Authors:László Mérai Pages: 193 - 203 Abstract: Let p be a prime and let \(\mathbf {E}\) be an elliptic curve defined over the finite field \(\mathbb {F}_p\) of p elements. For a point \(G\in \mathbf {E}(\mathbb {F}_p)\) the elliptic curve congruential generator (with respect to the first coordinate) is a sequence \((x_n)\) defined by the relation \(x_n=x(W_n)=x(W_{n-1}\oplus G)=x(nG\oplus W_0)\) , \(n=1,2,\ldots \) , where \(\oplus \) denotes the group operation in \(\mathbf {E}\) and \(W_0\) is an initial point. In this paper, we show that if some consecutive elements of the sequence \((x_n)\) are given as integers, then one can compute in polynomial time an elliptic curve congruential generator (where the curve possibly defined over the rationals or over a residue ring) such that the generated sequence is identical to \((x_n)\) in the revealed segment. It turns out that in practice, all the secret parameters, and thus the whole sequence \((x_n)\) , can be computed from eight consecutive elements, even if the prime and the elliptic curve are private. PubDate: 2017-06-01 DOI: 10.1007/s00200-016-0303-x Issue No:Vol. 28, No. 3 (2017)

Authors:Alexander Bors Pages: 205 - 214 Abstract: Aiming at a better understanding of finite groups as finite dynamical systems, we show that by a version of Fitting’s Lemma for groups, each state space of an endomorphism of a finite group is a graph tensor product of a finite directed 1-tree whose cycle is a loop with a disjoint union of cycles, generalizing results of Hernández-Toledo on linear finite dynamical systems, and we fully characterize the possible forms of state spaces of nilpotent endomorphisms via their “ramification behavior”. Finally, as an application, we will count the isomorphism types of state spaces of endomorphisms of finite cyclic groups in general, extending results of Hernández-Toledo on primary cyclic groups of odd order. PubDate: 2017-06-01 DOI: 10.1007/s00200-016-0304-9 Issue No:Vol. 28, No. 3 (2017)

Authors:Dabin Zheng; Zhen Chen Pages: 215 - 223 Abstract: This note presents two classes of permutation polynomials of the form \((x^{p^m}-x+\delta )^s+L(x)\) over the finite fields \({{\mathbb {F}}}_{p^{2m}}\) as a supplement of the recent works of Zha, Hu and Li, Helleseth and Tang. PubDate: 2017-06-01 DOI: 10.1007/s00200-016-0305-8 Issue No:Vol. 28, No. 3 (2017)

Authors:B. Panbehkar; H. Doostie Pages: 225 - 235 Abstract: For a finitely generated automatic semigroup \(S=\langle A\rangle \) we define a semigroup \(L_S\) of languages concerning the automatic structure of S, and study the automaticity of \(L_S\) . Also we investigate the natural question “when S is isomorphic to \(L_S\) ?”. Finally, we attempt to verify the equation \(L_S\cup L_T=L_{S\cup T}\) for two non-monoid semigroups \((S, *)\) and (T, o). PubDate: 2017-06-01 DOI: 10.1007/s00200-016-0306-7 Issue No:Vol. 28, No. 3 (2017)

Authors:Thierry Mefenza; Damien Vergnaud Pages: 237 - 255 Abstract: We prove lower bounds on the degree of polynomials interpolating the Naor–Reingold pseudo-random function over a finite field and over the group of points on an elliptic curve over a finite field. PubDate: 2017-06-01 DOI: 10.1007/s00200-016-0309-4 Issue No:Vol. 28, No. 3 (2017)

