Concrete constructions of weightwise perfectly balanced (2-rotation symmetric) functions with optimal algebraic immunity and high weightwise nonlinearity

Abstract

Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the well-known FLIP stream cipher proposed by Méaux et al. at the conference Eurocrypt 2016. After providing a security analysis on the FLIP cipher, those functions were nicely-investigated firstly by Carlet et al. in 2017 before taking a high interest by the community. Handling such Boolean functions and designing those with optimal characteristic cryptographic properties is no easy assignment. This article attempts to broaden the range of choices for these functions by offering two new concrete constructions of weightwise perfectly balanced (WPB) functions on $2^m$ variables (where m is a positive integer) with optimal algebraic immunity. It is worth noting that the second class of WPB functions can be linearly transformed to be 2-rotation symmetric. Simultaneously, the k-weight nonlinearities of these newly constructed WPB functions on $2^m$ variables are discussed for small values of m. Lastly, comparisons of the k-weight nonlinearities of all the known WPB functions are given, including the known results from computer investigations. The comparison to the current literature shows that despite its simplicity (an advantage from the implementation point of view), the WPB functions presented in this paper are the best in behavior from the algebraic immunity and the k-weight nonlinearities. Specifically, the even-weight nonlinearities of our second class of WPB functions are much higher than all the known WPB functions in the literature.