Authors:Şaban Alaca; Goldwyn Millar Pages: 665 - 682 Abstract: Abstract Binary sequences with good autocorrelation properties and large linear complexity are useful in stream cipher cryptography. The Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences have nearly optimal autocorrelation. However, the problem of determining the linear complexity of the SLCE sequences is still open. It is well known that one can gain insight into the linear complexity of a sequence if one can say something about the divisors of the gcd of a certain pair of polynomials associated with the sequence. Helleseth and Yang (IEEE Trans. Inf. Theory 49(6), 1548–1552 2002), Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) and Meidl and Winterhof (Des. Codes Crypt. 8, 159–178 2006) were able to obtain some results of this type for the SLCE sequences. Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) mention that it would be nice to obtain more such results. We derive new divisibility results for the SLCE sequences in this paper. Our approach is to exploit the fact that character values associated with the SLCE sequences can be expressed in terms of a certain type of Jacobi sum. By making use of known evaluations of Gauss and Jacobi sums in the “pure” and “small index” cases, we are able to obtain new insight into the linear complexity of the SLCE sequences. PubDate: 2017-11-01 DOI: 10.1007/s12095-016-0208-3 Issue No:Vol. 9, No. 6 (2017)

Authors:Vladimir Edemskiy; Xiaoni Du Pages: 683 - 691 Abstract: Abstract Based on the generalized cyclotomy theory, we design some classes of sequences with high linear complexity over the finite fields. First, we construct a new class of sequence from some generalized cyclotomic sequences of different orders with different prime powers period. Then we obtain the discrete Fourier transform, defining pairs and the linear complexity of the new sequences. Finally, we study the linear complexity of a special class of q−ary (q prime) sequences. PubDate: 2017-11-01 DOI: 10.1007/s12095-016-0209-2 Issue No:Vol. 9, No. 6 (2017)

Authors:Nian Li; Tor Helleseth Pages: 693 - 705 Abstract: Abstract Motivated by recent results on the constructions of permutation polynomials with few terms over the finite field \({\mathbb F}_{2^n}\) , where n is a positive even integer, we focus on the construction of permutation trinomials over \({\mathbb F}_{2^n}\) from Niho exponents. As a consequence, several new classes of permutation trinomials over \({\mathbb F}_{2^n}\) are constructed from Niho exponents based on some subtle manipulation of solving equations with low degrees over finite fields. PubDate: 2017-11-01 DOI: 10.1007/s12095-016-0210-9 Issue No:Vol. 9, No. 6 (2017)

Authors:Limengnan Zhou; Daiyuan Peng; Hongbin Liang; Changyuan Wang; Hongyu Han Pages: 707 - 728 Abstract: Abstract In a quasi-synchronous frequency-hopping multiple-access system, relative time delay between different users within a zone around the origin can be allowed. Therefore, frequency-hopping sequence (FHS) sets with low-hit-zone (LHZ) have attracted great interest of many related scholars. Moreover, on account of the limited synchronous time or hardware complexity, the periodic partial Hamming correlation (PPHC) plays a major role in determining the synchronization performance. In this paper, we first present three new generalized methods to construct LHZ-FHS sets via Cartesian product. Meanwhile, we pay our attention to the maximum periodic Hamming correlation (PHC) of the constructed LHZ-FHS sets in the first generalized method, and to the maximum PPHC of the constructed LHZ-FHS sets in the rest generalized methods. In addition, we also introduce five new classes of optimal LHZ-FHS sets based on these three generalized methods. PubDate: 2017-11-01 DOI: 10.1007/s12095-017-0211-3 Issue No:Vol. 9, No. 6 (2017)

Authors:J. Wolfmann Pages: 729 - 736 Abstract: Abstract We introduce infinite sequences of Boolean functions whose terms all are bent functions or all are near-bent functions. PubDate: 2017-11-01 DOI: 10.1007/s12095-017-0212-2 Issue No:Vol. 9, No. 6 (2017)

