Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
A proof of the Lucas-Lehmer test
American Mathematical Monthly
Number Theory for Computing
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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 Mp=2p–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.