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On cryptographic properties of the cosets of $$R(1,m)$$. (English) Zbl 1021.94014
A new approach for the study of weight distribution of cosets of the Reed-Muller code of order 1 is presented. First, some algebraic properties of Boolean functions of $$m$$ variables are studied; this study leads to the study of binary codewords of length $$2^m$$ and their relation with Reed-Muller codes. These tools are then used to study the maximal odd-weighting subspace of a given Boolean function $$f$$, a concept known to be linked with the nonlinearity of $$f$$. Here also the existence of maximal odd-weighting subspaces for any $$f$$ is proved.
Then, the nonlinearity of $$f$$ is investigated through the properties of weight polynomials of cosets of $$R(1,m)$$. General results on the weight polynomial of any binary linear code of length $$2^m$$ and dimension $$m+2$$ is established here as well as results for some subcases.
Finally, the propagation criterion and its relations with the nonlinearity are investigated. Besides other results, links between bent functions and three-valued almost-optimal functions are given here.

##### MSC:
 94A60 Cryptography 94B05 Linear codes, general
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