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International audienceIn this paper, we study the application of the function field sievealgorithm for computing discrete logarithms over finite fields ofthe form $\GF{q^n}$ when $q$ is a medium-sized prime power. Thisapproach is an alternative to a recent paper of Granger andVercauteren for computing discrete logarithms in tori, usingefficient torus representations. We show that when $q$ is not toolarge, a very efficient $L(1/3)$ variation of the function fieldsieve can be used. Surprisingly, using this algorithm, discretelogarithms computations over some of these fields are even easierthan computations in the prime field and characteristic two fieldcases.  We also show that this new algorithm has securityimplications on some existing cryptosystems, such as torus basedcryptography in $T_{30}$, short signature schemes in characteristic3 and cryptosystems based on supersingular abelian varieties. On theother hand, cryptosystems involving larger basefields and smallerextension degrees, typically of degree at most 6, such as LUC, XTRor $T_6$ torus cryptography, are not affected

The Function Field Sieve in the Medium Prime Case