SAO 1-Resilient Functions With Lower Absolute Indicator in Even Variables
Yanjun Li, Haibin Kan, Jie Peng, Chik How Tan · IEEE Access · 2020
In 2018, Tang and Maitra presented a class of balanced Boolean functions in$n$variables with the absolute indicator$\Delta _{f} 2^{n-1}-2^{n/2}$, that is,$f$is SAO (strictly almost optimal), for$n=2k\equiv 2\,\,({\mathrm {mod}\,\,}4)$and$n\geq 46$in [IEEE Ttans. Inf. Theory 64(1): 393-402, 2018]. However, there is no evidence to show that the absolute indicator of any 1-resilient function in$n$variables can be strictly less than$2^{\lfloor ({n+1})/{2}\rfloor }$, and the previously best known upper bound of which is$5\cdot 2^{n/2}-2^{n/4+2}+4$. In this paper, we concentrate on two directions. Firstly, to complete Tang and Maitra’s work for$k$being even, we present another class of balanced functions in$n$variables with the absolute indicator$\Delta _{f} 2^{n-1}-2^{n/2}$for$n\equiv 0~({\mathrm {mod}~}4)$and$n\geq 48$. Secondly, we obtain two new classes of 1-resilient functions possessing very high nonlinearity and very low absolute indicator, from bent functions and plateaued functions, respectively. Moreover, one class of them achieves the currently known highest nonlinearity$2^{n-1}-2^{n/2-1}-2^{n/4}$, and the absolute indicator of which is upper bounded by$2^{n/2}+2^{n/4+1}$that is a new upper bound of the minimum of absolute indicator of 1-resilient functions, as it is clearly optimal than the previously best known upper bound$5\cdot 2^{n/2}-2^{n/4+2}+4$.