Semi-bent functions with perfect three-level additive autocorrelation

Semi-bent functions have very high nonlinearity and hence they have many applications in symmetric-key cryptography, binary sequence design for communications, and combinatorics. In this paper, we focus on studying the additive autocorrelation of semi-bent functions. We provide a lower bound on the maximum additive autocorrelation absolute value of semi-bent functions with three-level additive autocorrelation. Semi-bent functions with three-level additive autocorrelation achieving this bound with equality are said to have perfect three-level additive autocorrelation. We present two classes of balanced semi-bent functions with optimal algebraic degree and perfect three-level additive autocorrelation. The functions in the first class are constructed from the generalized Maiorana-McFarland class. The functions in the second one are derived from the concatenation of two bent functions, and we obtain a necessary and sufficient condition on bent functions such that the constructed semi-bent functions have perfect three-level additive autocorrelation.