Cryptology ePrint Archive: Report 2006/248

Disguising tori and elliptic curves

Steven D. Galbraith

Abstract: Frey proposed the idea of disguising' an elliptic curve. This is a method to obtain a black box' representation of a group. We adapt this notion to finite fields and tori and study the question of whether such systems are secure.

Our main result is an algebraic attack which shows that it is not secure to disguise the torus $T_2$. We also show that some methods for disguising an elliptic curve are not secure. Finally, we present a method to disguise an elliptic curve which seems to resist our algebraic attack.

Category / Keywords: public-key cryptography / Elliptic curves, algebraic tori, weil descent