Asymptotic Behaviour of the Index of Regularity of Quadratic Semi-Regular Polynomial Systems

We compute the asymptotic expansion of the index of regularity for overdetermined quadratic semi-regular sequences of algebraic equations. This implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] family of algorithms used by the cryptographic community. 1 Motivations and Results The worst-case complexity of Gröbner bases has been the object of extensive studies. In the most general case, it is well known after work by Mayr and Meyer that the complexity is doubly exponential in the number of variables. For subclasses of polynomial systems, the complexity may be much smaller. Of particular importance is the class of regular sequences of polynomials. There, it is known that after a generic linear change of variables the complexity of the computation for the degree-reverse-lexicographic order is simply exponential in the number of variables. Moreover, in characteristic 0, these systems are generic. Our goal is to give similar complexity bounds for overdetermined systems, for a class of systems that we call semi-regular. The interest in overdetermined systems is not purely academic: there are a number of applications, such as error correcting codes (decoding of cyclic codes), robotics, calibration, cryptography,. . . . The security of many cryptographical primitives depends on the difficulty of system-solving. Sometimes (in the case of “multivariate public-key cryptosystems”) the public keys themselves become the system to be solved. Sometimes primitives can be cracked if one can find a solution to an associated overdetermined system of algebraic equations over a finite field. This is known as Algebraic Cryptanalysis and is currently one of the hot topics in cryptography. In most cases, only the solutions over a finite field are required, rather than solutions in the algebraic closure. Often the finite field is F2, and we may then think of the problem as LIP6, 8 rue du Capitaine Scott, F-75015 PARIS, {Magali.Bardet,Jean-Charles.Faugere}@lip6.fr INRIA Rocquencourt Bat. 9, Domaine de Voluceau, BP 105, F-78153 Le Chesnay Cedex, Bruno.Salvy@inria.fr Mathematics Department, Tamkang University, Tamsui, Taiwan 251-37, by@moscito.org