The Function Field Sieve in the Medium Prime Case
Résumé
In this paper, we study the application of the function field sieve
algorithm for computing discrete logarithms over finite fields of
the form $\GF{q^n}$ when $q$ is a medium-sized prime power. This
approach is an alternative to a recent paper of Granger and
Vercauteren for computing discrete logarithms in tori, using
efficient torus representations. We show that when $q$ is not too
large, a very efficient $L(1/3)$ variation of the function field
sieve can be used. Surprisingly, using this algorithm, discrete
logarithms computations over some of these fields are even easier
than computations in the prime field and characteristic two field
cases. We also show that this new algorithm has security
implications on some existing cryptosystems, such as torus based
cryptography in $T_{30}$, short signature schemes in characteristic
3 and cryptosystems based on supersingular abelian varieties. On the
other hand, cryptosystems involving larger basefields and smaller
extension degrees, typically of degree at most 6, such as LUC, XTR
or $T_6$ torus cryptography, are not affected.