Semiregular sequences and other random system of equations

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 Gr ̈obner bases, since computing the solutions of a polynomial system can be reduced to finding a lexicographic Gr ̈obner 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 ≤ k,n ≤ 100 and online we provide the values of the bounds for 2 ≤ k,n ≤ 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.