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:Yuan Cao; Qingguo Li Pages: 599 - 624 Abstract: Abstract Let \(R=\mathbb{Z}_{4}[u]/ \langle u^k \rangle=\mathbb{Z}_{4}+u \mathbb{Z}_{4}+\ldots+u^{k-1}\mathbb{Z}_{4}\) ( \(u^{k}=0\) ), where k ≥ 2 is an positive integer. For any odd positive integer n, it is known that cyclic codes of length n over R are identified with ideals of the ring \(R[x]/\langle x^{n}-1\rangle\) . In this paper, an explicit representation for each cyclic code over R of length n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes of length n over R is obtained. Precisely, the dual code of each cyclic code and self-dual cyclic codes of length n over R are investigated. As an application, some good quasi-cyclic codes of length 7k and index k over ℤ4 are obtained from cyclic codes over R = ℤ4 [u] / 〈u k 〉 when k = 2, 3, 4. PubDate: 2017-09-01 DOI: 10.1007/s12095-016-0204-7 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:Zhiqiang Lin; Dongdai Lin; Dingyi Pei Pages: 431 - 443 Abstract: Abstract Linear Feedback Shift Registers (LFSRs) and Feedback with Carry Shift Registers (FCSRs) are two pseudo-random generators which are widely used in many cryptographic applications. The Ring representation of them has been proposed using a matrix approach. In this paper, we show how to construct Ring LFSRs and Ring FCSRs with low diffusion delay (close to the expected value \(\sqrt {n}\) ) when considering other hardware cryptographic criteria. PubDate: 2017-07-01 DOI: 10.1007/s12095-016-0183-8 Issue No:Vol. 9, No. 4 (2017)

Authors:Zhixiong Chen Pages: 445 - 458 Abstract: Abstract We define a family of quaternary sequences over the residue class ring modulo 4 of length pq, a product of two distinct odd primes, using the generalized cyclotomic classes modulo pq and calculate the discrete Fourier transform (DFT) of the sequences. The DFT helps us to determine the exact values of linear complexity and the trace representation of the sequences. PubDate: 2017-07-01 DOI: 10.1007/s12095-016-0185-6 Issue No:Vol. 9, No. 4 (2017)

Authors:Madhu Raka; Leetika Kathuria; Mokshi Goyal Pages: 459 - 473 Abstract: Abstract Let \(\mathcal {R}=\mathbb {F}_{p}+u\mathbb {F}_{p}+u^{2}\mathbb {F}_{p}+u^{3}\mathbb {F}_{p}\) with u 4 = u be a finite non-chain ring, where p is a prime congruent to 1 modulo 3. In this paper we study (1−2u 3)-constacyclic codes over the ring \(\mathcal {R}\) , their equivalence to cyclic codes and find their Gray images. To illustrate this, examples of (1−2u 3)-constacyclic codes of lengths 2 m for p = 7 and of lengths 3 m for p = 19 are given. We also discuss quadratic residue codes over the ring \(\mathcal {R}\) and their extensions. A Gray map from \(\mathcal {R}\) to \(\mathbb {F}_{p}^{4}\) is defined which preserves self duality and gives self-dual and formally self-dual codes over \(\mathbb {F}_{p}\) from extended quadratic residue codes. PubDate: 2017-07-01 DOI: 10.1007/s12095-016-0184-7 Issue No:Vol. 9, No. 4 (2017)

Authors:Pramod Kumar Kewat; Priti Kumari Pages: 475 - 499 Abstract: Abstract Let \(n_{1}=df+1\) and \(n_{2}=df^{\prime }+1\) be two distinct odd primes with positive integers \(d,\ f,\ f^{\prime }\) and \(\gcd (f,f^{\prime })=1\) . In this paper, we compute the linear complexity and the minimal polynomial of the two-prime Whiteman’s generalized cyclotomic sequence of order \(d=6\) over \(\text {GF}(q)\) , where \(q=p^{m}\) and p is an odd prime and m is an integer. We employ this sequence of order 6 to construct several classes of cyclic codes over \(\text {GF}(q)\) with length \(n_{1}n_{2}\) . We also obtain lower bounds on the minimum distance of these cyclic codes. PubDate: 2017-07-01 DOI: 10.1007/s12095-016-0191-8 Issue No:Vol. 9, No. 4 (2017)

