Investigations on cc-Boomerang Uniformity and Perfect Nonlinearity

We defined in~\cite{EFRST20} a new multiplicative $c$-differential, and the corresponding $c$-differential uniformity and we characterized the known perfect nonlinear functions with respect to this new concept, as well as the inverse in any characteristic. The work was continued in~\cite{RS20}, investigating the $c$-differential uniformity for some further APN functions. Here, we extend the concept to the boomerang uniformity, introduced at Eurocrypt '18 by Cid et al.~\cite{Cid18}, to evaluate S-boxes of block ciphers, and investigate it in the context of perfect nonlinearity and related functions.Comment: 31 pages, 1 figur

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

Functions 3-to-1 and power APN S-boxes

Almost Perfect Nonlinear(APN) S-boxes are used in block ciphers to prevent differential attacks. The non-evidence of permutation APN S-box on even number of variables and the efficiency of power functions bring the importance of power APN S-boxes to use in block ciphers. We present a special class of 3-to-1 S-box on even number of variables. The power APN S-boxes on even number of variables fall in this class. We study some other known APN S-boxes on even number of variables whether they fall in this class. Then we present a necessary condition for power functions to be APN. Using this necessary condition we can filter out some non-APN power functions. Specifically, if the number of variables is multiple of small primes, then one can filter many non-APN functions

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 $\mathbb{F}_2^n$ 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 Derivative Imbalance and Ambiguity of Functions

In 2007, Carlet and Ding introduced two parameters, denoted by $Nb_F$ and $NB_F$, quantifying respectively the balancedness of general functions $F$ between finite Abelian groups and the (global) balancedness of their derivatives $D_a F(x)=F(x+a)-F(x)$, $a\in G\setminus\{0\}$ (providing an indicator of the nonlinearity of the functions). These authors studied the properties and cryptographic significance of these two measures. They provided for S-boxes inequalities relating the nonlinearity $\mathcal{NL}(F)$ to $NB_F$, and obtained in particular an upper bound on the nonlinearity which unifies Sidelnikov-Chabaud-Vaudenay's bound and the covering radius bound. At the Workshop WCC 2009 and in its postproceedings in 2011, a further study of these parameters was made; in particular, the first parameter was applied to the functions $F+L$ where $L$ is affine, providing more nonlinearity parameters. In 2010, motivated by the study of Costas arrays, two parameters called ambiguity and deficiency were introduced by Panario \emph{et al.} for permutations over finite Abelian groups to measure the injectivity and surjectivity of the derivatives respectively. These authors also studied some fundamental properties and cryptographic significance of these two measures. Further studies followed without that the second pair of parameters be compared to the first one. In the present paper, we observe that ambiguity is the same parameter as $NB_F$, up to additive and multiplicative constants (i.e. up to rescaling). We make the necessary work of comparison and unification of the results on $NB_F$, respectively on ambiguity, which have been obtained in the five papers devoted to these parameters. We generalize some known results to any Abelian groups and we more importantly derive many new results on these parameters

On the suitability of power functions as S-boxes for symmetric cryptosystems

textI present some results towards a classification of power functions that are Almost Perfect Nonlinear (APN), or equivalently differentially 2-uniform, over F2n for infinitely many positive integers n. APN functions are useful in constructing S-boxes in AES-like cryptosystems. An application of a theorem by Weil [20] on absolutely irreducible curves shows that a monomial x m is not APN over F2n for all sufficiently large n if a related two variable polynomial has an absolutely irreducible factor defined over F2. I will show that the latter polynomial’s singularities imply that except in five cases, all power functions have such a factor. Three of these cases are already known to be APN for infinitely many fields. The last two cases are still unproven. Some specific cases of power functions have already been known to be APN over only finitely many fields, but they also follow from the results below.Mathematic

Nonlinear time-series analysis revisited

In 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data---typically univariate---via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whether it contains noise. Moreover, the numerical algorithms that we use to instantiate these ideas are not perfect; they involve approximations, scale parameters, and finite-precision arithmetic, among other things. Even so, nonlinear time-series analysis has been used to great advantage on thousands of real and synthetic data sets from a wide variety of systems ranging from roulette wheels to lasers to the human heart. Even in cases where the data do not meet the mathematical or algorithmic requirements to assure full topological conjugacy, the results of nonlinear time-series analysis can be helpful in understanding, characterizing, and predicting dynamical systems