A large family of Boolean functions (Q2787084)

In a series of papers starting with \textit{D. Coppersmith} and \textit{I. Shparlinski} [J. Cryptology 13, No. 3, 339--360 (2000; Zbl 1038.94007)] several number-theoretic constructions of Boolean functions with good cryptographic properties were studied, namely, maximum Fourier coefficient, nonlinearity, average sensitivity, sparsity, collision and avalanche effects. This paper presents extensions of the results of [loc. cit.] and \textit{T. Lange} and the reviewer [Discrete Appl. Math. 128, No. 1, 193--206 (2003; Zbl 1039.94007)] on Boolean functions which characterize non-squares \(u\) in a finite field \(\mathbb F_q\) to functions characterizing non-squares \(f(u)\) for a polynomial \(f\) over \(\mathbb F_q\). The results are based on well-known bounds on character sums.NEWLINENEWLINENEWLINEReviewer's comment: From each finite binary sequence we can derive a Boolean function. General relations between the correlation measure of order \(k\) of the binary sequence and measures for the corresponding Boolean functions such as sparsity and nonlinearity, see \textit{G. Pirsic} and the reviewer [Lect. Notes Comput. Sci. 7280, 101--109 (2012; Zbl 1290.94171)] and references therein, combined with the bounds on the correlation measure of order \(k\) of \textit{L. Goubin} et al. [J. Number Theory 106, No. 1, 56--69 (2004; Zbl 1049.11089)] immediately give essentially the results of this paper.