Improvements to Lucas-sequence modular square roots and primality testing

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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 impro