**The complexity of solving Weil restriction systems**

*Alessio Caminata and Michela Ceria and Elisa Gorla*

**Abstract: **The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Groebner basis methods. In this paper, we consider polynomial systems that are obtained via Weil restriction of scalars. The latter is an arithmetic construction which, given a finite Galois field extension $k\hookrightarrow K$, associates to a system $\mathcal{F}$ defined over $K$ a system $\mathrm{Weil}(\mathcal{F})$ defined over $k$, in such a way that the solutions of $\mathcal{F}$ over $K$ and those of $\mathrm{Weil}(\mathcal{F})$ over $k$ are in natural bijection.
In this paper, we find upper bounds for the complexity of solving a polynomial system $\mathrm{Weil}(\mathcal{F})$ obtained via Weil restriction in terms of algebraic invariants of the system $\mathcal{F}$.

**Category / Keywords: **public-key cryptography / Weil restriction, solving degree, degree of regularity, Groebner basis

**Date: **received 20 Dec 2021

**Contact author: **caminata at dima unige it

**Available format(s): **PDF | BibTeX Citation

**Version: **20211221:122740 (All versions of this report)

**Short URL: **ia.cr/2021/1670

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