Semi-Regular Sequences and Other Random Systems of Equations

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