Authors:Yogesh Kumar; P. R. Mishra; N. Rajesh Pillai; R. K. Sharma Pages: 257 - 279 Abstract: In this paper, we explore further the non-linearity and affine equivalence as proposed by Mishra et al. (Non-linearity and affine equivalence of permutations. 2014. http://eprint.iacr.org/2014/974.pdf). We propose an efficient algorithm in order to compute affine equivalent permutation(s) of a given permutation of length n, of complexity \(O(n^4)\) in worst case and \(O(n^2)\) in best case. Also in the affirmative in a special case \(n = p\) , prime, it is of complexity \(O(n^3)\) . We also propose an upper bound of non-linearity of permutation(s) whose length satisfies a special condition. Further, behaviour of non-linearity on direct sum and skew sum of permutation has been analysed. Also the distance of an affine permutation from the other affine permutations has also been studied. The cryptographic implication of this work is on permutation based stream ciphers like RC4 and its variants. In this paper, we have applied this study on RC4 cipher. The analysis shows that increasing the key size for RC4 does not mean that increase in the security or saturation after a limit but security may falls as key size increases. PubDate: 2017-06-01 DOI: 10.1007/s00200-016-0307-6 Issue No:Vol. 28, No. 3 (2017)

Authors:Jiang Weng; Yunqi Dou; Chuangui Ma Pages: 99 - 108 Abstract: The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group \({\mathbb {G}}=\langle P\rangle \) of order p and some elements \(P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}\) , an attacker can recover \(\alpha \in {\mathbb {Z}}_p^*\) in the case of \(d (p\pm 1)\) with running time of \({\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)\) group operations by using \({\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})\) storage ( \(i=\frac{1}{2}\) or 1 for \(d (p-1)\) case or \(d (p+1)\) case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points \(P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}\) and multiplicative cyclic group \(K=\langle k \rangle \) are given, where d is a prime, \(\varphi (d)\) is the order of K and \(\varphi \) is the Euler totient function, the secret key \(\alpha \in {\mathbb {Z}}_p^*\) can be solved in \({\mathcal {O}}(\sqrt{(p-1)/d}+d)\) group operations by using \({\mathcal {O}}(\sqrt{(p-1)/d})\) storage. PubDate: 2017-03-01 DOI: 10.1007/s00200-016-0301-z Issue No:Vol. 28, No. 2 (2017)

Authors:Palash Sarkar; Shashank Singh Pages: 109 - 130 Abstract: Nagao proposed a decomposition method for divisors of hyperelliptic curves defined over a field \({\mathbb {F}}_{q^n}\) with \(n\ge 2\) . Joux and Vitse later proposed a variant which provided relations among the factor basis elements. Both Nagao’s and the Joux–Vitse methods require solving a multi-variate system of polynomial equations. In this work, we revisit Nagao’s approach with the idea of avoiding the requirement of solving a multi-variate system. While this cannot be done in general, we are able to identify special cases for which this is indeed possible. Our main result is for curves \(C:y^2=f(x)\) of genus g defined over \({\mathbb {F}}_{q^2}\) having characteristic >2. If there is no restriction on f(x), we show that it is possible to obtain a relation in \((4g+4)!\) trials. The number of trials, though high, quantifies the computation effort needed to obtain a relation. This is in contrast to the methods of Nagao and Joux–Vitse which are based on solving systems of polynomial equations, for which the computation effort is hard to precisely quantify. The new method combines well with a sieving technique proposed by Joux and Vitse. If f(x) has a special form, then the number of trials can be significantly lower. For example, if f(x) has at most g consecutive coefficients which are in \({\mathbb {F}}_{q^2}\) while the rest are in \({\mathbb {F}}_q\) , then we show that it is possible to obtain a single relation in about \((2g+3)!\) trials. Our implementation of the resulting algorithm provides examples of factor basis relations for \(g=5\) and \(g=6\) . To the best of our knowledge, none of the previous methods known in the literature can provide such relations faster than our method. Other than obtaining such decompositions, we also explore the applicability of our approach for \(n>2\) and for binary characteristic fields. PubDate: 2017-03-01 DOI: 10.1007/s00200-016-0299-2 Issue No:Vol. 28, No. 2 (2017)

