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On the Maximum Number of Bent Components of Vectorial Functions | IEEE Journals & Magazine | IEEE Xplore

On the Maximum Number of Bent Components of Vectorial Functions


Abstract:

In this paper, we show that the maximum number of bent component functions of a vectorial function F : GF(2)n → GF(2)n is 2n - 2n/2. We also show that it is very easy to ...Show More

Abstract:

In this paper, we show that the maximum number of bent component functions of a vectorial function F : GF(2)n → GF(2)n is 2n - 2n/2. We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form F ∈ GF(2n)[x], where F has only a few terms. The only known power functions having such a large number of bent components are xd, where d = 2n/2 + 1. In this paper, we show that the binomials Fi (x) = x2i (x + x(2n/2)) also have such a large number of bent components, and these binomials are inequivalent to the monomials x(2n/2+1) if 0 <; i <; n/2. In addition, the functions Fi have differential properties much better than x(2n/2+1). We also determine the complete Walsh spectrum of our functions when n/2 is odd and gcd(i, n/2) = 1.
Published in: IEEE Transactions on Information Theory ( Volume: 64, Issue: 1, January 2018)
Page(s): 403 - 411
Date of Publication: 13 September 2017

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