Primality Testing with Gaussian Periods

  title={Primality Testing with Gaussian Periods},
  author={Hendrik W. Lenstra},
  booktitle={Foundations of Software Technology and Theoretical Computer Science},
  • H. Lenstra
  • Published in
    Foundations of Software…
    12 December 2002
  • Mathematics
It was recognized in the mid-eighties, that several then current primality tests could be formulated in the language of Galois theory for rings. This made it possible to combine those tests for practical purposes. It turns out that the new polynomial time primality test due to Agrawal, Kayal, and Saxena can also be formulated in the Galois theory language. Whether the new formulation will allow the test to be combined with the older tests remains to be seen. It does lead to a primality test… 

On cyclotomic primality tests

In 1980, L. Adleman, C. Pomerance, and R. Rumely invented the first cyclotomic primality test, and shortly after, in 1981, a simplified and more efficient version was presented by H.W. Lenstra for

Cyclotomy Primality Proofs and their Certificates

An overview of the theoretical background and implementation specifics of CPP, such as the authors understand them in the year 2007 is given.

On the implementation of AKS-class primality tests

Algorithms of the new “cyclotomic AKS class” determine rigorously and in polynomial time whether an input integer is prime. Herein are discussed implementation issues, with a focus on techniques such

PPT: New Low Complexity Deterministic Primality Tests Leveraging Explicit and Implicit Non-Residues. A Set of Three Companion Manuscripts

This set of three companion manuscripts/articles unveils new results on primality testing and reveal new primalityTesting algorithms enabled by those results and reveals close connections between the state-of-the-art Miller-Rabin method and the renowned Euler-Criterion.

Dual Elliptic Primes and Applications to Cyclotomy Primality Proving

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.

An Empirical Study towards Refining the AKS Primality Testing Algorithm

From this analysis and observations, AKS algorithm is refined so that it runs in O(log n) time, based on the empirical results and statistics of certain parameters which control the asymptotic running time of the algorithm.

Kummer surfaces for primality testing

An algorithm is provided capable of proving the primality or compositeness of most of the integers in these families and the necessary steps to implement this algorithm in a computer are discussed.

De los números de Midy a la primalidad

We define the concept of q-pseudoprime to base b, which extends the idea of strong pseudoprime to base b. We stablish a new test of primality that refines the Pocklinton’s Theorem using some properties

Deterministic elliptic curve primality proving for a special sequence of numbers

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



Prime Numbers: A Computational Perspective

Prime numbers beckon to the beginner, the basic notion of primality being accessible to a child. Yet, some of the simplest questions about primes have stumped humankind for millennia. In this book,

Structure Theorem for Multiple Addition and the Frobenius Problem

Abstract Let A ⊆[0;  l ] be a set of n integers, and let h ⩾2. By how much does | hA | exceed |( h −1)  A | ? How can one estimate | hA | in terms of n ,  l ? We give sharp lower bounds extending and

The Brun-Titchmarsh Theorem on average

Throughout this paper a denotes a fixed non-zero integer and the letter p with or without subscript denotes a prime variable. As usual, for (q, a) = 1 we write $$ \pi \left( {x;\,q,\,a} \right)\,

Fast construction of irreducible polynomials over finite fields

  • V. Shoup
  • Computer Science, Mathematics
    SODA '93
  • 1993
The main result of this paper a new algorithm for constructing an irreducible polynomial of specified degree n over a finite field Fq . The algorithm is probabilistic, and is asymptotically faster

Finding irreducible polynomials over finite fields

Irreducible polynomials in Fp[X] are used to carry out the arithmetic in field extension of Fp to solve the random polynomial time problem of finding irreducibles of any degree over Fp.

I and i

There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.

Introduction to Analytic and Probabilistic Number Theory

Foreword Notation Part I. Elementary Methods: Some tools from real analysis 1. Prime numbers 2. Arithmetic functions 3. Average orders 4. Sieve methods 5. Extremal orders 6. The method of van der

Smooth Orders and Cryptographic Applications

We obtain rigorous upper bounds on the number of primes p ? x for which p-1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for

The large sieve

In this paper, we propose an integer quadratic optimization model to determine the optimal decision for a supplier selection problem. The decision is the optimal product volume that has to be