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

In 2018, Tang and Maitra presented a class of balanced Boolean functions in n variables with the absolute indicator Delta(f) < 2(n/2) and the nonlinearity NL (f) > 2(n-1) - 2(n/2), that is, f is SAO (strictly almost optimal), for n = 2k equivalent to 2 (mod 4) and n <= 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([(n+1)/2]), and the previously best known upper bound of which is 5 . 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/2) and the nonlinearity NL (f) > 2(n-1) - 2(n/2) for n equivalent to 0 (mod 4) and n >= 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 newupper bound of the minimum of absolute indicator of 1-resilient functions, as it is clearly optimal than the previously best known upper bound 5 x 2(n/2) - 2(n/4+2) + 4.