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Mathematics of Computation

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Modular exponentiation via the explicit Chinese remainder theorem


Authors: Daniel J. Bernstein and Jonathan P. Sorenson
Journal: Math. Comp. 76 (2007), 443-454
MSC (2000): Primary 11Y16; Secondary 68W10
Published electronically: September 14, 2006
MathSciNet review: 2261030
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Abstract: Fix pairwise coprime positive integers $ p_1,p_2,\dots,p_s$. We propose representing integers $ u$ modulo $ m$, where $ m$ is any positive integer up to roughly $ \sqrt{p_1p_2\cdots p_s}$, as vectors $ (u\bmod p_1,u\bmod p_2,\dots,u\bmod p_s)$. We use this representation to obtain a new result on the parallel complexity of modular exponentiation: there is an algorithm for the Common CRCW PRAM that, given positive integers $ x$, $ e$, and $ m$ in binary, of total bit length $ n$, computes $ x^e\bmod m$ in time $ O(n/{\lg\lg n})$ using $ n^{O(1)}$ processors. For comparison, a parallelization of the standard binary algorithm takes superlinear time; Adleman and Kompella gave an $ O((\lg n)^3)$ expected time algorithm using $ \exp( O(\sqrt{n\lg n}))$ processors; von zur Gathen gave an NC algorithm for the highly special case that $ m$ is polynomially smooth.


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Additional Information

Daniel J. Bernstein
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), The University of Illinois at Chicago, Chicago, IL 60607–7045
Email: djb@cr.yp.to

Jonathan P. Sorenson
Affiliation: Department of Computer Science and Software Engineering, Butler University, 4600 Sunset Avenue, Indianapolis, Indiana 46208
Email: sorenson@butler.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-06-01849-7
Received by editor(s): August 18, 2003
Received by editor(s) in revised form: June 15, 2005
Published electronically: September 14, 2006
Additional Notes: This paper combines and improves the preliminary papers \cite{1995/bernstein-mmecrt} by Bernstein and \cite{1999/sorenson} by Sorenson. Bernstein was supported by the National Science Foundation under grants DMS-9600083 and DMS–9970409. Sorenson was supported by the National Science Foundation under grant CCR–9626877. Sorenson completed part of this work while on sabbatical at Purdue University in Fall 1998.
Article copyright: © Copyright 2006 by the authors