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:Florian Luca; Pantelimon Stănică Pages: 1 - 10 Abstract: In this paper we find an exact formula for the number of affine equivalence classes under permutations for binary polynomials degree \(d=6\) invariant under the cyclic group (also, called monomial rotation symmetric), for a prime number of variables; this extends previous work for \(2\le d\le 5\) . PubDate: 2017-01-01 DOI: 10.1007/s00200-016-0294-7 Issue No:Vol. 28, No. 1 (2017)

Authors:Chengju Li; Qin Yue; Fang-Wei Fu Pages: 11 - 30 Abstract: Linear codes constructed from defining sets have been extensively studied and may have a few nonzero weights if the defining sets are well chosen. Let \({\mathbb {F}}_q\) be a finite field with \(q=p^m\) elements, where p is a prime and m is a positive integer. Motivated by Ding and Ding’s recent work (IEEE Trans Inf Theory 61(11):5835–5842, 2015), we construct p-ary linear codes \({\mathcal {C}}_D\) by $$\begin{aligned} {\mathcal {C}}_D=\{{\mathbf {c}}(a,b)=\big (\text {Tr}_m(ax+by)\big )_{(x,y)\in D}: a, b \in {\mathbb {F}}_q\}, \end{aligned}$$ where \(D \subset {\mathbb {F}}_q^2\) and \(\text {Tr}_m\) is the trace function from \({\mathbb {F}}_q\) onto \({\mathbb {F}}_p\) . In this paper, we will employ exponential sums to investigate the weight enumerators of the linear codes \({\mathcal {C}}_D\) , where \(D=\{(x, y) \in {\mathbb {F}}_q^2 \setminus \{(0,0)\}: \text {Tr}_m(x^{N_1}+y^{N_2})=0\}\) for two positive integers \(N_1\) and \(N_2\) . Several classes of two-weight and three-weight linear codes and their explicit weight enumerators are presented if \(N_1, N_2 \in \{1, 2, p^{\frac{m}{2}}+1\}\) . By deleting some coordinates, more punctured two-weight and three-weight linear codes \({\mathcal {C}}_{\overline{D}}\) which include some optimal codes are derived from \({\mathcal {C}}_D\) . PubDate: 2017-01-01 DOI: 10.1007/s00200-016-0297-4 Issue No:Vol. 28, No. 1 (2017)

Authors:Graham H. Norton Pages: 31 - 78 Abstract: Let \(\mathbbm {k}\) be a field. We simplify and extend work of Althaler and Dür on finite sequences over \(\mathbbm {k}\) by regarding \(\mathbbm {k}[x^{-1},z^{-1}]\) as a \(\mathbbm {k}[x,z]\) module and studying forms in \(\mathbbm {k}[x^{-1},z^{-1}]\) from first principles. Then we apply our results to finite sequences. First we define the annihilator ideal \(\mathcal {I}_F\) of a form \(F\in \mathbbm {k}[x^{-1},z^{-1}]\) of total degree \(m\le 0\) . This is a homogeneous ideal. We inductively construct an ordered pair ( \(f_1\) , \(f_2\) ) of forms in \(\mathbbm {k}[x,z]\) which generate \(\mathcal {I}_F\) ; our generators are special in that z does not divide the leading grlex monomial of \(f_1\) but z divides \(f_2\) , and the sum of their total degrees is always \(2-m\) . The corresponding algorithm is \(\sim m^2/2\) . We prove that the row vector obtained by accumulating intermediate forms of the construction gives a minimal grlex Gröbner basis for \(\mathcal {I}_F\) for no extra computational cost other than storage (this is based on a closed-form description of a ’form vector’ for F, an associated vector of total degrees and a syzygy triple derived from the construction. These imply that the remainder of the S polynomial of \(f_1,f_2\) is zero. Then we inductively apply Buchberger’s Criterion to show that the form vector yields a minimal Gb for \(\mathcal {I}_F\) ). We apply this to determining \(\dim _\mathbbm {k}(\mathbbm {k}[x,z] /\mathcal {I}_F)\) . We show that either the form vector is reduced or a monomial of \(f_1\) can be reduced by \(f_2\) . This enables us to efficiently construct the unique reduced Gröbner basis for \(\mathcal {I}_F\) from the vector extension of our algorithm. Then we specialise to the inverse form of a finite sequence, obtaining generator forms for its annihilator ideal and a corresponding algorithm. We compute the intersection of two annihilator ideals using syzygies in \(\mathbbm {k}[x,z]^5\) . This improves a result of Althaler and Dür. Finally we show that dehomogenisation induces a one-to-one correspondence ( \(f_1\) , PubDate: 2017-01-01 DOI: 10.1007/s00200-016-0295-6 Issue No:Vol. 28, No. 1 (2017)

