The GAC Property of a Class of 1‐Resilient Functions with High Nonlinearity

SAO 1-Resilient Functions With Lower Absolute Indicator in Even Variables

“…Secondly, we obtain two new classes of 1-resilient functions having very high nonlinearity and very low absolute indicator, from bent functions and from plateaued functions, respectively. Moreover, we prove that our 1-resilient functions from bent functions possess the currently highest known nonlinearity 2 n−1 − 2 n/2−1 − 2 n/4 and possess the currently known lowest absolute indicator 2 n/2 + 2 n/4+1 simultaneously, which breaks the previously best upper bound of the minimum absolute indicator of 1-resilient functions given by Ge et al in [10], and allows us to give another new smaller upper bound for the minimum absolute indicator of 1-resilient functions.…”

“…Remark 4. Note that the currently known highest nonlinearity of 1-resilient functions is 2 n−1 − 2 n/2−1 − 2 n/4 [30] and the previously best known upper bound of the minimum absolute indicator of 1-resilient functions is 5 • 2 n/2 − 2 n/4+2 + 4 [10]. Theorem 2 provides a class of 1-resilient functions with the currently highest nonlinearity and the currently lowest absolute indicator simultaneously.…”

“…In [10], the authors gave a table to compare the absolute indicator of their 1-resilient functions with [17]. In the following table, we compare our result with theirs.…”

“…Many works on resilient functions are devoted to estimating the nonlinearity or other cryptographic criteria of resilient functions, but seldom considering their absolute indicators (see [4], [5], [16], [23], [27]- [30] and the references therein). Until now, there are only a few works (see [10], [17]) on this topic and the best known upper bound of the minimum absolute indicator of 1-resilient functions on n-variables (n even) is 5 • 2 n/2 − 2 n/4+2 + 4, which was obtained by Ge et al [10] for the calculation of the absolute indicator of 1-resilient functions designed by Zhang et al in [30], and it turned out that those 1-resilient functions possess the currently highest nonlinearity 2 n−1 − 2 n/2−1 − 2 n/4 and lowest absolute indicator 5 • 2 n/2 − 2 n/4+2 + 4.…”

“…Many papers of constructing m-resilient functions are mainly focused on the discussions of the nonlinearity and other cryptographic criteria of resilient functions, but seldom consider their absolute indicators, mostly because the analysis of which is rather complicated. Until now, the best upper bound of the minimum absolute indicator of 1-resilient functions in n (n even) variables is 5 • 2 n/2 − 2 n/4+2 + 4, see [10,Theorem 2], which was obtained by computing the absolute indicator of 1-resilient functions obtained by Zhang et al in [30], and it turned out that the absolute indicator of which is smaller than that of derived by Maitra and Pasalic in [17]. In this section, we will break this limitation and give an even smaller upper bound, by constructing another new class of 1-resilient Boolean functions.…”