Authors:Jian Gao; Fang-Wei Fu; Yun Gao Pages: 131 - 153 Abstract: Some classes of linear codes over the ring \(\mathbb {Z}_4+v\mathbb {Z}_4\) with \(v^2=v\) are considered. Construction of Euclidean formally self-dual codes and unimodular complex lattices from self-dual codes over \(\mathbb {Z}_4+v\mathbb {Z}_4\) are studied. Structural properties of cyclic codes and quadratic residue codes are also considered. Finally, some good and new \(\mathbb {Z}_4\) -linear codes are constructed from linear codes over \(\mathbb {Z}_4+v\mathbb {Z}_4\) . PubDate: 2017-03-01 DOI: 10.1007/s00200-016-0300-0 Issue No:Vol. 28, No. 2 (2017)

Authors:Guangkui Xu; Xiwang Cao; Shanding Xu Pages: 155 - 176 Abstract: In this paper, several classes of Boolean functions with few Walsh transform values, including bent, semi-bent and five-valued functions, are obtained by adding the product of two or three linear functions to some known bent functions. Numerical results show that the proposed class contains cubic bent functions that are affinely inequivalent to all known quadratic ones. PubDate: 2017-03-01 DOI: 10.1007/s00200-016-0298-3 Issue No:Vol. 28, No. 2 (2017)

Authors:Haibo Hong; Licheng Wang; Haseeb Ahmad; Jun Shao; Yixian Yang Pages: 177 - 192 Abstract: As a special type of factorization of finite groups, logarithmic signature (LS) is used as the main component of cryptographic keys for secret key cryptosystems such as PGM and public key cryptosystems like \(MST_1\) , \(MST_2\) and \(MST_3\) . An LS with the shortest length, called a minimal logarithmic signature (MLS), is even desirable for cryptographic applications. The MLS conjecture states that every finite simple group has an MLS. Recently, the conjecture has been shown to be true for general linear groups \(GL_n(q)\) , special linear groups \(SL_n(q)\) , and symplectic groups \(Sp_n(q)\) with q a power of primes and for orthogonal groups \(O_n(q)\) with q a power of 2. In this paper, we present new constructions of minimal logarithmic signatures for the orthogonal group \(O_n(q)\) and \(SO_n(q)\) with q a power of an odd prime. Furthermore, we give constructions of MLSs for a type of classical groups—the projective commutator subgroup \(P{\varOmega }_n(q)\) . PubDate: 2017-03-01 DOI: 10.1007/s00200-016-0302-y Issue No:Vol. 28, No. 2 (2017)

Authors:Jacques Patarin Abstract: “Mirror Theory” is the theory that evaluates the number of solutions of affine systems of equalities \(({=})\) and non equalities ( \(\ne \) ) in finite groups. It is deeply related to the security and attacks of many generic cryptographic secret key schemes, for example random Feistel schemes (balanced or unbalanced), Misty schemes, Xor of two pseudo-random bijections to generate a pseudo-random function etc. In this paper we will assume that the groups are abelian. Most of time in cryptography the group is \(((\mathbb {Z}/2\mathbb {Z})^n, \oplus )\) and we will concentrate this paper on these cases. We will present here general definitions, some theorems, and many examples and computer simulations. PubDate: 2017-05-20 DOI: 10.1007/s00200-017-0326-y

Authors:Nuh Aydin; Nicholas Connolly; John Murphree Abstract: Explicit construction of linear codes with best possible parameters is one of the major problems in coding theory. Among all alphabets of interest, the binary alphabet is the most important one. In this work we use a comprehensive search strategy to find new binary linear codes in the well-known and intensively studied class of quasi-cyclic (QC) codes. We also introduce a generalization of an augmentation algorithm to obtain further new codes from those QC codes. Also applying the standard methods of obtaining new codes from existing codes, such as puncturing, extending and shortening, we have found a total of 62 new binary linear codes. PubDate: 2017-05-17 DOI: 10.1007/s00200-017-0327-x

