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• W2765254330 abstract "In this paper, we consider the maximum absolute value Δ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>f</sub> 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 Δ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>f</sub> = 0. The long standing open question (for two decades) in this area is to obtain a theoretical construction of balanced functions with Δ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>f</sub> <; 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n/2</sup> . 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 δ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</sub> = 2 <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n/2</sub> - 2( <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>(n+6)/4)</sup> , nonlinearity strictly greater than ρ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</sub> = 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n-1</sup> -2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n/2</sup> +2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n/2-3</sup> -5·2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>((n-2)/4)</sup> 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 Δ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>f</sub> <; 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n/2</sup> and nonlinearity > 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n-1</sup> - 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n/2</sup> for n = 18, 22, and 26." @default.
• W2765254330 created "2017-11-10" @default.
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• W2765254330 date "2018-01-01" @default.
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• W2765254330 title "Construction of $n$ -Variable ( $nequiv 2 bmod 4$ ) Balanced Boolean Functions With Maximum Absolute Value in Autocorrelation Spectra $" @default.
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