Euclid's Algorithm And The Lanczos Method Over Finite Fields

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Euclid's Algorithm And The Lanczos Method Over Finite Fields (1997)

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by Jeremy Teitelbaum

Venue:	Math. Comp
Citations:	6 - 0 self

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@ARTICLE{Teitelbaum97euclid'salgorithm,
    author = {Jeremy Teitelbaum},
    title = {Euclid's Algorithm And The Lanczos Method Over Finite Fields},
    journal = {Math. Comp},
    year = {1997},
    volume = {67},
    pages = {1665--1678}
}

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Abstract

. This paper shows that there is a close relationship between the Euclidean algorithm for polynomials and the Lanczos method for solving sparse linear systems, especially when working over finite fields. It uses this relationship to account rigorously for the appearance of self-orthogonal vectors arising in the course of the Lanczos algorithm. It presents an improved Lanczos method which overcomes problems with self-orthogonality and compares this improved algorithm with the Euclidean algorithm. Introduction The Lanczos method is an iterative algorithm for solving linear systems of the form Ax = b, where A is a square symmetric matrix and b is a known vector. In its original formulation, A was a matrix with real entries. Recently, however, the algorithm has been applied to solve the large, but very sparse, linear systems over finite fields which arise in the final stages of sieve methods for factoring integers and solving discrete logarithm problems. In this situation, the algorithm o...

Keyphrases

lanczos method finite field finite field lanczos method linear system euclidean algorithm close relationship form ax self-orthogonal vector known vector square symmetric matrix real entry lanczos algorithm sieve method improved algorithm sparse linear system iterative algorithm original formulation improved lanczos method final stage discrete logarithm problem

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