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This paper extends Joux-Naccache-Thomé's e -th root algorithm to the static Diffie-Hellman problem (sdhp ). The new algorithm can be adapted to diverse finite fields by customizing it with an nfs -like core or an ffs -like core. In both cases, after a number of non-adaptive sdhp oracle queries, the attacker builds-up the ability to solve new sdhp instances unknown before the query phase . While sub-exponential, the algorithm is still significantly faster than all currently known dlp and sdhp resolution methods. We explore the applicability of the technique to various cryptosystems.The attacks were implemented in ${\mathbb F}_{2^{1025}}$ and also in ${\mathbb F}_{p}$, for a 516-bit p .