Refinements of Miller's algorithm for computing the Weil/Tate pairing

Authors:
Ian F. Blake;V. Kumar Murty;Guangwu Xu
Affiliations:
Department of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario, Canada, M5S 3G4;Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3;Ganita Lab, University of Toronto at Mississauga, Mississauga, Ontario, Canada, L5L 1C6
Venue:
Journal of Algorithms
Year:
2006

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Abstract

The efficient computation of the Weil and Tate pairings is of significant interest in the implementation of certain recently developed cryptographic protocols. The standard method of such computations has been the Miller algorithm. Three refinements to Miller's algorithm are given in this work. The first refinement is an overall improvement. If the binary expansion of the involved integer has relatively high Hamming weight, the second improvement suggested shows significant gains. The third improvement is especially efficient when the underlying elliptic curve is over a finite field of characteristic three, which is a case of particular cryptographic interest. Comment on the performance analysis and characteristics of the refinements are given.