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Let {F n}n≥0 be the sequence of Fibonacci numbers. The aim of this paper is to give linear independence results over (5) for the infinite series n=1χj(n)/Fn with certain nonprincipal real Dirichlet characters χ j. We also deduce the irrationality results for the special principal Dirichlet characters and for other multiplicative functions.

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Hujter and Lángi defined the k-fold Borsuk number of a set S in Euclidean n-space of diameter d > 0 as the smallest cardinality of a family F of subsets of S, of diameters strictly less than d, such that every point of S belongs to at least k members of F.

We investigate whether a k-fold Borsuk covering of a set S in a finite dimensional real normed space can be extended to a completion of S. Furthermore, we determine the k-fold Borsuk number of sets in not angled normed planes, and give a partial characterization for sets in angled planes.

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Authors: Masoome Zahiri, Ahmad Moussavi and Rasul Mohammadi

In this paper we study rings R with the property that every finitely generated ideal of R consisting entirely of zero divisors has a nonzero annihilator. The class of commutative rings with this property is quite large; for example, noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose classical ring of quotients are von Neumann regular. We continue to study conditions under which right mininjective rings, right FP-injective rings, right weakly continuous rings, right extending rings, one sided duo rings, semiregular rings and semiperfect rings have this property.

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A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. It was conjectured in [10], that for any two graphs G and H, b(G[H]) ≦ b(G) − 1|V (H)| + Δ(H) + 1 and b(GH) ≦ max {b(G)(Δ(H) + 1), b(H) Δ(G) + 1)}, where G[H] and GH denotes the lexicographic product and the strong product of G and H, respectively. In this paper, we disprove both conjectures.

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A new concept of Walsh-Lebesgue points is introduced for higher dimensions and it is proved that almost every point is a modified Walsh-Lebesgue point of an integrable function. It is shown that the Walsh-Fejér means σ n f of a function fL 1[0, 1)d converge to f at each modified Walsh-Lebesgue point, whenever n→∞ and n is in a cone. The same is proved for other summability means, such as for the Weierstrass, Abel, Picard, Bessel, Cesàro, de La Vallée-Poussin, Rogosinski and Riesz summations.

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We propose a new two-parameter continuous model called the extended arcsine distribution restricted to the unit interval. It is a very competitive model to the beta and Kumaraswamy distributions for modeling percentages, rates, fractions and proportions. We provide a mathematical treatment of the new distribution including explicit expressions for the ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating and quantile functions, Shannon entropy and order statistics. Maximum likelihood is used to estimate the model parameters and the expected information matrix is determined. We demonstrate by means of two applications to proportional data that it can give consistently a better fit than other important statistical models.

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In this paper, we study dissipative q-Sturm—Liouville operators in Weyl’s limit circle case. We describe all maximal dissipative, maximal accretive, self adjoint extensions of q-Sturm—Liouville operators. Using Livšic’s theorems, we prove a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative q-Sturm—Liouville operators.

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In [14] we investigated some Vilenkin—Nörlund means with non-increasing coefficients. In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy space H 1/(1+α) to the space weak-L 1/(1+α), (0 < α ≦ 1). In this paper we construct a martingale in the space H 1/(1+α), which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin—Nörlund means with non-increasing coefficients are not bounded from the Hardy space H 1/(1+α) to the space L 1/(1+α). In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.

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We present here characterizations of the most recently introduced continuous univariate distributions based on: (i) a simple relationship between two truncated moments; (ii) truncated moments of certain functions of the 1th order statistic; (iii) truncated moments of certain functions of the n th order statistic; (iv) truncated moment of certain function of the random variable. We like to mention that the characterization (i) which is expressed in terms of the ratio of truncated moments is stable in the sense of weak convergence. We will also point out that some of these distributions are infinitely divisible via Bondesson’s 1979 classifications.

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A subgroup H of G is called M p-embedded in G, if there exists a p-nilpotent subgroup B of G such that H p ∈ Sylp(B) and B is M p-supplemented in G. In this paper, we use M p-embedded subgroups to study the structure of finite groups.

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In this paper we establish approximation properties of Cesàro (C, −α) means with α ∈ (0, 1) of Vilenkin—Fourier series. This result allows one to obtain a condition which is sufficient for the convergence of the means σ n α(f, x) to f(x) in the L p-metric.