Authors:Jon-Lark Kim; Nari Lee Pages: 79 - 97 Abstract: A secret sharing scheme (SSS) was introduced by Shamir in 1979 using polynomial interpolation. Later it turned out that it is equivalent to an SSS based on a Reed–Solomon code. SSSs based on linear codes have been studied by many researchers. However there is little research on SSSs based on additive codes. In this paper, we study SSSs based on additive codes over GF(4) and show that they require at least two steps of calculations to reveal the secret. We also define minimal access structures of SSSs from additive codes over GF(4) and describe SSSs using some interesting additive codes over GF(4) which contain generalized 2-designs. PubDate: 2017-01-01 DOI: 10.1007/s00200-016-0296-5 Issue No:Vol. 28, No. 1 (2017)

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

Authors:Ulrich Oberst Abstract: We complete the stability results of the paper Bourlès et al. (SIAM J Control Optim 53:2725–2761, 2015), and for this purpose use the linear time-varying (LTV) discrete-time behaviors and the exponential stability (e.s.) of this paper. In the main theorem we characterize the e.s. of an autonomous LTV system by standard spectral properties of a complex matrix connected with the system. We extend the theory of discrete-time LTV behaviors, developed in the quoted publication, from the coefficient field of rational functions to that of locally convergent Laurent series or even of Puiseux series. The stability test can and has to be applied in connection with the construction of stabilizing compensators. PubDate: 2017-02-15 DOI: 10.1007/s00200-017-0314-2

Authors:Riccardo Aragona; Riccardo Longo; Massimiliano Sala Abstract: In this paper we propose a tokenization algorithm of Reversible Hybrid type, as defined in PCI DSS guidelines for designing a tokenization solution, based on a block cipher with a secret key and (possibly public) additional input. We provide some formal proofs of security for it, which imply our algorithm satisfies the most significant security requirements described in PCI DSS tokenization guidelines. Finally, we give an instantiation with concrete cryptographic primitives and fixed length of the PAN, and we analyze its efficiency and security. PubDate: 2017-02-13 DOI: 10.1007/s00200-017-0313-3

Authors:T. Aaron Gulliver; Masaaki Harada Abstract: We study the performance of ternary isodual codes which are not self-dual and ternary self-dual codes, as measured by the decoding error probability with bounded distance decoding. We compare the performance of ternary double circulant and double twistulant codes which are not self-dual with ternary extremal self-dual codes. We also investigate the performance of ternary self-dual codes having large minimum weights. PubDate: 2017-01-16 DOI: 10.1007/s00200-017-0312-4

Authors:Katsusuke Nabeshima; Shinichi Tajima Pages: 451 - 467 Abstract: Algebraic local cohomology classes associated with parametric semi-quasihomogeneous hypersurface isolated singularities are considered in the context of symbolic computation. The motivations for this paper are computer calculations of complete lists of Tjurina numbers of semi-quasihomogeneous polynomials with isolated singularity. A new algorithm, that utilizes parametric local cohomology systems, is proposed to compute Tjurina stratifications associated with \(\mu \) -constant deformations of weighted homogeneous isolated singularities. The resulting algorithm gives in particular a suitable decomposition of the parameter space depending on the structure of the parametric local cohomology systems. An efficient algorithm of computing parametric standard bases of relevant ideals is also given as an application of parametric local cohomology systems. PubDate: 2016-12-01 DOI: 10.1007/s00200-016-0289-4 Issue No:Vol. 27, No. 6 (2016)

Authors:Katsusuke Nabeshima; Shinichi Tajima Pages: 451 - 467 Abstract: Algebraic local cohomology classes associated with parametric semi-quasihomogeneous hypersurface isolated singularities are considered in the context of symbolic computation. The motivations for this paper are computer calculations of complete lists of Tjurina numbers of semi-quasihomogeneous polynomials with isolated singularity. A new algorithm, that utilizes parametric local cohomology systems, is proposed to compute Tjurina stratifications associated with \(\mu \) -constant deformations of weighted homogeneous isolated singularities. The resulting algorithm gives in particular a suitable decomposition of the parameter space depending on the structure of the parametric local cohomology systems. An efficient algorithm of computing parametric standard bases of relevant ideals is also given as an application of parametric local cohomology systems. PubDate: 2016-12-01 DOI: 10.1007/s00200-016-0289-4 Issue No:Vol. 27, No. 6 (2016)

