A generalisation of Miller's algorithm and applications to pairing computations on abelian varieties. (March 2015)

• Record Type: Journal Article
• Authors: Lubicz, David; Robert, Damien
• Is Part Of: Journal of symbolic computation. Volume 67(2015)
• Page Start: 68
• Page End: 92
• Publication Date: 2015-03
• Subjects: Weil pairing -- Tate pairing -- Pairing computations -- Abelian varieties -- Miller's algrorithm; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2014.08.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this paper, we use the theory of theta functions to generalise to all abelian varieties the usual Miller's algorithm to compute a function associated with a principal divisor. We also explain how to use the Frobenius morphism on abelian varieties defined over a finite field in order to shorten the loop of the Weil and Tate pairings algorithms. This extends preceding results about ate and twisted ate pairings to all abelian varieties. Then building upon the two preceding ingredients, we obtain a variant of optimal pairings on abelian varieties. Finally, by introducing new addition formulas, we explain how to compute optimal pairings on Kummer varieties. We compare in term of performance the resulting algorithms to the algorithms already known in the genus one and two case.