### Pseudo-Boolean functions valued on hypershere

#### Abstract

Fixing some ordering on the domain of real-valued functions of n Boolean variables (i. e. pseudo-Boolean functions) we can identify these functions (or rather tables of their values) with vectors in the Euclidean space R 2 n of dimension 2 n . From a perspective of the Boolean function theory the integer-valued pseudo-Boolean functions are of special interest. It is due to the fact that the Walsh–Hadamard transform of a Boolean function gives the integer-valued pseudo-Boolean function that identically corresponds to the Boolean function. If we represent such pseudo-Boolean functions by points of Euclidean space then all of them appear to be placed on the (2n−1)-dimensional sphere with radius 2 n . Previously the mapping of the n-variables Boolean function set on the Euclidean hypersphere in R 2 n was already studied. This paper represents an attempt to extend the results obtained in those settings to the subset of pseudo-Boolean functions corresponding to the points on the hypersphere. In particular, we consider new concepts of curvature and nonlinearity of such pseudo-Boolean functions. We set relations between them and express curvature value via some metric parameters related to the described geometric representation of the pseudo-Boolean functions. One of the aims of this investigation is to work out an approach to bounding maximum nonlinearity of Boolean functions with odd number of variables.

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PDF (Russian)#### References

Logachev O. A., Fedorov S. N., Yashchenko V. V. Boolean functions as points on the hypersphere in the Euclidean space // Discrete Mathematics and Applications. 2019. Vol. 29, no. 2. P. 89–101.

Logachev O. A., Salnikov A. A., Yashchenko V. V. Boolean functions in coding theory and cryptography. Providence (Rhode Island, USA) : American Mathematical Society, 2011.

Tokareva N. N. Nonlinear Boolean functions : bent functions and their generalizations. Saarbrücken (Germany) : LAMBERT Academic Publishing, 2011 [in Russian].

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