## Section: New Results

### New results for solving the discrete logarithm problem

Recent results of R. Barbulescu, P. Gaudry, A. Joux, and E. Thomé seem to indicate that solving the discrete logarithm problem over finite fields of small characteristic is easier than was precedently thought. F. Morain and A. Guillevic, joined by R. Barbulescu and P. Gaudry, embarked on an attempt to assess the security of the discrete logarithm problem in a closely related context: that of finite fields with large characteristic and small degree. Improving on the methods of A. Joux, R. Lercier and others, they found new algorithms to select polynomials for the Number Field Sieve – the algorithm of choice in this setting. Moreover, a clever study of the algebraic properties of the fields used (e.g., algebraic units), enabled them to break the world record for the case of $GF\left({p}^{2}\right)$, soon to be followed by new cases. This work is described in [31] , and part of it is currently submitted.