# Testing Primality with Elliptic Curves

W34

#### Abstract

Efficiently distinguishing prime and composite numbers is a fundamental problem in number theory and cryptography. Fermat's Little Theorem gives a criterion for testing primality, but some composite numbers also pass: a Fermat pseudoprime for a base b is a composite number N which satisfies b^(N - 1) = 1 (mod N). A Carmichael number N is a Fermat pseudoprime for all b with gcd(b, N) = 1. In 1987, D. Gordon 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 number of points on a curve for which a fixed composite number N can be a strong pseudoprime. In 2012, J. Silverman extended Gordon's notion of elliptic pseudoprimes and elliptic Carmichael numbers to elliptic curves without CM. Using techniques from analytic number theory, we provide probabilistic bounds for whether a fixed composite number N is an elliptic Carmichael number for a randomly chosen elliptic curve.

Testing Primality with Elliptic Curves

Efficiently distinguishing prime and composite numbers is a fundamental problem in number theory and cryptography. Fermat's Little Theorem gives a criterion for testing primality, but some composite numbers also pass: a Fermat pseudoprime for a base b is a composite number N which satisfies b^(N - 1) = 1 (mod N). A Carmichael number N is a Fermat pseudoprime for all b with gcd(b, N) = 1. In 1987, D. Gordon 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 number of points on a curve for which a fixed composite number N can be a strong pseudoprime. In 2012, J. Silverman extended Gordon's notion of elliptic pseudoprimes and elliptic Carmichael numbers to elliptic curves without CM. Using techniques from analytic number theory, we provide probabilistic bounds for whether a fixed composite number N is an elliptic Carmichael number for a randomly chosen elliptic curve.