Strongly nonzero points and elliptic pseudoprimes

Abstract

Efficiently distinguishing prime and composite numbers is one of the fundamental problems in number theory. A Fermat pseudoprime is a composite number $N$ which satisfies Fermat’s little theorem for a specific base $b$: $b^{N-1}\equiv 1modN$. A Carmichael number $N$ is a Fermat pseudoprime for all $b$ with $gcd\left(b,N\right)=1$. D. Gordon (1987) introduced analogues of Fermat pseudoprimes and Carmichael numbers for elliptic curves with complex multiplication (CM): elliptic pseudoprimes, strong elliptic pseudoprimes and elliptic Carmichael numbers. It has previously been shown that no CM curve has a strong elliptic Carmichael number. We give bounds on the fraction of points on a curve for which a fixed composite number $N$ can be a strong elliptic pseudoprime. J. Silverman (2012) extended Gordon’s notion of elliptic pseudoprimes and elliptic Carmichael numbers to arbitrary elliptic curves. We provide probabilistic bounds for whether a fixed composite number $N$ is an elliptic Carmichael number for a randomly chosen elliptic curve.