On Polynomial Systems Arising from a Weil Descent

In the last two decades, many computational problems arising in cryptography have been successfully reduced to various systems of polynomial equations. In this paper, we revisit a class of polynomial systems introduced by Faugere, Perret, Petit and Renault. Based on new experimental results and heuristic evidence, we conjecture that their degrees of regularity are only slightly larger than the original degrees of the equations, resulting in a very low complexity compared to generic systems. We then revisit the application of these systems to the elliptic curve discrete logarithm problem (ECDLP) for binary curves. Our heuristic analysis suggests that an index calculus variant due to Diem requires a subexponential number of bit operations $(O2^{c\,n^{2/3}\log n})$ over the binary field ${\mathbb F}{2^n}$, where c is a constant smaller than 2. According to our estimations, generic discrete logarithm methods are outperformed for any n>N where N≈2000, but elliptic curves of currently recommended key sizes (n≈160) are not immediately threatened. The analysis can be easily generalized to other extension fields.