Primality proofs with elliptic curves: heuristics and analysis Wieb Bosma * , Eric Cator † , Antal J´ arai ‡ ,Gy¨ongyv´ er Kiss § September 11, 2014 Abstract This paper deals with the heuristic running time analysis of the elliptic curve primality proving (ECPP) algorithm of Atkin and Morain. Our aim is to collect assumptions and the fastest possible algorithms to reduce the heuristic running time and to show that under these assumptions and some plausible conditions the heuristic running time can be reduced down to o(ln 4 n) bit operation for input possible prime n. 1 Introduction In the work of Atkin and Morain [1] the background and an exact implemen- tation of elliptic curve primality proving (ECPP) algorithm is described. A heuristic running time analysis is given by Lenstra, Lenstra [20] and Morain [22]. They have found that the running time is O(ln 6+ε n) for any positive ε for input n. Using asymptotically fast methods for multiplication, division, polyno- mial calculation, etc. it is possible to reduce this down to O(ln 5+ε n). With a trick attributed to J. O. Shallit (building up discriminants from small primes), it can be expected to run in time O(ln 4+ε n). In this paper we are investigating the heuristic running time of ECPP using the fastest known algorithms to compute the various parts and prove that under some conditions the heuristic running time can be reduced down to o(ln 4 n). Moreover we are summarizing questions and assumptions that are related to this topic. There are projects of one of the authors, Gy. Kiss, on implementations that are using these assumptions in practice. One of them is finished, the details can * Department of Algebra and Topology, Radboud University, Nijmegen, The Netherlands. E-mail: bosma@math.ru.nl † Department of Applied Stochastics, Radboud University, Nijmegen, The Netherlands. E-mail: e.cator@science.ru.nl ‡ DepartmentofComputerAlgebra, E¨otv¨osLor´andUniversity, Budapest, Hungary. E-mail: ajarai@moon.inf.elte.hu § DepartmentofComputerAlgebra, E¨otv¨osLor´andUniversity, Budapest, Hungary. E-mail: kissgyongyver@gmail.com 1