Construction of n-variable (n ≡ 2 mod 4) balanced boolean functions with maximum absolute value in autocorrelation spectra <2n2

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Research Article

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IEEE Transactions on Information Theory


In this paper, we consider the maximum absolute value f in the autocorrelation spectrum (not considering the zero point) of a function f . In an even number of variables n, bent functions possess the highest nonlinearity with f = 0. The long standing open question (for two decades) in this area is to obtain a theoretical construction of balanced functions with f < 2n/2. So far, there are only a few examples of such functions for n = 10, 14, but no general construction technique is known. In this paper, we mathematically construct an infinite class of balanced Boolean functions on n variables having absolute indicator strictly lesser than δn = 2n/2 − 2((n+6)/4), nonlinearity strictly greater than ρn = 2n−1 −2n/2 +2n/2−3 −5·2((n−2)/4) and algebraic degree n − 1, where n ≡ 2 (mod 4) and n ≥ 46. While the bound n ≥ 46 is required for proving the generic result, our construction starts from n = 18, and we could obtain balanced functions with f < 2n/2 and nonlinearity > 2n−1 − 2n/2 for n = 18, 22, and 26.

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