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In this paper, we concern the Principal Ideal Theorem (PIT) for w-Noetherian rings. Let R be a w-Noetherian ring and a be a nonzero nonunit element of R. If p is a prime ideal of R minimal over (a), then ht p ≦ 1.

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In this paper, we study the k-th order Kantorovich type modication of Szász—Mirakyan operators. We first establish explicit formulas giving the images of monomials and the moments up to order six. Using this modification, we present a quantitative Voronovskaya theorem for differentiated Szász—Mirakyan operators in weighted spaces. The approximation properties such as rate of convergence and simultaneous approximation by the new constructions are also obtained.

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Authors: Muhammad Ahsan Binyamin, Junaid Alam Khan, Faira Kanwal Janjua and Naveed Hussain

In this article we characterize the classification of stably simple curve singularities given by V. I. Arnold, in terms of invariants. On the basis of this characterization we describe an implementation of a classifier for stably simple curve singularities in the computer algebra system SINGULAR.

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We investigate the pointwise and uniform convergence of the symmetric rectangular partial (also called Dirichlet) integrals of the double Fourier integral of a function that is Lebesgue integrable and of bounded variation over ℝ2. Our theorem is a two-dimensional extension of a theorem of Móricz (see Theorem 3 in [10]) concerning the single Fourier integrals, which is more general than the two-dimensional extension given by Móricz himself (see Theorem 3 in [11]).

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In this paper, we shall establish some Hadamard-type inequalities for differentiable coordinated convex functions in a rectangle from the plane in two variables. Through these inequalities, more precise estimates could be obtained. Some examples and applications to cubature formulas are also provided.

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Let α be an infinite ordinal. Let RCAα denote the variety of representable cylindric algebras of dimension α. Modifying Andréka’s methods of splitting, we show that the variety RQEAα of representable quasi-polyadic equality algebras of dimension α is not axiomatized by a set of universal formulas containing only finitely many variables over the variety RQAα of representable quasi-polyadic algebras of dimension α. This strengthens a seminal result due to Sain and Thompson, answers a question posed by Andréka, and lifts to the transfinite a result of hers proved for finite dimensions > 2. Using the modified method of splitting, we show that all known complexity results on universal axiomatizations of RCAα (proved by Andréka) transfer to universal axiomatizations of RQEAα. From such results it can be inferred that any algebraizable extension of L ω,ω is severely incomplete if we insist on Tarskian square semantics. Ways of circumventing the strong non-negative axiomatizability results hitherto obtained in the first part of the paper, such as guarding semantics, and /or expanding the signature of RQEAω by substitutions indexed by transformations coming from a finitely presented subsemigroup of (ω ω, ○) containing all transpositions and replacements, are surveyed, discussed, and elaborated upon.

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In this paper, using a Darbo type fixed point theorem associated with the measure of noncompactness we prove a theorem on the existence of asymptotically stable solutions of some nonlinear functional integral equations in the space of continuous and bounded functions on R+ = [0,∞). We also give some examples satisfying the conditions our existence theorem.

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Authors: Usama A. Aburawash and Muhammad Saad

The consistent way of investigating rings with involution, briefly *-rings, is to study them in the category of *-rings with morphisms preserving also involution. In this paper we continue the study of *-rings and the notion of *-reduced *-rings is introduced and their properties are studied. We introduce also the class of *-Baer *-rings. This class is defined in terms of *-annihilators and principal *-biideals, and it naturally extends the class of Baer *-rings. The use of *-biideals makes this concept more consistent with the involution than the use of right ideals in the notion of Baer *-rings. We prove that each *-Baer *-ring is semiprime. Moreover, we show that the property of *-Baer extends to both the *-corner and the center of the *-ring. Finally, we discuss the relation between *-Baer and quasi-Baer *-rings; the generalization of Baer *-ring.

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In this paper we introduce differential subordination and superordination properties for certain subclasses of analytic functions involving certain linear operator, and obtain sandwich-type results for the functions belonging to these classes.

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Let forb(m, F) denote the maximum number of columns possible in a (0, 1)-matrix A that has no repeated columns and has no submatrix which is a row and column permutation of F. We consider cases where the configuration F has a number of columns that grows with m. For a k × l matrix G, define s · G to be the concatenation of s copies of G. In a number of cases we determine forb(m, m α · G) is Θ(m k). Results of Keevash on the existence of designs provide constructions that can be used to give asymptotic lower bounds. An induction idea of Anstee and Lu is useful in obtaining upper bounds.

