplier, an n-point NTT design with DARM is implemented with various configurations. FPGA implementation results demonstrate that compared with the prior arts, the proposed multiplier has 1.12-1.89× speedups with the least DSP utilization. For the case of ⌈log
2
q⌉ = 60 and n = 4096, the NTT implementation with DARM achieves up to 41.2% and 61.2% reductions in LUTs and DSPs, respectively.