Authors:László Mérai; Harald Niederreiter; Arne Winterhof Pages: 501 - 509 Abstract: Abstract The linear complexity is a measure for the unpredictability of a sequence over a finite field and thus for its suitability in cryptography. In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. We study the relationship between linear complexity and expansion complexity. In particular, we show that for purely periodic sequences both figures of merit provide essentially the same quality test for a sufficiently long part of the sequence. However, if we study shorter parts of the period or nonperiodic sequences, then we can show, roughly speaking, that the expansion complexity provides a stronger test. We demonstrate this by analyzing a sequence of binomial coefficients modulo p. Finally, we establish a probabilistic result on the behavior of the expansion complexity of random sequences over a finite field. PubDate: 2017-07-01 DOI: 10.1007/s12095-016-0189-2 Issue No:Vol. 9, No. 4 (2017)

Authors:Hongyu Han; Daiyuan Peng; Udaya Parampalli Pages: 511 - 522 Abstract: Abstract In quasi-synchronous frequency-hopping multiple-access systems where relative delays are restricted within a certain correlation zone, low-hit-zone frequency-hopping sequences (LHZ-FHSs) are commonly employed to minimize multiple-access interferences. In this paper, we present two classes of optimal LHZ-FHS sets with respect to the Peng-Fan-Lee bound, which are obtained from an m-sequence and its decimated sequence, respectively. The parameters of these LHZ-FHS sets are new and flexible. PubDate: 2017-07-01 DOI: 10.1007/s12095-016-0192-7 Issue No:Vol. 9, No. 4 (2017)

Authors:Santanu Sarkar; Prakash Dey; Avishek Adhikari; Subhamoy Maitra Pages: 523 - 543 Abstract: Abstract Differential Fault Attack (DFA) considers injection of faults and the most general set-up should take care of faults at random location and random time. Then one should be able to identify the exact location as well as the exact timing of the fault (including the multi bit ones) with the help of fault signatures. In this paper we solve the problem of DFA under a general frame-work, introducing the idea of probabilistic signatures. The method considers the Maximum Likelihood approach related to probability distributions. Our techniques subsume all the existing DFAs against the Grain family, MICKEY 2.0 and Trivium. In the process we provide improved fault attacks for all the versions of Grain family and also for MICKEY 2.0. Our generalized method successfully takes care of the cases where certain parts of the keystream bits are missing (this situation may arise for authentication purpose). In particular, we show that the unsolved problem of identifying the faults in random time for Grain 128a can be solved in this manner. Moreover, for MICKEY 2.0, our method not only provides improvement in fault identification probability but also reduces the required faults by 60 %, compared to the best known result. PubDate: 2017-07-01 DOI: 10.1007/s12095-016-0197-2 Issue No:Vol. 9, No. 4 (2017)

Authors:Qiuyan Wang; Kelan Ding; Dongdai Lin; Rui Xue Pages: 315 - 322 Abstract: Abstract Recently, linear codes with few weights have been constructed through defining sets. Results show that some optimal codes can be obtained if the defining sets were well chosen. In this paper, we investigate the linear codes constructed from the absolute trace function. It is shown that the constructed codes are binary linear codes with three weights. The dual codes of the proposed linear codes are also studied and proved to be optimal or almost optimal. PubDate: 2017-05-01 DOI: 10.1007/s12095-015-0180-3 Issue No:Vol. 9, No. 3 (2017)

Authors:Ziling Heng; Qin Yue Pages: 323 - 343 Abstract: Abstract Complete weight distribution can be used to study authentication codes and the Walsh transform of monomial functions over finite fields. Also, the Hamming weight distribution of a code can be obtained from its complete weight distribution. In this paper, we investigate the complete weight distributions of two classes of cyclic codes. We explicitly present the complete weight enumerators of the cyclic codes. Particularly, we partly solve an open problem proposed in Luo and Feng (IEEE Trans. Inf. Theory 54(12), 5345–5353 (2008)). PubDate: 2017-05-01 DOI: 10.1007/s12095-015-0177-y Issue No:Vol. 9, No. 3 (2017)

