I am looking for the best know attack against HFE cryptosystem. Reading this paper DEGREE OF REGULARITY FOR HFE I found the next claim
However, Faugere and Joux demonstrated that we can solve and break these systems easily in the case when $q = 2$ and $D$ small [15] using the Grobner basis algorithm F4. Furthermore their experimental results imply that such algorithms should finish at a degree of order $log_q (D)$, such that the highest degree polynomials we need to process are of a degree of order $log_q(D)$. Therefore they conclude that the complexity of the algorithm is roughly $O(n^{log_q(D)})$.
But, using the first HFE challenge, i.e. ($n = 80$, $D = 96$) and $q=2^n$, and replace these in that formula I get $80^{log_{2^{80}}(96)}$ that is < $2^{7*0.08}$. I thinking that I make mistake replacing the values of the first HFE challenge, or maybe I am interpreting wrong the claim. Please Could you help?
