## Inverting HFE Systems is Quasipolynomial for all Fields

**Jintai Ding and Timothy Hodges**

*
South China University of Technology, Guangzhou, China +
Department of Mathematical Sciences, University of Cincinnati, Ohio, USA; and
Department of Mathematical Sciences, University of Cincinnati, Ohio, USA*
**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;
- 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 quasi-polynomial
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.