26 research outputs found
Non-Boolean almost perfect nonlinear functions on non-Abelian groups
The purpose of this paper is to present the extended definitions and
characterizations of the classical notions of APN and maximum nonlinear Boolean
functions to deal with the case of mappings from a finite group K to another
one N with the possibility that one or both groups are non-Abelian.Comment: 17 page
A new characterization of group action-based perfect nonlinearity
International audienceThe left-regular multiplication is explicitly embedded in the notion of perfect nonlinearity. But there exist many other group actions. By replacing translations by another group action the new concept of group action-based perfect nonlinearity has been introduced. In this paper we show that this generalized concept of nonlinearity is actually equivalent to a new bentness notion that deals with functions defined on a finite Abelian group G that acts on a finite set X and with values in the finite-dimensional vector space of complex-valued functions defined on X
Non Abelian Bent Functions
International audiencePerfect nonlinear functions from a finite group to another one are those functions such that for all nonzero , the derivative is balanced. In the case where both and are Abelian groups, is perfect nonlinear if and only if is bent {\it i.e.} for all nonprincipal character of , the (discrete) Fourier transform of has a constant magnitude equals to . In this paper, using the theory of linear representations, we exhibit similar bentness-like characterizations in the cases where and/or are (finite) non Abelian groups. Thus we extend the concept of bent functions to the framework of non Abelian groups
C-DIFFERENTIALS AND GENERALIZED CRYPTOGRAPHIC PROPERTIES OF VECTORIAL BOOLEAN AND P-ARY FUNCTIONS
This dissertation investigates a newly defined cryptographic differential, called a c-differential, and its relevance to the nonlinear substitution boxes of modern symmetric block ciphers. We generalize the notions of perfect nonlinearity, bentness, and avalanche characteristics of vectorial Boolean and p-ary functions using the c-derivative and a new autocorrelation function, while capturing the original definitions as special cases (i.e., when c=1). We investigate the c-differential uniformity property of the inverse function over finite fields under several extended affine transformations. We demonstrate that c-differential properties do not hold in general across equivalence classes typically used in Boolean function analysis, and in some cases change significantly under slight perturbations. Thus, choosing certain affine equivalent functions that are easy to implement in hardware or software without checking their c-differential properties could potentially expose an encryption scheme to risk if a c-differential attack method is ever realized. We also extend the c-derivative and c-differential uniformity into higher order, investigate some of their properties, and analyze the behavior of the inverse function's second order c-differential uniformity. Finally, we analyze the substitution boxes of some recognizable ciphers along with certain extended affine equivalent variations and document their performance under c-differential uniformity.Commander, United States NavyApproved for public release. Distribution is unlimited
Doubly Perfect Nonlinear Boolean Permutations
Due to implementation constraints the XOR operation is widely used in order
to combine plaintext and key bit-strings in secret-key block ciphers. This
choice directly induces the classical version of the differential attack by the
use of XOR-kind differences. While very natural, there are many alternatives to
the XOR. Each of them inducing a new form for its corresponding differential
attack (using the appropriate notion of difference) and therefore block-ciphers
need to use S-boxes that are resistant against these nonstandard differential
cryptanalysis. In this contribution we study the functions that offer the best
resistance against a differential attack based on a finite field
multiplication. We also show that in some particular cases, there are robust
permutations which offers the best resistant against both multiplication and
exponentiation base differential attacks. We call them doubly perfect nonlinear
permutations
-differential bent functions and perfect nonlinearity
Drawing inspiration from Nyberg's paper~\cite{Nyb91} on perfect nonlinearity
and the -differential notion we defined in~\cite{EFRST20}, in this paper we
introduce the concept of -differential bent functions in two different ways
(thus extending Kumar et al.~\cite{Ku85} classical definition). We further
extend the notion of perfect -nonlinear introduced in~\cite{EFRST20}, also
in two different ways, and show that, in both cases, the concepts of
-differential bent and perfect -nonlinear are equivalent (under some
natural restriction of the parameters). Some constructions of functions with
these properties are also provided; one such construction provides a large
class of PcN functions with respect to all in some subfield of the field
under consideration. We also show that both our classes of -differential
bents are supersets of permutation polynomials, and that Maiorana-McFarland
bent functions are not differential bent (of the first kind).Comment: 24 page
C -differential bent functions and perfect nonlinearity
17 USC 105 interim-entered record; under review.The article of record as published may be found at https://doi.org/10.1016/j.dam.2021.10.010Drawing inspiration from Nyberg’s paper (Nyberg, 1991) on perfect nonlinearity and the c-differential notion we defined in Ellingsen et al. (2020), in this paper we introduce the concept of c-differential bent functions in two different ways (thus extending Kumar et al. (1985) classical definition). We further extend the notion of perfect c-nonlinear introduced in Ellingsen et al. (2020), also in two different ways, and show that, in both cases, the concepts of c-differential bent and perfect c-nonlinear are equivalent (under some natural restriction of the parameters). Some constructions of functions with these properties are also provided; one such construction provides a large class of PcN functions with respect to all c in some subfield of the field under consideration. We also show that both our classes of 0-differential bents are supersets of permutation polynomials, and that Maiorana–McFarland bent functions are not differential bent (of the first kind).Pantelimon Stănică acknowledges the sabbatical support from Naval Postgraduate School from September 2020 to July 2021