# American Institute of Mathematical Sciences

August  2010, 4(3): 307-321. doi: 10.3934/amc.2010.4.307

## Invalid-curve attacks on (hyper)elliptic curve cryptosystems

 1 Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada, N2L 3G1, Canada 2 NTT Information Sharing Platform Laboratories, 3-9-11, Midori-cho Musashino-shi, Tokyo 180-8585, Japan

Received  May 2009 Revised  January 2010 Published  August 2010

We extend the notion of an invalid-curve attack from elliptic curves to genus 2 hyperelliptic curves. We also show that invalid singular (hyper)elliptic curves can be used in mounting invalid-curve attacks on (hyper)elliptic curve cryptosystems, and make quantitative estimates of the practicality of these attacks. We thereby show that proper key validation is necessary even in cryptosystems based on hyperelliptic curves. As a byproduct, we enumerate the isomorphism classes of genus g hyperelliptic curves over a finite field by a new counting argument that is simpler than the previous methods.
Citation: Koray Karabina, Berkant Ustaoglu. Invalid-curve attacks on (hyper)elliptic curve cryptosystems. Advances in Mathematics of Communications, 2010, 4 (3) : 307-321. doi: 10.3934/amc.2010.4.307
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