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We investigate some equivalent conditions for the reverse order laws (ab)# = b a # and (ab)# = b # a in rings with involution. Similar results for (ab)# = b # a* and (ab)# = b*a # are presented too.

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In Bayesian statistics, one frequently encounters priors and posteriors that are product of two probability density functions. In this paper, we discuss three such priors/posteriors, provide motivation and derive expressions for their moments, median and mode. Forty seven motivating examples are discussed. We expect that this paper could serve as a useful reference for practitioners of Bayesian statistics. It could also encourage further research in this area.

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A characterization of power function distribution based on the distribution of spacings is presented here extending the existing characterizations of the uniform distribution in this direction.

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The oscillatory properties of half-linear second order Euler type differential equations are studied, where the coefficients of the considered equations can be unbounded. For these equations, we prove an oscillation criterion and a non-oscillation one. We also mention a corollary which shows how our criteria improve the known results. In the corollary, the criteria give an explicit oscillation constant.

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By making use of the critical point theory, the existence of periodic solutions for fourth-order nonlinear p-Laplacian difference equations is obtained. The main approach used in our paper is a variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions of fourth-order nonlinear p-Laplacian difference equations. The results obtained successfully generalize and complement the existing one.

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The paper focuses on existence of solutions of a system of nonlocal resonant boundary value problems x=f(t,x),x(0)=0,x(1)=01x(s)dg(s), where f : [0, 1] × ℝk → ℝk is continuous and g : [0, 1] → ℝk is a function of bounded variation. Imposing on the function f the following condition: the limit limλ→∞ f(t, λ a) exists uniformly in aS k−1, we have shown that the problem has at least one solution.

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In this paper, using a Darbo type fixed point theorem associated with the measure of noncompactness we prove a theorem on the existence of solutions of some nonlinear functional integral equations in the space of continuous functions on interval [0, a]. We give also some examples which show that the obtained results are applicable

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A basic model in financial mathematics was introduced by Black, Scholes and Merton in 1973. A classical discrete approximation in distribution is the binomial model given by Cox, Ross and Rubinstein in 1979. In this work we give a strong (almost sure, pathwise) discrete approximation of the BSM model using a suitable nested sequence of simple, symmetric random walks. The approximation extends to the stock price process, the value process, the replicating portfolio, and the greeks. An important tool in the approximation is a discrete version of the Feynman-Kac formula as well.

Our aim is to show that from an elementary discrete approach, by taking simple limits, one may get the continuous versions. We think that such an approach can be advantageous for both research and applications. Moreover, it is hoped that this approach has pedagogical merits as well: gives insight and seems suitable for teaching students whose mathematical background may not contain e.g. measure theory or stochastic analysis.

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In this paper we deduce some tight Turán type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature. We also give alternative proofs for some already established Turán type inequalities. Moreover, by using these Turán type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometric functions of the second kind. The key tool in the proof of the Turán type inequalities is an integral representation for a quotient of Tricomi confluent hypergeometric functions, which arises in the study of the infinite divisibility of the Fisher-Snedecor F distribution.

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A semigroup is called eventually regular if each of its elements has a regular power. In this paper we study certain fundamental congruences on an eventually regular semigroup. We generalize some results of Howie and Lallement (1966) and LaTorre (1983). In particular, we give a full description of the semilattice of group congruences (together with the least such a congruence) on an arbitrary eventually regular (orthodox) semigroup. Moreover, we investigate UBG-congruences on an eventually regular semigroup. Finally, we study the eventually regular subdirect products of an E-unitary semigroup and a Clifford semigroup.

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The set of Cohen-Macaulay monomial ideals with a given radical contains the so-called Cohen-Macaulay modifications. Not all Cohen-Macaulay squarefree monomial ideals admit nontrivial Cohen-Macaulay modifications. We present classes of Cohen-Macaulay squarefree monomial ideals with infinitely many nontrivial Cohen-Macaulay modifications.

