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Compact Representations: Applications and Recent Results

Banff International Research Station for Mathematical Innovation and Discovery

Compact representations are explicit representations of algebraic numbers or functions, with size polynomial in the logarithm of their height or, respectively, degree. These representations enable much more efficient manipulations of large algebraic numbers or functions than would be possible using a standard representation, and have proved to be useful in a variety of applications. In this talk, we will describe two such applications - how compact representations are essential for short certificates of the unit group and ideal class group of a number field, and how they can be used to speed the resolution of certain Diophantine equations. We will also present recent improvements that reduce the size of compact representations, efforts to generalize these to hyperelliptic function fields, and applications of the latter to speeding the computation of bilinear pairings.

Mathematics; Number theory; Group theory and generalizations

video/mp4

Author affiliation: University of Calgary

2017-02-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59464

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/