His own use of it was inimitable; in terms of later techniques, it is recognised as a prototype of the large sieve method in its application of bilinear forms, and also as an exploitation of combinatorial structure.
12.
He also invented the block Lanczos algorithm for finding nullspace of a matrix over a finite field, which is very widely used for the quadratic sieve and number field sieve methods of factorization; he has been involved in the computations which set a number of integer factorization records.
13.
He is best known for Montgomery's pair correlation conjecture, his development of the large sieve methods and for co-authoring ( with Ivan M . Niven and Herbert Zuckerman ) one of the standard introductory number theory texts, " An Introduction to the Theory of Numbers ", now in its fifth edition ( ISBN 0471625469 ).
14.
A result of S . D . Cohen, based on the large sieve method, extends this result, counting points by height function and showing, in a strong sense, that a thin set contains a low proportion of them ( this is discussed at length in Serre's " Lectures on the Mordell-Weil theorem " ).
15.
His algorithms include : Baby-step giant-step algorithm for computing the discrete logarithm, which is useful in public-key cryptography; Shanks'square forms factorization, an integer factorization method that generalizes Fermat's factorization method; and the Tonelli Shanks algorithm that finds square roots moduli a prime, which is useful for the quadratic sieve method of integer factorization.
