A Faster Software Implementation of SQIsign

Isogeny-based cryptography is famous for its short key size. As one of the most compact digital signatures, SQIsign (Short Quaternion and Isogeny Signature) is attractive among post-quantum cryptography, but it is inefficient compared to other post-quantum competitors because of complicated procedures in the ideal-to-isogeny translation, which is the efficiency bottleneck of the signing phase. In this paper, we recall the current implementation of SQIsign and mainly focus on how to improve the execution of the ideal-to-isogeny translation in SQIsign. Specifically, we demonstrate how to utilize the reduced Tate pairing to save one of the two elliptic curve discrete logarithms. In addition, the efficient implementation of the remainder discrete logarithm computation is explored. We speed up other procedures in the ideal-to-isogeny translation with various techniques as well. It should be noted that our improvements also benefit the performance of key generation and verification in SQIsign. In the instantiation with <inline-formula> <tex-math notation="LaTeX">$ {p_{1973}}$ </tex-math></inline-formula>, the improvements lead to a speedup of 5.47%, 8.80% and 25.34% for key generation, signature and verification, respectively.