Testing Mersenne Primes with Elliptic Curves

Abstract

The current primality test in use for Mersenne primes continues to be the Lucas-Lehmer test, invented by Lucas in 1876 and proved by Lehmer in 1935. In this paper, a practical approach to an elliptic curve test of Gross for Mersenne primes, is discussed and analyzed. The most important advantage of the test is that, unlike the Lucas-Lehmer test which requires ${\cal O}(p)$ arithmetic operations and ${\cal O}(p^3)$ bit operations in order to determine whether or not M _p =2^p–1 is prime, it only needs ${\cal O}(\lambda)$ arithmetic operations and ${\cal O}(\lambda^3)$ bit operations, with λ≪p. Hence it is more efficient than the Lucas-Lehmer test, but is still as simple, elegant and practical.

The authors would like to thank Prof Benedict Gross of the Department of Mathematics at Harvard University for his advice to work on the computational aspects of the elliptic curve test for Mersenne primes. This paper was written while the first author visited the Dept of Computer Science at the University of Toronto hosted by Prof Stephen Cook and supported by the Royal Society London.