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Incomplete exponential sums and Diffie–Hellman triples

Published online by Cambridge University Press:  22 February 2006

WILLIAM D. BANKS
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A. e-mail: bbanks@math.missouri.edu
JOHN B. FRIEDLANDER
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada. e-mail: frdlndr@math.toronto.edu
SERGEI V. KONYAGIN
Affiliation:
Department of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia. e-mail: konyagin@ok.ru
IGOR E. SHPARLINSKI
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia. e-mail: igor@ics.mq.edu.au

Abstract

Let $p$ be a prime and $\vartheta$ an integer of order $t$ in the multiplicative group modulo $p$. In this paper, we continue the study of the distribution of Diffie–Hellman triples$(\vartheta^x, \vartheta^y, \vartheta^{xy})$ by considering the closely related problem of estimating exponential sums formed from linear combinations of the entries in such triples. We show that the techniques developed earlier for complete sums can be combined, modified and developed further to treat incomplete sums as well. Our bounds imply uniformity of distribution results for Diffie–Hellman triples as the pair $(x,y)$ varies over small boxes.

Type
Research Article
Copyright
2006 Cambridge Philosophical Society

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