## Cryptology ePrint Archive: Report 2004/118

Fast addition on non-hyperelliptic genus $3$ curves

Stéphane Flon and Roger Oyono and Christophe Ritzenthaler

Abstract: We present a fast addition algorithm in the Jacobian of a genus $3$ non-hyperelliptic curve over a field of any characteristic. When the curve has a rational flex and $\textrm{char}(k) > 5$, the computational cost for addition is $148M+15SQ+2I$ and $165M+20SQ+2I$ for doubling. An appendix focuses on the computation of flexes in all characteristics. For large odd $q$, we also show that the set of rational points of a non-hyperelliptic curve of genus $3$ can not be an arc.

Category / Keywords: public-key cryptography / Jacobians, non-hyperelliptic curves, algebraic curves cryptography, discrete logarithm problem