Authors:Haode Yan Abstract: BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. Recently, several classes of BCH codes with length \(n=q^m-1\) and designed distances \(\delta =(q-1)q^{m-1}-1-q^{\lfloor (m-1)/2\rfloor }\) and \(\delta =(q-1)q^{m-1}-1-q^{\lfloor (m+1)/2\rfloor }\) were widely studied, where \(m\ge 4\) is an integer. In this paper, we consider the case \(m=3\) . The weight distribution of a class of primitive BCH codes with designed distance \(q^3-q^2-q-2\) is determined, which solves an open problem put forward in Ding et al. (Finite Fields Appl 45:237–263, 2017). PubDate: 2017-05-16 DOI: 10.1007/s00200-017-0320-4

Authors:Xiaoni Du; Yunqi Wan Abstract: Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime power q, we present a class of linear codes over finite fields \(F_q\) with quadratic forms via a general construction and then determine the explicit complete weight enumerators of these linear codes. Our construction covers some related ones via quadratic form functions and the linear codes may have applications in cryptography and secret sharing schemes. PubDate: 2017-05-15 DOI: 10.1007/s00200-017-0319-x

Authors:Serhii Dyshko Abstract: The minimal code length for which there exists an unextendable Hamming isometry of a linear code defined over a matrix module alphabet is found. An extension theorem for MDS codes over module alphabets is proved. An extension theorem for the case of MDS group codes is observed. PubDate: 2017-05-15 DOI: 10.1007/s00200-017-0324-0

Authors:Gerardo Vega Abstract: The purpose of this work is to use an already known identity among the weight enumerator polynomials, in order to present an improved method for determining the weight distribution of a family of q-ary reducible cyclic codes, that generalize, in an easier way, the results in Yu and Liu (Des Codes Cryptogr 78:731–745, 2016). PubDate: 2017-04-18 DOI: 10.1007/s00200-017-0318-y

Authors:Krzysztof Ziemiański Abstract: The spaces of directed paths on the geometric realizations of pre-cubical sets, called also \(\square \) -sets, can be interpreted as the spaces of possible executions of Higher Dimensional Automata, which are models for concurrent computations. In this paper we construct, for a sufficiently good pre-cubical set K, a CW-complex \(W(K)_v^w\) that is homotopy equivalent to the space of directed paths between given vertices v, w of K. This construction is functorial with respect to K, and minimal among all functorial constructions. Furthermore, explicit formulas for incidence numbers of the cells of \(W(K)_v^w\) are provided. PubDate: 2017-03-02 DOI: 10.1007/s00200-017-0316-0

Authors:Zohreh Rajabi; Kazem Khashyarmanesh Abstract: Cyclic codes are an important class of linear codes. The objectives of this paper are to earn and extend earlier results over cyclic codes from some monomials. In fact, we determine the dimension and the generator polynomial of the code \({\mathcal {C}}_s\) defined by the monomial \(f(x)=x^{\frac{p^h+1}{2}}\) over \({\mathrm {GF}}(p^m)\) , where p is an odd prime and h is an integer. Also, we provide some answers for Open Problems 5.26 and 5.30 in Ding (SIAM J Discrete Math 27:1977–1994, 2013). Moreover, we study the code \({\mathcal {C}}_s\) defined by the monomial \(f(x)=x^{\frac{q^h-1}{q-1}}\) over \(\mathrm {GF}(q^m)\) , where h is an integer, without any restriction on h (see Section 5.3 in the above mentioned paper). PubDate: 2017-03-01 DOI: 10.1007/s00200-017-0317-z

Authors:Yun Gao; Jian Gao; Tingting Wu; Fang-Wei Fu Abstract: In this paper, we study 1-generator quasi-cyclic and generalized quasi-cyclic codes over the ring \(R=\frac{{{\mathbb {Z}_4}[u]}}{{\left\langle {{u^2} - 1} \right\rangle }}\) . We determine the structure of the generators and the minimal generating sets of 1-generator QC and GQC codes. We also give a lower bound for the minimum distance of free 1-generator quasi-cyclic and generalized quasi-cyclic codes over this ring, respectively. Furthermore, some new \(\mathbb {Z}_4\) -linear codes via the Gray map which have better parameters than the best known \(\mathbb {Z}_4\) -linear codes are presented. PubDate: 2017-02-20 DOI: 10.1007/s00200-017-0315-1