Authors:Jian Liu; Sihem Mesnager; Lusheng Chen Pages: 345 - 361 Abstract: Abstract For multi-output Boolean functions (also called S-boxes), various measures of nonlinearity have been widely discussed in the literature but many problems are left open in this topic. The purpose of this paper is to present a new approach to estimating the nonlinearity of S-boxes. A more fine-grained view on the notion of nonlinearity of S-boxes is presented and new connections to some linear codes are established. More precisely, we mainly study the nonlinearity indicator (denoted by \(\mathcal {N}_{\mathrm {v}}\) ) for S-boxes from a coding theory point of view. Such a cryptographic parameter \(\mathcal {N}_{\mathrm {v}}\) is more related to best affine approximation attacks on stream ciphers. We establish a direct link between \(\mathcal {N}_{\mathrm {v}}\) and the minimum distance of the corresponding linear code. We exploit that connection to derive the first general lower bounds on \(\mathcal {N}_{\mathrm {v}}\) of non-affine functions from \(\mathbb {F}_{2^{n}}\) to \(\mathbb {F}_{2^{m}}\) for m dividing n. Furthermore, we show that \(\mathcal {N}_{\mathrm {v}}\) can be determined directly by the weight distribution of the corresponding linear code. PubDate: 2017-05-01 DOI: 10.1007/s12095-015-0176-z Issue No:Vol. 9, No. 3 (2017)

Authors:Jie Peng; Chik How Tan Pages: 363 - 378 Abstract: Abstract Permutations over \(\mathbb {F}_{2^{2k}}\) with low differential uniformity, high algebraic degree and high nonlinearity are of great cryptographic importance since they can be chosen as the substitution boxes (S-boxes) for many block ciphers with SPN (Substitution Permutation Network) structure. A well known example is that the S-box of the famous Advanced Encryption Standard (AES) is derived from the inverse function on \(\mathbb {F}_{2^{8}}\) , which has been proved to be a differentially 4-uniform permutation with the optimal algebraic degree and known best nonlinearity. Recently, Zha et al. proposed two constructions of differentially 4-uniform permutations over \(\mathbb {F}_{2^{2k}}\) , say G t and G s, t with T r(s −1) = 1, by applying affine transformations to the inverse function on some subfields of \(\mathbb {F}_{2^{2k}}\) (Zha et al. Finite Fields Appl. 25, 64–78, 2014). In this paper, we generalize their method by applying other types of EA (extended affine) equivalent transformations to the inverse function on some subfields of \(\mathbb {F}_{2^{2k}}\) and present two new constructions of differentially 4-uniform permutations, say F α and F β, α with T r(β −1) = 1. Furthermore, we prove that all the functions G t with different t are CCZ (Carlet-Charpin-Zinoviev) equivalent to our subclass F 0, while all the functions G s, t with different t are CCZ-equivalent to our subclass F s,0. In addition, both our two constructions give many new CCZ-inequivalent classes of such functions, as checked by computer in small numbers of variables. Moreover, all these newly constructed permutations are proved to have the optimal algebraic degree and high nonlinearity. PubDate: 2017-05-01 DOI: 10.1007/s12095-016-0181-x Issue No:Vol. 9, No. 3 (2017)

Authors:Minquan Cheng; Jing Jiang; Xiaohu Tang Pages: 397 - 405 Abstract: Abstract Multimedia fingerprinting is an effective technique to trace the sources of pirate copies of copyrighted multimedia information. Separable codes can be used to construct fingerprints resistant to the averaging collusion attack on multimedia contents. In this paper, we first show an equivalent condition of a \(\overline {2}\) -SC (4,M,q), and then construct two infinite families of \(\overline {2}\) -SCs of length 4, one of which is asymptotically optimal. PubDate: 2017-05-01 DOI: 10.1007/s12095-016-0182-9 Issue No:Vol. 9, No. 3 (2017)

Authors:Miao Liang; Lijun Ji; Jingcai Zhang Pages: 407 - 430 Abstract: Abstract Optimal restricted strong partially balanced t-design can be used to construct splitting authentication codes which achieve combinatorial lower bounds or information-theoretic lower bounds. In this paper, we investigate the existence of optimal restricted strong partially balanced 2-designs ORSPBD (v, k×c,1), and show that there exists an ORSPBD (v,2×c,1) for any positive integer v≡ v 0 (mod 2c 2) and \(v_{0}\in \{1\leq x\leq 2c^{2}:\ \gcd (x,c)=1\ \text {or} \ \gcd (x,c)=c \} \setminus \) \(\{c^{2}+1\leq x\leq (c+1)^{2} :\gcd (x,c)=1\ \text {and}\ \gcd (x,2)=2\}\) . Furthermore, we determine the existence of an ORSPBD (v,k×c,1) for any integer v≥k c with (k,c)=(2,4), (2,5), (3,2) or for any even integer v≥k c with (k,c)=(4,2). As their applications, we obtain six new infinite classes of 2-fold optimal or perfect c-splitting authentication codes. PubDate: 2017-05-01 DOI: 10.1007/s12095-015-0179-9 Issue No:Vol. 9, No. 3 (2017)