51 research outputs found
Binary linear codes with few weights from two-to-one functions
In this paper, we apply two-to-one functions over in two
generic constructions of binary linear codes. We consider two-to-one functions
in two forms: (1) generalized quadratic functions; and (2)
with and . Based on the study of the Walsh transforms of those functions
or their related-ones, we present many classes of linear codes with few nonzero
weights, including one weight, three weights, four weights and five weights.
The weight distributions of the proposed codes with one weight and with three
weights are determined. In addition, we discuss the minimum distance of the
dual of the constructed codes and show that some of them achieve the sphere
packing bound. { Moreover, several examples show that some of our codes are
optimal and some have the best known parameters.
New Results about the Boomerang Uniformity of Permutation Polynomials
In EUROCRYPT 2018, Cid et al. \cite{BCT2018} introduced a new concept on the
cryptographic property of S-boxes: Boomerang Connectivity Table (BCT for short)
for evaluating the subtleties of boomerang-style attacks. Very recently, BCT
and the boomerang uniformity, the maximum value in BCT, were further studied by
Boura and Canteaut \cite{BC2018}. Aiming at providing new insights, we show
some new results about BCT and the boomerang uniformity of permutations in
terms of theory and experiment in this paper. Firstly, we present an equivalent
technique to compute BCT and the boomerang uniformity, which seems to be much
simpler than the original definition from \cite{BCT2018}. Secondly, thanks to
Carlet's idea \cite{Carlet2018}, we give a characterization of functions
from to itself with boomerang uniformity by
means of the Walsh transform. Thirdly, by our method, we consider boomerang
uniformities of some specific permutations, mainly the ones with low
differential uniformity. Finally, we obtain another class of -uniform BCT
permutation polynomials over , which is the first binomial.Comment: 25 page
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