48,609 research outputs found

    A Highly Nonlinear Differentially 4 Uniform Power Mapping That Permutes Fields of Even Degree

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    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

    On the Derivative Imbalance and Ambiguity of Functions

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    In 2007, Carlet and Ding introduced two parameters, denoted by NbFNb_F and NBFNB_F, quantifying respectively the balancedness of general functions FF between finite Abelian groups and the (global) balancedness of their derivatives DaF(x)=F(x+a)−F(x)D_a F(x)=F(x+a)-F(x), a∈G∖{0}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 NL(F)\mathcal{NL}(F) to NBFNB_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+LF+L where LL 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 NBFNB_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 NBFNB_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 Equivalence of Known Families of APN Functions in Small Dimensions

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    In this extended abstract, we computationally check and list the CCZ-inequivalent APN functions from infinite families on F2n\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

    Further Results of the Cryptographic Properties on the Butterfly Structures

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    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 e=3e=3. It is conjecture that the nonlinearity is also optimal for every odd kk, which was proposed as a open problem. In this paper, we further study the butterfly structures and show that these structure with exponent e=2i+1e=2^i+1 have also very good cryptographic properties. More importantly, we prove in theory the nonlinearity is optimal for every odd kk, 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

    High-frequency oscillations in low-dimensional conductors and semiconductor superlattices induced by current in stack direction

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    A narrow energy band of the electronic spectrum in some direction in low-dimensional crystals may lead to a negative differential conductance and N-shaped I-V curve that results in an instability of the uniform stationary state. A well-known stable solution for such a system is a state with electric field domain. We have found a uniform stable solution in the region of negative differential conductance. This solution describes uniform high-frequency voltage oscillations. Frequency of the oscillation is determined by antenna properties of the system. The results are applicable also to semiconductor superlattices.Comment: 8 pages, 3 figure
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