Paper 2021/1670

The complexity of solving Weil restriction systems

Alessio Caminata, Michela Ceria, and Elisa Gorla


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 $\mathcal{F}$.

Available format(s)
Public-key cryptography
Publication info
Preprint. MINOR revision.
Weil restrictionsolving degreedegree of regularityGroebner basis
Contact author(s)
caminata @ dima unige it
2021-12-21: received
Short URL
Creative Commons Attribution


      author = {Alessio Caminata and Michela Ceria and Elisa Gorla},
      title = {The complexity of solving Weil restriction systems},
      howpublished = {Cryptology ePrint Archive, Paper 2021/1670},
      year = {2021},
      note = {\url{}},
      url = {}
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