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Implementing the asymptotically fast version of the elliptic curve primality proving algorithm
 Math. Comp
"... Abstract. The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time Õ((log N)5) to prove the primality of N ..."
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Cited by 27 (1 self)
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Abstract. The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time Õ((log N)5) to prove the primality of N. An asymptotically fast version of it, attributed to J. O. Shallit, runs in time Õ((log N)4). The aim of this article is to describe this version in more details, leading to actual implementations able to handle numbers with several thousands of decimal digits. 1.
Primality testing
, 1992
"... Abstract For many years mathematicians have searched for a fast and reliable primality test. This is especially relevant nowadays, because the RSA publickey cryptosystem requires very large primes in order to generate secure keys. I will describe some efficient randomised algorithms that are useful ..."
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Cited by 3 (1 self)
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Abstract For many years mathematicians have searched for a fast and reliable primality test. This is especially relevant nowadays, because the RSA publickey cryptosystem requires very large primes in order to generate secure keys. I will describe some efficient randomised algorithms that are useful in practice, but have the defect of occasionally giving the wrong answer, or taking a very long time to give an answer. Recently Agrawal, Kayal and Saxena found a deterministic polynomialtime primality test. I will describe their algorithm, mention some improvements by Bernstein and Lenstra, and explain why this is not the end of the story.
Uncertainty can be Better than Certainty: Some Algorithms for Primality Testing ∗
, 2006
"... First, some notation As usual, we say that f(n) = O(n k) if, for some c and n0, for all n ≥ n0, We say that if, for all ε> 0, f(n) ≤ cn k. f(n) = �O(n k) f(n) = O(n k+ε). The “ � O ” notation is useful to avoid terms like log n and log log n. For example, when referring to the SchönhageStra ..."
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First, some notation As usual, we say that f(n) = O(n k) if, for some c and n0, for all n ≥ n0, We say that if, for all ε> 0, f(n) ≤ cn k. f(n) = �O(n k) f(n) = O(n k+ε). The “ � O ” notation is useful to avoid terms like log n and log log n. For example, when referring to the SchönhageStrassen algorithm for nbit integer multiplication, it is easier to write than the (more precise) �O(n) O(nlog nlog log n).
Primality testing
, 2003
"... We consider the classical problem of testing if a given (large) number n is prime or composite. First we outline some of the efficient randomised algorithms for solving this problem. For many years it has been an open question whether a deterministic polynomial time algorithm exists for primality ..."
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We consider the classical problem of testing if a given (large) number n is prime or composite. First we outline some of the efficient randomised algorithms for solving this problem. For many years it has been an open question whether a deterministic polynomial time algorithm exists for primality testing, i.e. whether "PRIMES is in P". Recently Agrawal, Kayal and Saxena answered this question in the affirmative. They gave a surprisingly simple deterministic algorithm. We describe their algorithm, mention some improvements by Bernstein and Lenstra, and consider whether the algorithm is useful in practice. Finally, as a topic for future research, we mention a conjecture that, if proved, would give a fast and practical deterministic primality test.