## Section: New Results

### Discrete logarithms

Participant : Andreas Enge.

In [34] , 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 + )
for any >0 . In [27] ,
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. The article has been accepted by
*Journal of Cryptology* .