Authors:Heiko Dietrich; Nathan Jolly Pages: 737 - 748 Abstract: Abstract Arrays with low autocorrelation are widely sought in applications; important examples are arrays whose periodic autocorrelation is zero for all nontrivial cyclic shifts, so-called perfect arrays. In 2001, Arasu and de Launey defined almost perfect arrays: these have size 2u×v and autocorrelation arrays with only two nonzero entries, namely 2u v and −2u v in positions (0,0) and (u,0), respectively. In this paper we present a new class of arrays with low autocorrelation: for an integer n≥1, we call an array n-perfect if it has size n u×v and if its autocorrelation array has only n nonzero entries, namely n u v λ i in position (i u,0) for i=0,1,…,n−1, where λ is a primitive n-th root of unity. Thus, an array is 1-perfect (2-perfect) if and only if it is (almost) perfect. We give examples and describe a recursive construction of families of n-perfect arrays of increasing size. PubDate: 2017-11-01 DOI: 10.1007/s12095-017-0214-0 Issue No:Vol. 9, No. 6 (2017)

Authors:Eric Férard Pages: 749 - 767 Abstract: Abstract Let t be an integer ≥ 3 such that t ≡ 1 mod 4. The absolute irreducibility of the polynomial \(\phi _{t}(x, y) = \frac {x^{t} + y^{t} + 1 + (x + y + 1)^{t}}{(x + y)(x + 1)(y + 1)}\) (over \(\mathbb {F}_{2}\) ) plays an important role in the study of APN functions. We prove that this polynomial is absolutely irreducible under the assumptions that the largest odd integer which divides t − 1 is large enough and can not be written in a specific form. PubDate: 2017-11-01 DOI: 10.1007/s12095-017-0213-1 Issue No:Vol. 9, No. 6 (2017)

Authors:Xianfang Wang; Jian Gao; Fang-Wei Fu Pages: 545 - 562 Abstract: Abstract In this paper, we give the complete weight enumerators of two classes of linear codes over the finite field \(\mathbb {F}_{p}\) , where p is a prime. These linear codes are the torsion codes of MacDonald codes over the finite non-chain ring \(\mathbb {F}_{p}+v\mathbb {F}_{p}\) , where v 2 = v. We also employ these linear codes to construct systematic authentication codes with new parameters. PubDate: 2017-09-01 DOI: 10.1007/s12095-016-0198-1 Issue No:Vol. 9, No. 5 (2017)

Authors:Shuai Xue; Wen-Feng Qi; Xiao-Yuan Yang Pages: 563 - 580 Abstract: Abstract In this paper, the best linear approximations of addition modulo 2 n are studied. Let x = (x n−1, x n−2,…,x 0) and y = (y n−1, y n−2,…,y 0) be any two n-bit integers, and let z = x + y (mod 2 n ). Firstly, all the correlations of a single bit z i approximated by x j ’s and y j ’s (0 ≤ i, j ≤ n − 1) are characterized, and similar results are obtained for the linear approximation of the xoring of the neighboring bits of z i ’s. Then the maximum correlations and the best linear approximations are presented when these z j ’s (0 ≤ j ≤ n − 1) are xored in any given means. PubDate: 2017-09-01 DOI: 10.1007/s12095-016-0203-8 Issue No:Vol. 9, No. 5 (2017)

Authors:Thalia M. Laing; Keith M. Martin; Maura B. Paterson; Douglas R. Stinson Pages: 581 - 597 Abstract: Abstract A localised multisecret sharing scheme is a multisecret sharing scheme for an ordered set of players in which players in the smallest sets who are authorised to access secrets are close together in the underlying ordering. We define threshold versions of localised multisecret sharing schemes, we provide lower bounds on the share size of perfect localised multisecret sharing schemes in an information theoretic setting, and we give explicit constructions of schemes to show that these bounds are tight. We then analyse a range of approaches to relaxing the model that provide trade-offs between the share size and the level of security guarantees provided by the scheme, in order to permit the construction of schemes with smaller shares. We show how these techniques can be used in the context of an application to key distribution for RFID-based supply-chain management motivated by the proposal of Juels, Pappu and Parno from USENIX 2008. PubDate: 2017-09-01 DOI: 10.1007/s12095-016-0202-9 Issue No:Vol. 9, No. 5 (2017)