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The Bn (k) poly-Bernoulli numbers — a natural generalization of classical Bernoulli numbers (B n = Bn (1)) — were introduced by Kaneko in 1997. When the parameter k is negative then Bn (k) is a nonnegative number. Brewbaker was the first to give combinatorial interpretation of these numbers. He proved that Bn (−k) counts the so called lonesum 0–1 matrices of size n × k. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combinatorial explanation of Kaneko’s recursive formula for poly-Bernoulli numbers.

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Over the (1, 1)-dimensional real supercircle, we consider the K(1)-modules D λ,μ k of linear differential operators of order k acting on the superspaces of weighted densities, where K(1) is the Lie superalgebra of contact vector fields. We give, in contrast to the classical setting, a classification of these modules. This work is the simplest superization of a result by Gargoubi and Ovsienko.

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A ring R is called right SSP (SIP) if the sum (intersection) of any two direct summands of R R is also a direct summand. Left SSP (SIP) rings are defined similarly. There are several interesting results on rings with SSP. For example, R is right SSP if and only if R is left SSP, and R is a von Neumann regular ring if and only if Mn(R) is SSP for some n > 1. It is shown that R is a semisimple ring if and only if the column finite matrix ring ℂFM(R) is SSP, where ℕ is the set of natural numbers. Some known results are proved in an easy way through idempotents of rings. Moreover, some new results on SSP rings are given.

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Let G be a finite group. A subgroup H of G is said to be s-permutable in G if H permutes with all Sylow subgroups of G. Let H be a subgroup of G and let HsG be the subgroup of H generated by all those subgroups of H which are s-permutable in G. A subgroup H of G is called n-embedded in G if G has a normal subgroup T such that HG = HT and HTHsG, where HG is the normal closure of H in G. We investigate the influence of n-embedded subgroups of the p-nilpotency and p-supersolvability of G.

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In 1944, Santaló asked about the average number of normals through a point of a given convex body. Since then, numerous results appeared in the literature about this problem. The aim of this paper is to add to this list some new, recent developments. We point out connections of the problem to static equilibria of rigid bodies as well as to geometric partial differential equations of surface evolution.

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S. Banach in [1] proved that for any function fL 2(0, 1), f ≁ 0, there exists an ONS (orthonormal system) such that the Fourier series of this function is not summable a.e. by the method (C, α), α > 0.

D. Menshov found the conditions which should be satisfied by the Fourier coefficients of the function for the summability a.e. of its Fourier series by the method (C, α), α > 0.

In this paper the necessary and sufficient conditions are found which should be satisfied by the ONS functions (φ n(x)) so that the Fourier coefficients (by this system) of functions from class Lip 1 or A (absolutely continuous) satisfy the conditions of D. Menshov.

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Motivated by the well known Kadec-Pełczynski disjointification theorem, we undertake an analysis of the supports of non-zero functions in strongly embedded subspaces of Banach functions spaces. The main aim is to isolate those properties that bring additional information on strongly embedded subspaces. This is the case of the support localization property, which is a necessary condition fulfilled by all strongly embedded subspaces. Several examples that involve Rademacher functions, the Volterra operator, Lorentz spaces or Orlicz spaces are provided.

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An extension of von Neumann’s characterization of essentially selfadjoint operators is given among not necessarily densely defined symmetric operators which are assumed to be closable. Our arguments are of algebraic nature and follow the idea of [1].

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Authors: Jaroslav Hančl, Katarína Korčeková and Lukáš Novotný

We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

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The present note will present a different and direct way to generalize the convexity while keep the classical results for L 1-convergence still alive.

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The Separation Problem, originally posed by K. Bezdek in [1], asks for the minimum number s(O, K) of hyperplanes needed to strictly separate an interior point O in a convex body K from all faces of K. It is conjectured that s(O, K) ≦ 2d in d-dimensional Euclidean space. We prove this conjecture for the class of all totally-sewn neighbourly 4-dimensional polytopes.

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A remarkable class of quadratic irrational elements having both explicit Engel series and continued fraction expansions in the field of Laurent series, mimicking the case of real numbers discovered by Sierpiński and later extended by Tamura, is constructed. Linear integer-valued polynomials which can be applied to construct such class are determined. Corresponding results in the case of real numbers are mentioned.

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R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings.

We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the δ-vector of integrally closed lattice polytopes. Finally we apply our results for reflexive integrally closed and order polytopes.

