Extending Lenstra's Primality Test to CM elliptic curves and a new quasi-quadratic Las Vegas algorithm for primality

Abstract

For an elliptic curve with CM by $K$ defined over its Hilbert class field, $E/H$, we extend Lenstra's finite fields test to generators of norms of certain ideals in $\mathcal{O}_H$, yielding a sufficient $\widetilde{O}(\log^3 N)$ primality test and partially answering an open question of Lemmermeyer in the case of CM elliptic curves. Letting $\iota,\gamma, b\in \mathcal{O}_K$, $(\iota)$ prime, and $b$ a primitive $k$-th root of unity modulo $(\iota)^n$ we specialize this test to rational integers of the form $N_{K/\mathbb{Q}}(\gamma\iota^n+b)$ with the norm of $\gamma$ small, giving a Las Vegas test for primality with average runtime $\widetilde{O}(\log^2 N)$, that further certifies primality of such integers in $\widetilde{O}(\log^2 N)$ for nearly all choices of input parameters. The integers tested were not previously amenable to quasi-quadratic heuristic primality certification.