Paper 2025/2083

Improvements to Lucas-sequence modular square roots and primality testing

Abstract

Lucas sequences are a helpful tool in mathematical and cryptographic calculations, providing in particular an efficient way to exponentiate in a quotient ring $R[x]/(x^2 - Px + Q)$. As with exponentiation in other finite rings and fields, we can use the periodic nature of these sequences to find roots of polynomials. Since they behave differently in the ring $\mathbb{Z}/N$ depending on whether $N$ is prime, Lucas sequences are also useful for primality testing. In this paper, we discuss improvements to Lucas-sequence algorithms for square roots and heuristic primality testing. Our first application is modular square roots. It is straightforward to take square roots modulo primes $p\equiv \{3,5,7\}$ mod 8. When $p\equiv 1$ mod 8, and especially when $p-1$ is divisible by many powers of 2, Müller's algorithm and Kim-Koo-Kwon are attractive options. Both of these use Lucas sequences. Here we show how to simplify and speed up Kim-Koo-Kwon. We also show a variant on Müller's algorithm which works even when $p\equiv 3$ mod 4, which would be useful if $p$ were secret. Our second application is heuristic primality testing. The Baillie-PSW primality test combines a strong Fermat test with a strong Lucas test. The recent Baillie-Fiori-Wagstaff variant strengthens Baillie-PSW. Here we show an improved variant, $\mathtt{SuperBFPSW}$, which is stronger than Baillie-Fiori-Wagstaff, but also faster than the original Ballie-PSW.