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The modified method of estimation of the resistance of block ciphers to truncated byte differential attack is proposed. The previously known method estimate the truncated byte differential probability for Rijndael-like ciphers. In this paper we spread the sphere of application of that method on wider class of ciphers. The proposed method based on searching the most probable truncated byte differential characteristics and verification of sufficient conditions of effective byte differentials absence.

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Multiplicative complexity is the minimum number of AND-gates required to implement a given Boolean function in (AND, XOR) algebra. It is a good measure of a hardware complexity of an S-box, but an S-box cannot have too low multiplicative complexity due to security constraints. In this article we focus on generic constructions that can be used to find good n×n S-boxes with low multiplicative complexity. We tested these constructions in the specific case when n = 8. We were able to find 8 × 8 S-boxes with multiplicative complexity at most 16 (which is half of the known bound on multiplicative complexity of the AES S-box), while providing a reasonable resistance against linear and differential cryptanalysis.

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Since Gentry’s breakthrough result was introduced in the year 2009, the homomorphic encryption has become a very popular topic. The main contribution of Gentry’s thesis [5] was, that it has proven, that it actually is possible to design a fully homomorphic encryption scheme. However ground-breaking Gentry’s result was, the designs, that employ the bootstrapping technique suffer from terrible performance both in key generation and homomorphic evaluation of circuits. Some authors tried to design schemes, that could evaluate homomorphic circuits of arbitrarily many inputs without need of bootstrapping. This paper introduces the notion of symmetric homomorphic encryption, and analyses the security of four such proposals, published in three different papers ([2], [7], [10]). Our result is a known plaintext key-recovery attack on every one of these schemes.

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HaF is a family of hash functions developed in Poland at Poznán University of Technology, see [1, 2]. It is a classical Merkle-Damgård construction with the output sizes of 256, 512 or 1024 bits. In this paper we present a collision attack with negligible complexity (collisions can be found without using a computer) for all the members of HaF family. We have also shown that the improved function (without the critical transformation) is still insecure. It is possible to find a preimage for a short message with the complexity lower than the exhaustive search. We are also able to create some fixed points with a complexity of single compression function call.

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New multivariate cryptosystems are introduced. Sequences f(n) of bijective polynomial transformations of bijective multivariate transformations of affine spaces K n, n = 2, 3, ... , where K is a finite commutative ring with special properties, are used for the constructions of cryptosystems. On axiomatic level, the concept of a family of multivariate maps with invertible decomposition is proposed. Such decomposition is used as private key in a public key infrastructure. Requirements of polynomiality of degree and density allow to estimate the complexity of encryption procedure for a public user. The concepts of stable family and family of increasing order are motivated by studies of discrete logarithm problem in Cremona group. Statement on the existence of families of multivariate maps of polynomial degree and polynomial density with the invertible decomposition is formulated. We observe known explicit constructions of special families of multivariate maps. They correspond to explicit constructions of families of nonlinear algebraic graphs of increasing girth which appeared in Extremal Graph Theory. The families are generated by pseudorandom walks on graphs. This fact ensures the existence of invertible decomposition; a certain girth property guarantees the increase of order for the family of multivariate maps, good expansion properties of families of graphs lead to good mixing properties of graph based private key algorithms. We describe the general schemes of cryptographic applications of such families (public key infrastructure, symbolic Diffie—Hellman protocol, functional versions of El Gamal algorithm).

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We present an efficient endomorphism for the Jacobian of a curve C of genus 2 for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulæ for divisor doubling and addition of divisors with disjoint support in JF(C) using only base field operations. Explicit formulæ is presented for this third case and a different approach for divisor doubling.

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The main aim of this paper is to present the concept of fault-injection backdoors in Random Number Generators. Backdoors can be activated by fault-injection techniques. Presented algorithms can be used in embedded systems like smart-cards and hardware security modules in order to implement subliminal channels in random number generators.

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Authors: Tommi Meskanen, Valtteri Niemi and Noora Nieminen

The methods for secure outsourcing and secure one-time programs have recently been of great research interest. Garbling schemes are regarded as a promising technique for these applications while Bellare, Hoang and Rogaway introduced the first formal security notions for garbling schemes in [3, 4]. Ever since, even more security notions have been introduced and garbling schemes have been categorized in different security classes according to these notions. In this paper, we introduce new security classes of garbling schemes and build a hierarchy for the security classes including the known classes as well as classes introduced in this paper.

