Advisor - Prof. Alice Silverberg
Abstract - For any imaginary quadratic field $K$, number field $F$ containing the Hilbert class field of $K$,  elliptic curve $E / F$ with complex multiplication by $K$, and  sequence of numbers $F_k$ of "special" form we give an efficient deterministic primality (and compositeness) test on $F_k$.  These primality tests are an extension of the Lucas primality test to the setting of elliptic curves.  In particular, for every prime $p$ in the sequence $F_k$, we need to know $|E(\Z / p\Z)|$.  To satisfy this requirement we restrict to the setting of CM elliptic curves.