The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.

We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.

Abstract. In a series of papers, D. Gordon and C. Pomerance demonstrated that pseudoprimes on elliptic curves behave in many ways very similar to pseudoprimes related to Lucas sequences. In this paper we give an answer to a challenge that was posted by D. Gordon in 1989. The challenge was to either prove that a certain composite N ≡ 1mod4didnotexist, orto explicitly calculate such a number. In this paper, we both present such a specific composite (for Gordon’s curve with CM by Q ( √ −7)), as well as a proof of the non-existence (for curves with CM by Q ( √ −3)). We derive some criteria for the group structure of CM curves that allow testing for all composites, including N ≡ 3 mod 4 which had been excluded by Gordon. This gives rise to another type of examples of composites where strong elliptic pseudoprimes are not Euler elliptic pseudoprimes. 1.

this paper, we aim at presenting the most recent results achieved in the theory of pseudoprime numbers. First of all, we make a list of all pseudoprime varieties existing so far. This includes Lucas-pseudoprimes and the generalization to sequences generated by integer polynomials modulo N , elliptic pseudoprimes. We discuss the making of tables and the consequences on the design of very fast primality algorithms for small numbers. Then, we describe the recent work of Alford, Granville and Pomerance, in which they prove that there

Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1.