Paper 2020/1375

Semi-regular sequences and other random systems of equations

M. Bigdeli, E. De Negri, M. M. Dizdarevic, E. Gorla, R. Minko, and S. Tsakou

Abstract

The security of multivariate cryptosystems and digital signature schemes relies on the hardness of solving a system of polynomial equations over a finite field. Polynomial system solving is also currently a bottleneck of index-calculus algorithms to solve the elliptic and hyperelliptic curve discrete logarithm problem. The complexity of solving a system of polynomial equations is closely related to the cost of computing Gröbner bases, since computing the solutions of a polynomial system can be reduced to finding a lexicographic Gröbner basis for the ideal generated by the equations. Several algorithms for computing such bases exist: We consider those based on repeated Gaussian elimination of Macaulay matrices. In this paper, we analyze the case of random systems, where random systems means either semi-regular systems, or quadratic systems in $n$ variables which contain a regular sequence of $n$ polynomials. We provide explicit formulae for bounds on the solving degree of semi-regular systems with $m>n$ equations in $n$ variables, for equations of arbitrary degrees for $m=n+1$, and for any $m$ for systems of quadratic or cubic polynomials. In the appendix, we provide a table of bounds for the solving degree of semi-regular systems of $m=n+k$ quadratic equations in $n$ variables for $2\leq k,n\leq 100$ and online we provide the values of the bounds for $2\leq k,n\leq 500$. For quadratic systems which contain a regular sequence of $n$ polynomials, we argue that the Eisenbud-Green-Harris conjecture, if true, provides a sharp bound for their solving degree, which we compute explicitly.