## Cryptology ePrint Archive: Report 2020/1375

Semi-regular sequences and other random systems of equations

M. Bigdeli and E. De Negri and M. M. Dizdarevic and E. Gorla and 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 Gr{\"o}bner bases, since computing the solutions of a polynomial system can be reduced to finding a lexicographic Gr{\"o}bner 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.

Category / Keywords: public-key cryptography / Groebner basis, multivariate cryptography, solving degree, random system, semiregular sequence

Original Publication (in the same form): Women in Numbers Europe III: Research Directions in Number Theory A. Cojocaru, S. Ionica and E. Lorenzo Garcia Eds., Springer