Inverting HFE Systems Is Quasi-Polynomial for All Fields

Abstract

Abstract. In this paper, we present and prove the first closed formula bounding the degree of regularity of an HFE system over an arbitrary finite field. Though these bounds are not necessarily optimal, they can be used to deduce if D, the degree of the corresponding HFE polynomial, and q, the size of the corresponding finite field, are fixed, inverting HFE system is polynomial for all fields; 2. if D is of the scale O(n α ) where n is the number of variables in an HFE system, and q is fixed, inverting HFE systems is quasipolynomial for all fields.We generalize and prove rigorously similar results by Granboulan, Joux and Stern in the case when q = 2 that were communicated at Crypto 2006.

Inverting square systems algebraically is exponential

“…This proves the conjecture in [8], though it does not answer the question about the cases other than Square systems. However the common senses tells us that the conjecture is very likely to be true for all generic HFE cases, since Square systems are the simplest among all.…”

“…Inspired by the work of [12], and using a similar idea to that used in [15] -roughly that one can bound the degree of regularity of a system by finding a bound for certain simpler subsystems, in [8], a new closed formula was found for the degree regularities for all HFE systems for any field. However this bound is derived using a very different approach.…”

“…However this bound is derived using a very different approach. Previously all estimates on the degree of regularity were based on a dimension counting argument, while the new approach constructively proves the upper bound of the degree of regularity as an explicit function of q and D. Such explicit formulas enable [8] to draw conclusions about the upper bound complexity of inverting the system using Gröbner basis methods.…”

“…The contribution of this paper. In the paper [8], a very strong conjecture was presented on the lower bound of the degree of regularity for the case of that q is odd and q is the size of Ω(n), which implies that to invert the related systems algebraically is actually exponential.…”

“…We will first introduce briefly HFE and Square cryptosystems in the section below. In Section 3, we review the definition and basic properties of the degree of regularity from [12] [8]. In Section 4, we will prove and main theorem that degree of regularity of Square systems is indeed q and derive that the complexity of the Gröbner basis attacks on Square systems is indeed exponential.…”