Authors:Xianfang Wang; Can Xiang; Fang-Wei Fu Pages: 625 - 635 Abstract: Abstract In this paper, we devise ideal and probabilistic secret sharing schemes for two kinds of compartmented access structures. The first one is a compartmented access structures with hierarchical compartments. The second one is the compartmented access structures with strictly lower bounds. We propose ideal and probabilistic schemes for these two compartmented access structures by using the idea of bivariate interpolation. PubDate: 2017-09-01 DOI: 10.1007/s12095-016-0205-6 Issue No:Vol. 9, No. 5 (2017)

Authors:Minjia Shi; Rongsheng Wu; Yan Liu; Patrick Solé Pages: 637 - 646 Abstract: Abstract We construct an infinite family of three-Lee-weight codes of dimension 2m, where m is singly-even, over the ring \(\mathbb {F}_{p}+u\mathbb {F}_{p}\) with u 2=0. These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain an infinite family of abelian p-ary three-weight codes. When m is odd, and p≡3 (mod 4), we obtain an infinite family of two-weight codes which meets the Griesmer bound with equality. An application to secret sharing schemes is given. PubDate: 2017-09-01 DOI: 10.1007/s12095-016-0206-5 Issue No:Vol. 9, No. 5 (2017)

Authors:Chunming Tang; Yanfeng Qi Pages: 647 - 664 Abstract: Abstract This paper is devoted to the characterization of hyper-bent functions. Several classes of hyper-bent functions have been studied, such as Charpin and Gong’s family \(\sum \limits _{r\in R}\text {Tr}_{1}^{n} (a_{r}x^{r(2^{m}-1)})\) and Mesnager’s family \(\sum \limits _{r\in R}\text {Tr}_{1}^{n}(a_{r}x^{r(2^{m}-1)}) +\text {Tr}_{1}^{2}(bx^{\frac {2^{n}-1}{3}})\) . In this paper, we generalize these results by considering the following class of Boolean functions over \(\mathbb {F}_{2^{n}}\) : $$\sum\limits_{r\in R}\sum\limits_{i=0}^{2}T{r^{n}_{1}}(a_{r,i} x^{r(2^{m}-1)+\frac{2^{n}-1}{3}i}) +T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), $$ where \(n=2m\) , m is odd, \(b\in \mathbb {F}_{4}\) , and \(a_{r,i}\in \mathbb {F}_{2^{n}}\) . With the restriction of \(a_{r,i}\in \mathbb {F}_{2^{m}}\) , we present a characterization of hyper-bentness of these functions in terms of crucial exponential sums. For some special cases, we provide explicit characterizations for some hyper-bent functions in terms of Kloosterman sums and cubic sums. Finally, we explain how our results on binomial, trinomial and quadrinomial hyper-bent functions can be generalized to the general case where the coefficients \(a_{r,i}\) belong to the whole field \(\mathbb {F}_{2^{n}}\) . PubDate: 2017-09-01 DOI: 10.1007/s12095-016-0207-4 Issue No:Vol. 9, No. 5 (2017)

Authors:Tao Bai; Yongbo Xia Abstract: Abstract Permutation polynomials over finite fields are an interesting subject due to their important applications in the areas of mathematics and engineering. In this paper, we investigate the trinomial f(x) = x (p−1)q+1 + x p q − x q+(p−1) over the finite field \(\mathbb {F}_{q^{2}}\) , where p is an odd prime and q = p k with k being a positive integer. It is shown that when p = 3 or 5, f(x) is a permutation trinomial of \(\mathbb {F}_{q^{2}}\) if and only if k is even. This property is also true for a more general class of polynomials g(x) = x (q+1)l+(p−1)q+1 + x (q+1)l + p q − x (q+1)l + q+(p−1), where l is a nonnegative integer and \(\gcd (2l+p,q-1)=1\) . Moreover, we also show that for p = 5 the permutation trinomials f(x) proposed here are new in the sense that they are not multiplicative equivalent to previously known ones of similar form. PubDate: 2017-10-17 DOI: 10.1007/s12095-017-0263-4

