Abstract
We give an overview of a method for using elliptic curves with complex multiplication to give efficient deterministic polynomial time primality tests for the integers in sequences of a special form. This technique has been used to find the largest proven primes $N$ for which there was no known significant partial factorization of $N-1$ or $N+1$.
Highlights
In this article we will make some remarks on a technique for using elliptic curves to give efficient deterministic primality tests for integers in very special sequences
In his 1985 Masters thesis “Primality testing using elliptic curves” [8], Wieb Bosma gave sufficient conditions for primality of numbers of special forms, using elliptic curve analogues of Lucas’ test, where arithmetic in the group (Z/nZ)× is replaced by arithmetic in the redu√ction mod n of an elliptic curve with complex multiplication (CM) by Q(i) or Q( −3)
We present two proofs in parallel, in order to show the relationship between a classical primality test and the elliptic curve tests we are concerned with in this paper
Summary
In this article we will make some remarks on a technique for using elliptic curves to give efficient deterministic primality tests for integers in very special sequences. The implementations run in quasi-quadratic time, and are useful for proving the primality of large primes in certain sequences to which classical p ± 1 tests do not apply. We state Gross’s result in §4, and give a proof of it that runs parallel to the proof of Pepin’s primality test for Fermat numbers. The primality testing theorems have been phrased in parallel ways to try to make clear how they are all related
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