**Solving multivariate polynomial systems and an invariant from commutative algebra**

*Alessio Caminata and Elisa Gorla*

**Abstract: **The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree.
In this paper, we discuss how to rigorously estimate the solving degree of a system, focusing on systems arising within public-key cryptography. In particular, we show that it is upper bounded by, and often equal to, the Castelnuovo-Mumford regularity of the ideal generated by the homogenization of the equations of the system, or by the equations themselves in case they are homogeneous. We discuss the underlying commutative algebra and clarify under which assumptions the commonly used results hold. In particular, we discuss the assumption of being in generic coordinates (often required for bounds obtained following this type of approach) and prove that systems that contain the field equations or their fake Weil descent are in generic coordinates. We also compare the notion of solving degree with that of degree of regularity, which is commonly used in the literature. We complement the paper with some examples of bounds obtained following the strategy that we describe.

**Category / Keywords: **solving degree; degree of regularity; Castelnuovo-Mumford regularity; Groebner basis; multivariate cryptography; post-quantum cryptography

**Original Publication**** (with minor differences): **Lecture Notes in Computer Science, 2021, 12542 LNCS, pp. 3-36
**DOI: **10.1007/978-3-030-68869-1_1

**Date: **received 20 Jun 2017, last revised 16 Jul 2021

**Contact author: **alessiocaminata87 at gmail com, caminata at dima unige it

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

**Note: **Final version. Theorem numbering adjusted to match the published version.

**Version: **20210716:145742 (All versions of this report)

**Short URL: **ia.cr/2017/593

[ Cryptology ePrint archive ]