Elliptic curve primality

In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin the same year. The algorithm was altered and improved by several collaborators subsequently, and notably by Atkin and de, in 1993. The concept of using elliptic curves in factorization had been developed by H. W. Lenstra in 1985, and the implications for its use in primality testing (and proving) followed quickly. Primality testing is a field that has been around since the time of Fermat, in whose time most algorithms were based on factoring, which become unwieldy with large input; modern algorithms treat the problems of determining whether a number is prime and what its factors are separately. It became of practical importance with the advent of modern cryptography. Although many current tests result in a probabilistic output (N is either shown composite, or probably prime, such as with the Baillie–PSW primality test or the Miller–Rabin test), the elliptic curve test proves primality (or compositeness) with a quickly verifiable certificate. Previously-known prime-proving methods such as the Pocklington primality test required at least partial factorization of in order to prove that is prime. As a result, these methods required some luck and are generally slow in practice. It is a general-purpose algorithm, meaning it does not depend on the number being of a special form. ECPP is currently in practice the fastest known algorithm for testing the primality of general numbers, but the worst-case execution time is not known. ECPP heuristically runs in time: for some . This exponent may be decreased to for some versions by heuristic arguments. ECPP works the same way as most other primality tests do, finding a group and showing its size is such that is prime. For ECPP the group is an elliptic curve over a finite set of quadratic forms such that is trivial to factor over the group.