Section: New Results

Discrete logarithms

Participant : Andreas Enge.

In [10] , we presented for the first time an algorithm for the discrete logarithm problem in certain algebraic curves that runs in subexponential time less than $L(1/2)$, namely, $L(1/3+\epsilon )$ for any $\epsilon >0$. In [13] , we lower this complexity to $L(1/3)$, showing that the corresponding algebraic curves (essentially $C_{ab}$ curves of genus $g$ growing at least quadratically with the logarithmic size of the finite field of definition, $logq$) result in cryptosystems that are as easily attacked as RSA or tradtional cryptosystems based on discrete logarithms in finite fields. We provide a complete classification of all the curves to which the attack applies.