The Complexity of MinRank

Caminata, Alessio; Gorla, Elisa

doi:10.1007/978-3-030-77700-5_5

Cited by 2 publications

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“…We consider polynomial systems coming from the MinRank Problem. For more bounds on the complexity of the MinRank Problem, see [CG20].…”

Section: Solving Degree Of Ideals Of Minors and The Minrank Problemmentioning

confidence: 99%

Solving Multivariate Polynomial Systems and an Invariant from Commutative Algebra

Caminata¹,

Gorla²

2021

Arithmetic of Finite Fields

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The complexity of computing the solutions of a system of multivariate polynomial equations by means of Gröbner 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.

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“…We consider polynomial systems coming from the MinRank Problem. For more bounds on the complexity of the MinRank Problem, see [CG20].…”

Section: Solving Degree Of Ideals Of Minors and The Minrank Problemmentioning

confidence: 99%