# Primality testing using elliptic curves

@article{Goldwasser1999PrimalityTU, title={Primality testing using elliptic curves}, author={Shafi Goldwasser and Joe Kilian}, journal={J. ACM}, year={1999}, volume={46}, pages={450-472} }

We present a primality proving algorithm—a probablistic primality test that produces short certificates of primality on prime inputs. We prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes. As a corollary, we obtain an algorithm for generating large certified primes with distribution statistically close to uniform. Under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size, the test is shown to…

## 60 Citations

### DUAL ELLIPTIC PRIMES AND APPLICATIONS TO CYCLOTOMY PRIMALITY PROVING

- 2007

Mathematics, Computer Science

By extending to elliptic curves some notions of galois theory of rings used in the cyclotomy primality tests, one obtains a new algorithm which has heuristic cubic run time and generates certificates that can be verified in quadratic time.

### Dual Elliptic Primes and Applications to Cyclotomy Primality Proving

- 2007

Mathematics, Computer Science

By extending to elliptic curves some notions of galois theory of rings used in the cyclotomy primality tests, one obtains a new algorithm which has heuristic cubic run time and generates certificates that can be verified in quadratic time.

### Deterministic elliptic curve primality proving for a special sequence of numbers

- 2013

Mathematics, Computer Science

We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring…

### Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

- 2007

Computer Science, Mathematics

Math. Comput.

The elliptic curve primality proving algorithm is one of the current fastest practical algorithms for proving the primality of large numbers, and an asymptotically fast version, attributed to J. O. Shallit, is described.

### Primality tests for 2^kn-1 using elliptic curves

- 2009

Mathematics

We propose some primality tests for 2^kn-1, where k, n in Z, k>= 2 and n odd. There are several tests depending on how big n is. These tests are proved using properties of elliptic curves.…

### A New Deterministic Algorithm for Testing Primality Based on a New Property of Prime Numbers

- 2012

Mathematics

SNDS

The paper proposes new theorems by which any prime number can be calculated from the knowledge of any other prime number of lower value in a simple way and proves to be a common thread through which all the prime numbers of a number system can be related.

### An RSA Scheme based on Improved AKS Primality Testing Algorithm

- 2016

Computer Science

This paper proves the necessary and sufficient condition for AKS primality test, and an improved AKS algorithm is proposed using Fermat’s Little Theorem, which becomes an enhanced Miller-Rabin probabilistic algorithm, which can generate primes as fast as the Miller- Rabin algorithm does.

### Some remarks on primality proving and elliptic curves

- 2014

Mathematics

Adv. Math. Commun.

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 used to find the largest proven primes.

### A framework for deterministic primality proving using elliptic curves with complex multiplication

- 2016

Mathematics, Computer Science

Math. Comput.

We provide a framework for using elliptic curves with complex multiplication to determine the primality or compositeness of integers that lie in special sequences, in deterministic quasi-quadratic…

### Probabilistic Search Algorithms with Unique Answers and Their Cryptographic Applications

- 2011

Computer Science, Mathematics

Electron. Colloquium Comput. Complex.

A new type of probabilistic search algorithm, which is guaranteed to run in expected polynomial time, and to produce a correct and unique solution with high probability is introduced, called the Bellagio algorithm.

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