Efficiently computable distortion maps for supersingular curves

Authors:
Katsuyuki Takashima
Affiliations:
Information Technology R&D Center, Mitsubishi Electric Corporation, Kamakura, Kanagawa, Japan
Venue:
ANTS-VIII'08 Proceedings of the 8th international conference on Algorithmic number theory
Year:
2008

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Abstract

Efficiently computable distortion maps are useful in cryptography.Galbraith-Pujolàs-Ritzenthaler-Smith [6] considered them forsupersingular curves of genus 2. They showed that there exists a distortionmap in a specific set of efficiently computable endomorphisms forevery pair of nontrivial divisors under some unproven assumptions fortwo types of curves. In this paper, we prove that this result holds usinga different method without these assumptions for both curves with r 5and r 19 respectively, where r is the prime order of the divisors. Inother words, we solve an open problem in [6]. Moreover, we successfullygeneralize this result to the case C : Y2 = X2g+1 + 1 over Fp for any gs.t. 2g+1 is prime. In addition, we provide explicit bases of JacC[r] witha new property that seems interesting from the cryptographic viewpoint.