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Constructing abelian surfaces for cryptography via Rosenhain invariants

Part of: Finite fields and commutative rings (number-theoretic aspects) Computational number theory Algebraic number theory: global fields Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 August 2014

Craig Costello ,
Alyson Deines-Schartz ,
Kristin Lauter  and
Tonghai Yang
Craig Costello
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA email [email protected]
Alyson Deines-Schartz
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195, USA email [email protected]
Kristin Lauter
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA email [email protected]
Tonghai Yang
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA email [email protected]

Abstract

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This paper presents an algorithm to construct cryptographically strong genus $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ curves and their Kummer surfaces via Rosenhain invariants and related Kummer parameters. The most common version of the complex multiplication (CM) algorithm for constructing cryptographic curves in genus 2 relies on the well-studied Igusa invariants and Mestre’s algorithm for reconstructing the curve. On the other hand, the Rosenhain invariants typically have much smaller height, so computing them requires less precision, and in addition, the Rosenhain model for the curve can be written down directly given the Rosenhain invariants. Similarly, the parameters for a Kummer surface can be expressed directly in terms of rational functions of theta constants. CM-values of these functions are algebraic numbers, and when computed to high enough precision, LLL can recognize their minimal polynomials. Motivated by fast cryptography on Kummer surfaces, we investigate a variant of the CM method for computing cryptographically strong Rosenhain models of curves (as well as their associated Kummer surfaces) and use it to generate several example curves at different security levels that are suitable for use in cryptography.

MSC classification

Secondary: 11G10: Abelian varieties of dimension $> 1$ 11R37: Class field theory 11G15: Complex multiplication and moduli of abelian varieties 11G20: Curves over finite and local fields 11T71: Algebraic coding theory; cryptography 11Y16: Algorithms; complexity
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Special Issue A: Algorithmic Number Theory Symposium XI , 2014 , pp. 157 - 180
Copyright
© The Author(s) 2014 

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