An improved Las Vegas primality test

Authors:
E. Kaltofen;T. Valente;N. Yui
Affiliations:
Department of Computer Science, Rensselaer Polytechnic Institute, Troy, New York;Department of Computer Science, Rensselaer Polytechnic Institute, Troy, New York;Department of Mathematics, Queen's University, Kingston, Ontario, Canada K7L3N6
Venue:
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
Year:
1989

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Abstract

We present a modification of the Goldwasser-Kilian-Atkin primality test, which, when given an input n, outputs either prime or composite, along with a certificate of correctness which may be verified in polynomial time. Atkin's method computes the order of an elliptic curve whose endomorphism ring is isomorphic to the ring of integers of a given imaginary quadratic field Q(√—D). Once an appropriate order is found, the parameters of the curve are computed as a function of a root modulo n of the Hilbert class equation for the Hilbert class field of Q(√—D). The modification we propose determines instead a root of the Watson class equation for Q(√—D) and applies a transformation to get a root of the corresponding Hilbert equation. This is a substantial improvement, in that the Watson equations have much smaller coefficients than do the Hilbert equations.