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A Binary Algorithm for the Jacobi Symbol (1993)
| Venue: | ACM SIGSAM Bulletin |
| Citations: | 6 - 1 self |
BibTeX
@ARTICLE{Shallit93abinary,
author = {Jeffrey Shallit and Jonathan Sorenson},
title = {A Binary Algorithm for the Jacobi Symbol},
journal = {ACM SIGSAM Bulletin},
year = {1993},
volume = {27},
pages = {4--11}
}
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Abstract
We present a new algorithm to compute the Jacobi symbol, based on Stein's binary algorithm for the greatest common divisor, and we determine the worst-case behavior of this algorithm. Our implementation of the algorithm runs approximately 7--25% faster than traditional methods on inputs of size 100--1000 decimal digits. 1 Introduction Efficient computation of the Jacobi symbol \Gamma a n \Delta is an important component of the Monte Carlo primality test of Solovay and Strassen [9]. Algorithms for computing the Jacobi symbol can also be found on symbolic algebra systems such as Mathematica and Maple. Several efficient algorithms modeled on Euclid's algorithm for computing the greatest common divisor (gcd) have been proposed and analyzed; see, for example, [12, 3, 8]. Indeed, it is possible to compute \Gamma a n \Delta in O((log a)(log n)) bit operations using the "naive arithmetic" model. Using Schonhage's result [7], it is possible (see [1]) to compute \Gamma a n \Delta (...
Keyphrases
jacobi symbol binary algorithm common divisor new algorithm symbolic algebra system several efficient algorithm monte carlo primality test introduction efficient computation bit operation decimal digit important component traditional method naive arithmetic model jacobi symbol gamma worst-case behavior
