5,526 research outputs found
On Equivalence of Known Families of APN Functions in Small Dimensions
In this extended abstract, we computationally check and list the
CCZ-inequivalent APN functions from infinite families on for n
from 6 to 11. These functions are selected with simplest coefficients from
CCZ-inequivalent classes. This work can simplify checking CCZ-equivalence
between any APN function and infinite APN families.Comment: This paper is already in "PROCEEDING OF THE 20TH CONFERENCE OF FRUCT
ASSOCIATION
On the Fourier Spectra of the Infinite Families of Quadratic APN Functions
It is well known that a quadratic function defined on a finite field of odd
degree is almost bent (AB) if and only if it is almost perfect nonlinear (APN).
For the even degree case there is no apparent relationship between the values
in the Fourier spectrum of a function and the APN property. In this article we
compute the Fourier spectrum of the new quadranomial family of APN functions.
With this result, all known infinite families of APN functions now have their
Fourier spectra and hence their nonlinearities computed.Comment: 12 pages, submitted to Adavances in the Mathematics of communicatio
Resurgence of the Euler-MacLaurin summation formula
The Euler-MacLaurin summation formula relates a sum of a function to a
corresponding integral, with a remainder term. The remainder term has an
asymptotic expansion, and for a typical analytic function, it is a divergent
(Gevrey-1) series. Under some decay assumptions of the function in a half-plane
(resp. in the vertical strip containing the summation interval), Hardy (resp.
Abel-Plana) prove that the asymptotic expansion is a Borel summable series, and
give an exact Euler-MacLaurin summation formula.
Using a mild resurgence hypothesis for the function to be summed, we give a
Borel summable transseries expression for the remainder term, as well as a
Laplace integral formula, with an explicit integrand which is a resurgent
function itself. In particular, our summation formula allows for resurgent
functions with singularities in the vertical strip containing the summation
interval.
Finally, we give two applications of our results. One concerns the
construction of solutions of linear difference equations with a small
parameter. And another concerns the problem of proving resurgence of formal
power series associated to knotted objects.Comment: AMS-LaTeX, 15 pages with 2 figure
Linear Codes from Some 2-Designs
A classical method of constructing a linear code over \gf(q) with a
-design is to use the incidence matrix of the -design as a generator
matrix over \gf(q) of the code. This approach has been extensively
investigated in the literature. In this paper, a different method of
constructing linear codes using specific classes of -designs is studied, and
linear codes with a few weights are obtained from almost difference sets,
difference sets, and a type of -designs associated to semibent functions.
Two families of the codes obtained in this paper are optimal. The linear codes
presented in this paper have applications in secret sharing and authentication
schemes, in addition to their applications in consumer electronics,
communication and data storage systems. A coding-theory approach to the
characterisation of highly nonlinear Boolean functions is presented
A Highly Nonlinear Differentially 4 Uniform Power Mapping That Permutes Fields of Even Degree
Functions with low differential uniformity can be used as the s-boxes of
symmetric cryptosystems as they have good resistance to differential attacks.
The AES (Advanced Encryption Standard) uses a differentially-4 uniform function
called the inverse function. Any function used in a symmetric cryptosystem
should be a permutation. Also, it is required that the function is highly
nonlinear so that it is resistant to Matsui's linear attack. In this article we
demonstrate that a highly nonlinear permutation discovered by Hans Dobbertin
has differential uniformity of four and hence, with respect to differential and
linear cryptanalysis, is just as suitable for use in a symmetric cryptosystem
as the inverse function.Comment: 10 pages, submitted to Finite Fields and Their Application
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