Construction and search of balanced Boolean functions on even number of variables towards excellent autocorrelation profile

Article Type

Research Article

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Designs, Codes, and Cryptography


In a very recent work by Tang and Maitra (IEEE Ttans Inf Theory 64(1):393–402, 2018], a theoretical construction of balanced functions f on n-variables (n≡2mod4) with very good autocorrelation and Walsh spectra values (Δfnl(f)>2n-1-2n2+2n2-3-5·2n-24) has been presented. The theoretical bounds could be satisfied for all such n≥ 46. The case for n≡0mod4 could not be solved in the said paper and it has also been pointed out that though theoretically not proved, such constructions may provide further interesting examples of Boolean functions. In this follow-up work, we concentrate in two directions. First we present a construction method for balanced functions f on n-variables (n≡0mod4 and n≥ 52) with Δfnl(f)>2n-1-2n2). Secondly, we apply search methods in suitable places to obtain balanced functions on even variables in the interval [10 , … , 26] with improved parameters that could never be achieved before. As a consequence, for the first time we could provide examples of balanced Boolean functions f having Δfn≡0mod4, where n= 12 , 16 , 20 , and 24. Whatever functions we present in this paper have nonlinearity greater than 2n-1-2n2.

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