Authors:Wilfried Meidl; Ísabel Pirsic Abstract: Abstract Depending on the parity of n and the regularity of a bent function f from \({{\mathbb F}_{p}^{n}}\) to \({\mathbb F}_{p}\) , f can be affine on a subspace of dimension at most n/2, (n − 1)/2 or n/2 − 1. We point out that many p-ary bent functions take on this bound, and it seems not easy to find examples for which one can show a different behaviour. This resembles the situation for Boolean bent functions of which many are (weakly) n/2-normal, i.e. affine on a n/2-dimensional subspace. However applying an algorithm by Canteaut et.al., some Boolean bent functions were shown to be not n/2-normal. We develop an algorithm for testing normality for functions from \({{\mathbb F}_{p}^{n}}\) to \({\mathbb F}_{p}\) . Applying the algorithm, for some bent functions in small dimension we show that they do not take on the bound on normality. Applying direct sum of functions this yields bent functions with this property in infinitely many dimensions. PubDate: 2017-10-17 DOI: 10.1007/s12095-017-0259-0

Authors:László Mérai; Arne Winterhof Abstract: Abstract We study the pseudorandomness of automatic sequences in terms of well-distribution and correlation measure of order 2. We detect non-random behavior which can be derived either from the functional equations satisfied by their generating functions or from their generating finite automatons, respectively. PubDate: 2017-10-13 DOI: 10.1007/s12095-017-0260-7

Authors:Subhabrata Samajder; Palash Sarkar Abstract: Abstract This work considers statistical analysis of attacks on block cyphers using several linear approximations. A general and unified approach is adopted. To this end, the general key randomisation hypotheses for multidimensional and multiple linear cryptanalysis are introduced. Expressions for the success probability in terms of the data complexity and the advantage are obtained using the general key randomisation hypotheses for both multidimensional and multiple linear cryptanalysis and under the settings where the plaintexts are sampled with or without replacement. Particularising to standard/adjusted key randomisation hypotheses gives rise to success probabilities in 16 different cases out of which in only five cases expressions for success probabilities have been previously reported. Even in these five cases, the expressions for success probabilities that we obtain are more general than what was previously obtained. A crucial step in the analysis is the derivation of the distributions of the underlying test statistics. Whilst we carry out the analysis formally to the extent possible, there are certain inherently heuristic assumptions that need to be made. In contrast to previous works which have implicitly made such assumptions, we carefully highlight these and discuss why they are unavoidable. Finally, we provide a complete characterisation of the dependence of the success probability on the data complexity. PubDate: 2017-09-27 DOI: 10.1007/s12095-017-0257-2

Authors:Lucky Galvez; Jon-Lark Kim; Nari Lee; Young Gun Roe; Byung-Sun Won Abstract: Abstract A linear code with a complementary dual (or An LCD code) is defined to be a linear code C whose dual code C ⊥ satisfies C ∩ C ⊥= \(\left \{ \mathbf {0}\right \} \) . Let L D (n, k) denote the maximum of possible values of d among [n, k, d] binary LCD codes. We give the exact values of L D (n, k) for k = 2 for all n and some bounds on L D (n, k) for other cases. From our results and some direct search we obtain a complete table for the exact values of L D (n, k) for 1 ≤ k ≤ n ≤ 12. As a consequence, we also derive bounds on the dimensions of LCD codes with fixed lengths and minimum distances. PubDate: 2017-09-26 DOI: 10.1007/s12095-017-0258-1

Authors:Aleksandr Tuxanidy; Qiang Wang Abstract: Abstract We introduce a new type of sequences using the sum of coefficients of characteristic polynomials for elements (in particular, primitive elements) in a finite field. These sequences are nonlinear filtering sequences of the well-known m-sequences. We show that they have large linear complexity and large period. We also provide some examples of such binary sequences with good autocorrelation values. PubDate: 2017-09-23 DOI: 10.1007/s12095-017-0256-3

Authors:Houda Ferradi; Rémi Géraud; Diana Maimuţ; David Naccache; Hang Zhou Abstract: Abstract This paper describes a new multiplication algorithm, particularly suited to lightweight microprocessors when one of the operands is known in advance. The method uses backtracking to find a multiplication-friendly encoding of the operand known in advance. A 68hc05 microprocessor implementation shows that the new algorithm indeed yields a twofold speed improvement over classical multiplication for 128-byte numbers. PubDate: 2017-09-05 DOI: 10.1007/s12095-017-0254-5