The degree of all these functions is known, and they have been proven to reach the optimal algebraic immunity, but still very few is known about its fast algebraic immunity. For a function in $n=2^m+j$ variables, an upper bound is known for all $m$ and $j$, proving that these functions do not reach the optimal fast algebraic immunity. However, the exact fast algebraic immunity is known only for very few families indexed by $j$, where the parameter is exhibited for all members of the family since $m$ is big enough. Recent works gave exact values for $j=0$ and $j=1$ (in the first case), and for $j=2$ and $j=3$ with $m\geq2$ (in the second case). In this work, we determine the exact fast algebraic immunity for all possible values of $j$, for all member of the family assuming $m\geq 1+\log_2(j+1)$.
Category / Keywords: secret-key cryptography / Boolean Functions, Fast Algebraic Attacks, Symmetric Functions, Majority Functions Original Publication (with minor differences): Latincrypt 2019 Date: received 3 Sep 2019 Contact author: pierrick meaux at uclouvain be Available format(s): PDF | BibTeX Citation Version: 20190905:072632 (All versions of this report) Short URL: ia.cr/2019/999