3,762 research outputs found
On the Equivalence of Quadratic APN Functions
Establishing the CCZ-equivalence of a pair of APN functions is generally
quite difficult. In some cases, when seeking to show that a putative new
infinite family of APN functions is CCZ inequivalent to an already known
family, we rely on computer calculation for small values of n. In this paper we
present a method to prove the inequivalence of quadratic APN functions with the
Gold functions. Our main result is that a quadratic function is CCZ-equivalent
to an APN Gold function if and only if it is EA-equivalent to that Gold
function. As an application of this result, we prove that a trinomial family of
APN functions that exist on finite fields of order 2^n where n = 2 mod 4 are
CCZ inequivalent to the Gold functions. The proof relies on some knowledge of
the automorphism group of a code associated with such a function.Comment: 13 p
Differentially 4-uniform functions
We give a geometric characterization of vectorial boolean functions with
differential uniformity less or equal to 4
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
Further Results of the Cryptographic Properties on the Butterfly Structures
Recently, a new structure called butterfly introduced by Perrin et at. is
attractive for that it has very good cryptographic properties: the differential
uniformity is at most equal to 4 and algebraic degree is also very high when
exponent . It is conjecture that the nonlinearity is also optimal for
every odd , which was proposed as a open problem. In this paper, we further
study the butterfly structures and show that these structure with exponent
have also very good cryptographic properties. More importantly, we
prove in theory the nonlinearity is optimal for every odd , which completely
solve the open problem. Finally, we study the butter structures with trivial
coefficient and show these butterflies have also optimal nonlinearity.
Furthermore, we show that the closed butterflies with trivial coefficient are
bijective as well, which also can be used to serve as a cryptographic
primitive.Comment: 20 page
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