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The complexity of solving a system of equations of the same degree

Giulia Gaggero, Elisa Gorla

TL;DR

The paper investigates the solvability of cryptographic polynomial systems where all equations share the same degree $D$ by deriving provable upper bounds on the degree of regularity $d_{reg}$, which in turn bound the solving degree for Gröbner-basis computations. It introduces the notion of systems regular in degree $D$ (top-degree part containing a regular sequence) and leverages the Eisenbud-Green-Harris conjecture to compare with $(D,\

Abstract

Many systems of interest in cryptography consist of equations of the same degree. Under the assumption that the degree of regularity is finite, we prove upper bounds on the degree of regularity of a system of equations of the same degree, with or without adding the field equations to the system. The bounds translate into upper bounds on the solving degree of the systems, and hence on the complexity of solving them via Gröbner bases methods. Our bounds depend on the number of equations in the system, the number of variables, and the degree of the equations.

The complexity of solving a system of equations of the same degree

TL;DR

The paper investigates the solvability of cryptographic polynomial systems where all equations share the same degree by deriving provable upper bounds on the degree of regularity , which in turn bound the solving degree for Gröbner-basis computations. It introduces the notion of systems regular in degree (top-degree part containing a regular sequence) and leverages the Eisenbud-Green-Harris conjecture to compare with $(D,\

Abstract

Many systems of interest in cryptography consist of equations of the same degree. Under the assumption that the degree of regularity is finite, we prove upper bounds on the degree of regularity of a system of equations of the same degree, with or without adding the field equations to the system. The bounds translate into upper bounds on the solving degree of the systems, and hence on the complexity of solving them via Gröbner bases methods. Our bounds depend on the number of equations in the system, the number of variables, and the degree of the equations.
Paper Structure (4 sections, 13 theorems, 59 equations)

This paper contains 4 sections, 13 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Systems that contain a regular sequence in a given degree
  4. Bounds on the degree of regularity

Key Result

Theorem 3

Let $\mathcal{F} = \{f_1, \ldots, f_m, x_{1}^{q}-x_{1}, \ldots , x_{n}^{q}-x_{n}\} \subseteq R$ be a polynomial system. If $d_{\mathop{\mathrm{reg}}\nolimits}(\mathcal{F}) \geq \max\{q, \deg(f_1),\ldots, \deg(f_m)\}$, then

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Theorem 3: T19 and ST21
  • Theorem 4: Sal23
  • Definition 5
  • Proposition 6
  • proof
  • Definition 7
  • Definition 8
  • Proposition 9
  • ...and 31 more