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 currently 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, is expected to run in time Õ((log N)4). We describe this version in more details, leading to actual implementations able to handle numbers with several thousands of decimal digits. 1.

We present an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite. 1

Abstract For many years mathematicians have searched for a fast and reliable primality test. This is especially relevant nowadays, because the RSA public-key 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 polynomial-time 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.

Abstract. This paper offers numerical evidence for a conjecture that primality proving may be done in (log N) 3+o(1) operations by examining the growth rate of quantities known as pseudosquares and pseudocubes. In the process, a novel method of solving simultaneous congruences— doubly-focused enumeration — is examined. This technique, first described by D. J. Bernstein, allowed us to obtain record-setting sieve computations in software on general purpose computers. 1

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 Schönhage-Strassen algorithm for n-bit integer multiplication, it is easier to write than the (more precise) �O(n) O(nlog nlog log n).

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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.

Abstract. Let p denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of p − 1 exceeding (p − 1) 1 2 in which the constants are effectively computable. As a result we prove that it is possible to calculate a value x0 such that for every x>x0 there is a p<xwith the greatest prime factor of p − 1 exceeding x 3 5. The novelty of our approach is the avoidance of any appeal to Siegel’s Theorem on primes in arithmetic progression. 1.