Split-Radix Based Compact Hardware Architecture for CRYSTALS-Kyber

Facing the threat of large-scale quantum computers to traditional public-key cryptography, the National Institute of Standards and Technology has conducted Post-Quantum Cryptography algorithms evaluation for a long time, and CRYSTALS-Kyber has been selected to enter the standardization process. In the previous literature, hardware designs can significantly improve the performance of CRYSTALS-Kyber, and the most time-consuming operations are Number Theoretic Transform (NTT) and point-wise multiplication (PWM). However, the split-radix algorithm, which has a lower theoretical complexity in the FFT, has rarely been studied in the NTT. In this paper, we studied whether there are advantages of introducing split-radix algorithms into the NTT defined by CRYSTALS-Kyber and detailed derived the split-radix algorithms for the forward and inverse NTT without pre- or post-processing. By further optimizing the split-radix algorithm for the forward NTT, one of the three modular multipliers in the
$\boldsymbol{L}$
-shaped butterfly unit is replaced by shifting-and-addition, which will reduce the hardware resource consumption. Besides, we proposed a recombined formula for PWM, which reduces the capacity of the intermediate data RAM for PWM by 25%. Together with the proposed hardware scheduling method, the above algorithms can improve performance and save hardware resources.