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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 certainideals in $\mathcal{O}_H$, yielding a sufficient $\widetilde{O}(\log^3 N)$primality test and partially answering an open question of Lemmermeyer in thecase of CM elliptic curves. Letting $\iota,\gamma, b\in \mathcal{O}_K$,$(\iota)$ prime, and $b$ a primitive $k$-th root of unity modulo $(\iota)^n$ wespecialize this test to rational integers of the form$N_{K/\mathbb{Q}}(\gamma\iota^n+b)$ with the norm of $\gamma$ small, giving aLas 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 notpreviously amenable to quasi-quadratic heuristic primality certification.

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