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Construction of <span class="MathJax_Preview">n</span><script type="math/tex" id="MathJax-Element-1">n</script> -Variable ( <span class="MathJax_Preview">n\equiv 2 \bmod 4</span><script type="math/tex" id="MathJax-Element-2">n\equiv 2 \bmod 4</script> ) Balanced Boolean Functions With Maximum Absolute Value in Autocorrelation Spectra <span class="MathJax_Preview">< 2^{\frac {n}2}</span><script type="math/tex" id="MathJax-Element-3">< 2^{\frac {n}2}</script> | IEEE Journals & Magazine | IEEE Xplore

Construction of n -Variable ( n\equiv 2 \bmod 4 ) Balanced Boolean Functions With Maximum Absolute Value in Autocorrelation Spectra < 2^{\frac {n}2}

Publisher: IEEE

Abstract:

In this paper, we consider the maximum absolute value Δ f in the autocorrelation spectrum (not considering the zero point) of a function f. In an even number of variable...View more

Abstract:

In this paper, we consider the maximum absolute value Δ 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 Δ f = 0. The long standing open question (for two decades) in this area is to obtain a theoretical construction of balanced functions with Δ f <; 2 n/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 δ n = 2 n/2 - 2( (n+6)/4) , nonlinearity strictly greater than ρ n = 2 n-1 -2 n/2 +2 n/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 Δ f <; 2 n/2 and nonlinearity > 2 n-1 - 2 n/2 for n = 18, 22, and 26.
Published in: IEEE Transactions on Information Theory ( Volume: 64, Issue: 1, January 2018)
Page(s): 393 - 402
Date of Publication: 02 November 2017

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Publisher: IEEE

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