On the last fall degree of zero-dimensional Weil descent systems

In this article we will discuss a new, mostly theoretical, method for solving (zero-dimensional) polynomial systems, which lies in between Gr\"obner basis computations and the heuristic first fall degree assumption and is not based on any heuristic. This method relies on the new concept of last fall degree. Let $k$ be a finite field of cardinality $q^n$ and let $k'$ be its subfield of cardinality $q$. Let $\mathcal{F} \subset k[X_0,\ldots,X_{m-1}]$ be a finite subset generating a zero-dimensional ideal. We give an upper bound of the last fall degree of the Weil descent system of $\mathcal{F}$, which depends on $q$, $m$, the last fall degree of $\mathcal{F}$, the degree of $\mathcal{F}$ and the number of solutions of $\mathcal{F}$, but not on $n$. This shows that such Weil descent systems can be solved efficiently if $n$ grows. In particular, we apply these results for multi-HFE and essentially show that multi-HFE is insecure. Finally, we discuss that the degree of regularity (or last fall degree) of Weil descent systems coming from summation polynomials to solve the elliptic curve discrete logarithm problem might depend on $n$, since such systems without field equations are not zero-dimensional.