634 research outputs found
Regular complete permutation polynomials over quadratic extension fields
Let be any positive integer which is relatively prime to and
. Let be any permutation polynomials over
is an invertible linear map over
and . In this paper,
we prove that, for suitable and , the map
could be -regular complete permutation polynomials over quadratic extension
fields.Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:2212.1286
A note on constructions of bent functions from involutions
Bent functions are maximally nonlinear Boolean functions. They are important
functions introduced by Rothaus and studied rstly by Dillon and next by many researchers
for four decades. Since the complete classication of bent functions seems
elusive, many researchers turn to design constructions of bent functions. In this note,
we show that linear involutions (which are an important class of permutations) over
nite elds give rise to bent functions in bivariate representations. In particular, we
exhibit new constructions of bent functions involving binomial linear involutions whose
dual functions are directly obtained without computation
A general construction of regular complete permutation polynomials
Let be a positive integer and the finite field with
elements. In this paper, we consider the -regular complete permutation
property of maps with the form where
is a PP over an extension field and is an
invertible linear map over . We give a general construction
of -regular PPs for any positive integer . When is additive, we
give a general construction of -regular CPPs for any positive integer .
When is not additive, we give many examples of regular CPPs over the
extension fields for and for arbitrary odd positive integer .
These examples are the generalization of the first class of -regular CPPs
constructed by Xu, Zeng and Zhang (Des. Codes Cryptogr. 90, 545-575 (2022)).Comment: 24 page
Anomalies and Vector Space Search: Tools for S-Box Analysis
International audienceS-boxes are functions with an input so small that the simplest way to specify them is their lookup table (LUT). How can we quantify the distance between the behavior of a given S-box and that of an S-box picked uniformly at random? To answer this question, we introduce various "anomalies". These real numbers are such that a property with an anomaly equal to should be found roughly once in a set of random S-boxes. First, we present statistical anomalies based on the distribution of the coefficients in the difference distribution table, linear approximation table, and for the first time, the boomerang connectivity table. We then count the number of S-boxes that have block-cipher like structures to estimate the anomaly associated to those. In order to recover these structures, we show that the most general tool for decomposing S-boxes is an algorithm efficiently listing all the vector spaces of a given dimension contained in a given set, and we present such an algorithm. Combining these approaches, we conclude that all permutations that are actually picked uniformly at random always have essentially the same cryptographic properties and the same lack of structure
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