Advances in Applied Mathematics
Almost all primes can be quickly certified
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Recognizing primes in random polynomial time
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
An improved Las Vegas primality test
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
Uses of randomness in algorithms and protocols
Uses of randomness in algorithms and protocols
Faster primality testing (extended abstract)
EUROCRYPT '89 Proceedings of the workshop on the theory and application of cryptographic techniques on Advances in cryptology
Cyclotomy Primality Proving - Recent Developments
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
Algorithms in number theory
ACM Transactions on Algorithms (TALG)
Algorithms and theory of computation handbook
Spreading alerts quietly and the subgroup escape problem
ASIACRYPT'05 Proceedings of the 11th international conference on Theory and Application of Cryptology and Information Security
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We present a primality proving algorithm—a probablistic primality test that produces short certificates of primality on prime inputs. We prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes. As a corollary, we obtain an algorithm for generating large certified primes with distribution statistically close to uniform. Under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size, the test is shown to run in expected polynomial time for all primes, yielding a Las Vegas primality test.Our test is based on a new methodology for applying group theory to the problem of prime certification, and the application of this methodology using groups generated by elliptic curves over finite fields.We note that our methodology and methods have been subsequently used and improved upon, most notably in the primality proving algorithm of Adleman and Huang using hyperelliptic curves and in practical primality provers using elliptic curves.