Paper 2026/1031

Compact Quaternion Algorithms for SQIsign

Abstract

SQIsign is an isogeny-based post-quantum signature scheme whose public keys and signatures are remarkably compact. However, since SQIsign relies on arithmetic in quaternion algebras over the field of rational numbers, no fixed-precision integer arithmetic for SQIsign had been established until recently, hindering constant-time implementation and deployment on memory-constrained devices. Recent work by Kim et al. instantiated an SQIsign implementation with fixed-precision integer arithmetic by deriving uniform worst-case bounds for the quaternion algorithms used in key generation and signing. Nevertheless, the resulting precision budget remains large, exceeding 13 times the public key size. Consequently, this forces implementations to reserve wide integer buffers throughout the computation. This increases the memory footprint and reduces the suitability of fixed-precision SQIsign for constrained platforms. In this work, we present compact quaternion algorithms that substantially reduce the fixed-precision memory requirements of SQIsign. First, we modify and analyze quaternion algorithms for SQIsign, in which large intermediate integer values appear. Then, we derive the improved uniform worst-case size bound on integers during the key generation and signing procedures. As a result, we reduce the required precision budgets from 7026/10713/14150 bits to 1665/2521/3319 bits for the NIST-I/III/V security levels, respectively, corresponding to improvements of $76.30\%$, $76.47\%$, and $76.54\%$. We also provide a fixed-precision implementation of SQIsign applying these improved precision budgets and accelerating previous inefficient implementation of fixed-precision integer arithmetic. Compared with the previous fixed-precision implementation, our implementation achieves performance improvements larger than $99\%$ for both key generation and signing procedures at each security level.