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In this paper we introduce a novel block cipher based on the composition of abstract finite automata and Latin cubes. For information encryption and decryption the apparatus uses the same secret keys, which consist of key-automata based on composition of abstract finite automata such that the transition matrices of the component automata form Latin cubes. The aim of the paper is to show the essence of our algorithms not only for specialists working in compositions of abstract automata but also for all researchers interested in cryptosystems. Therefore, automata theoretical background of our results is not emphasized. The introduced cryptosystem is important also from a theoretical point of view, because it is the first fully functioning block cipher based on automata network.

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Authors: Nicolas T. Courtois, Theodosis Mourouzis, Anna Grocholewska-Czuryło and Jean-Jacques Quisquater

Differential Cryptanalysis (DC) is one of the oldest known attacks on block ciphers. DC is based on tracking of changes in the differences between two messages as they pass through the consecutive rounds of encryption. However DC remains very poorly understood. In his textbook written in the late 1990s Schneier wrote that against differential cryptanalysis, GOST is “probably stronger than DES”. In fact Knudsen have soon proposed more powerful advanced differential attacks however the potential space of such attacks is truly immense. To this day there is no method which allows to evaluate the security of a cipher against such attacks in a systematic way. Instead, attacks are designed and improved in ad-hoc ways with heuristics [6–13,21]. The best differential attack known has time complexity of 2179 [13].

In this paper we show that for a given block cipher there exists an optimal size for advanced differential properties. This new understanding allows to considerably reduce the space to be searched for “good” truncated differential properties suitable for an attack.

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This paper deals with the phase-shift fault analysis of cipher Trivium. So far, only bit-flipping technique has been presented in the literature. The best fault attack on Trivium [13] combines bit-flipping with algebraic cryptanalysis and needs to induce 2 one-bit faults and to generate 420 bits per each keystream. Our attack combines phase-shifting and algebraic cryptanalysis and needs to phase-shift 2 registers of the cipher and to generate 120 bits per each keystream.

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Authors: Péter Kasza, Péter Ligeti and Ádám Nagy

In this paper we propose a decentralized privacy-preserving system which is able to share sensible data in a way, that only predefined subsets of authorized entities can recover the data after getting an additional alarm message. The protocol uses two main communication channels: a P2P network where the encrypted information is stored, and a smaller private P2P network, which consists of the authorized parties called friend-to-friend network. We describe the communication protocol fulfilling the desired security requirements. The proposed protocol achieves unconditional security. The main cryptographic building blocks of the protocol are symmetric encryption schemes and secret sharing schemes.

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We present an algorithm to compute the primary decomposition of a submodule N of the free module ℤ[x 1,...,x n]m. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of N, i.e. the minimal associated primes of the ideal Ann (ℤ[x 1,...,x n]m/N) in ℤ[x 1,...,x n] and then compute the primary components using pseudo-primary decomposition and extraction, following the ideas of Shimoyama-Yokoyama. The algorithms are implemented in Singular.

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Let A 1,...,A N and B 1,...,B M be two sequences of events and let ν N(A) and ν M(B) be the number of those A i and B j, respectively, that occur. Based on multivariate Lagrange interpolation, we give a method that yields linear bounds in terms of S k,t, k+tm on the distribution of the vector (ν N(A), ν M(B)). For the same value of m, several inequalities can be generated and all of them are best bounds for some values of S k,t. Known bivariate Bonferroni-type inequalities are reconstructed and new inequalities are generated, too.

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M. Giusti’s classification of the simple complete intersection singularities is characterized in terms of invariants. This is a basis for the implementation of a classifier in the computer algebra system Singular.

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Authors: Ruy Fabila-Monroy, Clemens Huemer and Dieter Mitsche

Let S be a set of n points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of S in its interior. We show that the expected number of empty non-convex four-gons with vertices from S is 12n 2logn + o(n 2logn) and the expected number of empty convex four-gons with vertices from S is Θ(n 2).

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This paper attempts an exposition of the connection between valuation theory and hyperstructure theory. In this regards, by considering the notion of totally ordered canonical hypergroup we define a hypervaluation of a hyperfield onto a totally ordered canonical hypergroup and obtain some related basic results.