Adleman–Pomerance–Rumely primality test

Adleman–Pomerance–Rumely primality test
Name	Adleman–Pomerance–Rumely primality test
Authors	Leonard Adleman, Carl Pomerance, Robert Rumely
First published	1983
Type	Deterministic primality test
Complexity	subexponential (heuristic)
Related	Miller–Rabin test, AKS primality test, Elliptic curve primality proving

Adleman–Pomerance–Rumely primality test is a deterministic algorithm for proving primality of integers developed by Leonard Adleman, Carl Pomerance, and Robert Rumely in the early 1980s. The test builds on ideas from analytic number theory and algebraic number fields to replace probabilistic checks such as those used by Joseph Silverman, Gary Miller, and Michael Rabin with deterministic criteria inspired by work of John Selfridge, Hugh C. Williams, and D. H. Lehmer. It served as an important bridge between classical primality criteria associated with Évariste Galois, Adrien-Marie Legendre, and Carl Friedrich Gauss and later deterministic results such as the AKS primality test of Manindra Agrawal, Neeraj Kayal, and Nitin Saxena.

Introduction

The Adleman–Pomerance–Rumely primality test arises from a collaboration among Leonard Adleman, Carl Pomerance, and Robert Rumely and was published after prior contributions from Andrew Odlyzko, Stephen Bach, and Hendrik Lenstra. It provides a deterministic primality proof for integers by combining algebraic number theory from Richard Dedekind, David Hilbert, and Emil Artin with computational techniques influenced by Peter Shor, Michael Rabin, and John Pollard. The test is closely related to algorithms studied by Alan Turing, Srinivasa Ramanujan, and Édouard Lucas, and it influenced subsequent work at institutions like Bell Labs, AT&T, and the Massachusetts Institute of Technology.

Background and mathematical foundations

The foundations of the test draw on cyclotomic fields studied by Carl Friedrich Gauss, Niels Henrik Abel, and Ernst Kummer, and on reciprocity laws formalized by Emil Artin and Helmut Hasse. It uses properties of multiplicative order and character sums developed in the tradition of Bernhard Riemann, Peter Gustav Lejeune Dirichlet, and André Weil. Core inputs include group-theoretic concepts from Évariste Galois, representation theory connected to Hermann Weyl, and algebraic integers treated by David Hilbert and Emmy Noether. Analytic estimates employ bounds reminiscent of those used by G. H. Hardy, John Littlewood, and Atle Selberg, while computational number theory tools reflect the influence of Carl Gauss, Leopold Kronecker, and Kurt Gödel.

Algorithm description

The algorithm constructs explicit tests using cyclotomic extensions and computes residue class orders in manner akin to algorithms by Édouard Lucas and Derrick Lehmer, while harnessing computational techniques associated with John von Neumann, Donald Knuth, and Richard Brent. It iteratively evaluates congruences and norm conditions in rings of integers of number fields related to Kummer extensions studied by Ernst Kummer and Heinrich Weber. The procedure verifies multiplicative order relations similar to those used by Gary Miller and Michael Rabin but replaces randomized bases with deterministic choices informed by works of Manuel Blum, Silvio Micali, and Adi Shamir. Subroutines depend on polynomial arithmetic and fast Fourier ideas connected to James Cooley, John Tukey, and Norbert Wiener.

Complexity and performance

The original complexity analysis by Adleman, Pomerance, and Rumely produced subexponential time bounds reminiscent of results by Carl Pomerance, Andrew Odlyzko, and Hugh C. Williams. Heuristic improvements and average-case analyses reference methods from Alan T. G. Crawford, Lenstra, and Bach, while concrete performance comparisons invoke practical implementations influenced by Peter Montgomery, Richard Brent, and Daniel Bernstein. The test is faster than trial division and deterministic versions of the Miller test for large classes of inputs but is outperformed in asymptotic simplicity by the AKS algorithm of Agrawal, Kayal, and Saxena and in practical speed by elliptic curve primality proving developed by Atkin and Morain.

Variants and improvements

Subsequent refinements by Henri Cohen, Hendrik Lenstra, and Andrew Granville adapted the test to specific families of integers and exploited heuristics similar to those used by Carl Pomerance and Andrew Odlyzko. Improvements integrated fast integer arithmetic techniques from Peter Montgomery, Alexander Schönhage, and Volker Strassen and leveraged sieving strategies influenced by Daniel Goldston and János Pintz. Connections to elliptic curve methods developed by John Cremona, Neal Koblitz, and Andrew Wiles created hybrid approaches that combine cyclotomic checks with elliptic certificates inspired by Atkin, Morain, and Benedict Gross.

Implementation and usage

Implementations of the Adleman–Pomerance–Rumely test have appeared in software libraries influenced by Richard Brent, Eric Bach, and Paul Zimmermann, and in computer algebra systems such as GNU, PARI/GP associated with Henri Cohen, and MAGMA associated with John Cannon. Practical usage was prominent at research centers like Bell Labs, IBM, and Microsoft Research before the rise of AKS and elliptic curve methods; applied projects at the National Institute of Standards and Technology and projects involving RSA Laboratories used related deterministic techniques. Implementers must handle algebraic number field arithmetic and rely on libraries developed by Tim Peters, William Stein, and Robert Beeler.

Historical impact and legacy

The Adleman–Pomerance–Rumely primality test influenced theoretical and practical developments in computational number theory, connecting classical figures like Carl Friedrich Gauss and Évariste Galois to modern researchers such as Manindra Agrawal, Neeraj Kayal, Nitin Saxena, and Hendrik Lenstra. It informed subsequent deterministic proofs and inspired work at academic institutions including Princeton University, Harvard University, Stanford University, and the University of Cambridge. The algorithm's legacy persists in ongoing research by Andrew Granville, Peter Sarnak, and Kannan Soundararajan and in modern cryptographic practice related to Whitfield Diffie, Ronald Rivest, Adi Shamir, and Leonard Adleman.

Category:Primality tests