Paper 2011/570

Degree of regularity for HFE-

Jintai Ding and Thorsten Kleinjung

Abstract

In this paper, we prove a closed formula for the degree of regularity of the family of HFE- (HFE Minus) multivariate public key cryptosystems over a finite field of size $q$. The degree of regularity of the polynomial system derived from an HFE- system is less than or equal to \begin{eqnarray*} \frac{(q-1)(\lfloor \log_q(D-1)\rfloor +a)}2 +2 & & \text{if $q$ is even and $r+a$ is odd,} \\ \frac{(q-1)(\lfloor \log_q(D-1)\rfloor+a+1)}2 +2 & & \text{otherwise.} \end{eqnarray*} Here $q$ is the base field size, $D$ the degree of the HFE polynomial, $r=\lfloor \log_q(D-1)\rfloor +1$ and $a$ is the number of removed equations (Minus number). This allows us to present an estimate of the complexity of breaking the HFE Challenge 2: \vskip .1in \begin{itemize} \item the complexity to break the HFE Challenge 2 directly using algebraic solvers is about $2^{96}$. \end{itemize}