Authors:Yangjiang Wei; Guangwu Xu; Yi Ming Zou Pages: 469 - 479 Abstract: The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous publication, the last two authors developed an efficient algorithm to determine whether a linear dynamical system over a finite commutative ring is a fixed point system or not. The algorithm can also be used to reduce the problem of finding the cycles of such a system to the case where the system is given by an automorphism. Here, we further analyze the cycle structure of such a system and develop a method to determine its cycles. PubDate: 2016-12-01 DOI: 10.1007/s00200-016-0290-y Issue No:Vol. 27, No. 6 (2016)

Authors:Yangjiang Wei; Guangwu Xu; Yi Ming Zou Pages: 469 - 479 Abstract: The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous publication, the last two authors developed an efficient algorithm to determine whether a linear dynamical system over a finite commutative ring is a fixed point system or not. The algorithm can also be used to reduce the problem of finding the cycles of such a system to the case where the system is given by an automorphism. Here, we further analyze the cycle structure of such a system and develop a method to determine its cycles. PubDate: 2016-12-01 DOI: 10.1007/s00200-016-0290-y Issue No:Vol. 27, No. 6 (2016)

Authors:Neha Goel; Indivar Gupta; M. K. Dubey; B. K. Dass Pages: 523 - 535 Abstract: D. Chaum and H. van Antwerpen first introduced the concept of an undeniable signature scheme where the verification step is verified with the signer’s co-operation. In this paper, first we discuss a combination of Discrete Logarithm Problem (DLP) and Conjugacy Search Problem (CSP) analysing its security. Then we propose an undeniable signature scheme in a non-abelian group over group ring whose security relies on difficulty of the combination of the DLP and the CSP. The complexity and security of our proposed scheme has also been discussed. PubDate: 2016-12-01 DOI: 10.1007/s00200-016-0293-8 Issue No:Vol. 27, No. 6 (2016)

Authors:Neha Goel; Indivar Gupta; M. K. Dubey; B. K. Dass Pages: 523 - 535 Abstract: D. Chaum and H. van Antwerpen first introduced the concept of an undeniable signature scheme where the verification step is verified with the signer’s co-operation. In this paper, first we discuss a combination of Discrete Logarithm Problem (DLP) and Conjugacy Search Problem (CSP) analysing its security. Then we propose an undeniable signature scheme in a non-abelian group over group ring whose security relies on difficulty of the combination of the DLP and the CSP. The complexity and security of our proposed scheme has also been discussed. PubDate: 2016-12-01 DOI: 10.1007/s00200-016-0293-8 Issue No:Vol. 27, No. 6 (2016)

Authors:Miriam Abdón; Robert Rolland Abstract: For any finite field \({\mathbb {F}}_q\) with q elements, we study the set \({\mathscr {F}}_{(q,m)}\) of functions from \({\mathbb {F}}_q^m\) into \({\mathbb {F}}_q\) from geometric, analytic and algorithmic points of view. We determine a linear system of \(q^{m+1}\) equations and \(q^{m+1}\) unknowns, which has for unique solution the Hamming distances of a function in \({\mathscr {F}}_{(q,m)}\) to all the affine functions. Moreover, we introduce a Fourier-like transform which allows us to compute all these distances at a cost \(O(mq^m)\) and which would be useful for further problems. PubDate: 2016-12-24 DOI: 10.1007/s00200-016-0311-x

Authors:W. Fish Abstract: Let \(n, m \ge 2\) be integers. The cartesian, categorical and lexicographic products of m copies of the n-cycle denoted by \(C_n\) all have as their vertex-set \(\{0, 1, \ldots , n-1\}^m\) , with adjacency defined variously. In this paper the binary codes generated by the row span of adjacency matrices of the cartesian, categorical and lexicographic products of m copies of \(C_n\) are examined. Full and partial PD-sets were also found for the various codes. PubDate: 2016-12-20 DOI: 10.1007/s00200-016-0310-y