Author(s)： Jintai Ding and Thorsten Kleinjung

J. Math-for-Ind. 4B (2012) 97-104.

• File： JMI2012B-3.pdf (141KB)

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
\[
\begin{array}{rl}
\dfrac{(q-1)(\lfloor \log_q(D-1)\rfloor +a)}2 +2 &
\text{if $q$ is even and $r+a$ is odd,}
\\
\dfrac{(q-1)(\lfloor \log_q(D-1)\rfloor+a+1)}2 +2 &
\text{otherwise.}
\end{array}
\]
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:
$\bullet\ $ the complexity to break the HFE Challenge 2 directly using algebraic solvers is about $2^{97}$.

Keyword(s).　 HFE, degree of regularity, Minus