Authors:Haode Yan Pages: 1 - 11 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: 2018-01-01 DOI: 10.1007/s00200-017-0320-4 Issue No:Vol. 29, No. 1 (2018)

Authors:Yuan Cao; Yonglin Cao; Li Dong Pages: 13 - 39 Abstract: Let \({\mathbb {F}}_{3^m}\) be a finite field of cardinality \(3^m\) , \(R={\mathbb {F}}_{3^m}[u]/\langle u^4\rangle \) which is a finite chain ring, and n be a positive integer satisfying \(\mathrm{gcd}(3,n)=1\) . For any \(\delta ,\alpha \in {\mathbb {F}}_{3^m}^{\times }\) , an explicit representation for all distinct \((\delta +\alpha u^2)\) -constacyclic codes over R of length 3n is given, formulas for the number of all such codes and the number of codewords in each code are provided, respectively. Moreover, the dual code for each of these codes is determined explicitly. PubDate: 2018-01-01 DOI: 10.1007/s00200-017-0328-9 Issue No:Vol. 29, No. 1 (2018)

Authors:Dongyoung Roh; I-Yeol Kim; Sang Geun Hahn Pages: 41 - 57 Abstract: There are many variants of the computational Diffie–Hellman problem that are necessary to provide security of many cryptographic schemes. Two of them are the square Diffie–Hellman problem and the square root Diffie–Hellman problem. Recently, the first and third authors proved that these two problems are polynomial-time equivalent under a certain condition (Roh and Hahn in Des Codes Cryptogr 62(2):179–187, 2011). In this paper, we generalize this result. We introduce the l-th power Diffie–Hellman problem and the l-th root Diffie–Hellman problem and show that these two problems are polynomial-time equivalent for \(l = O (\log p)\) under a condition similar to that of Roh and Hahn (2011), where p is the order of the underlying group. PubDate: 2018-01-01 DOI: 10.1007/s00200-017-0321-3 Issue No:Vol. 29, No. 1 (2018)

Authors:Jaehyun Ahn; Dongseok Ka Pages: 59 - 76 Abstract: Recently, linear codes constructed from defining sets have been studied widely and they have many applications. For an odd prime p, let \(q=p^{m}\) for a positive integer m and \(\mathrm {Tr}_{m}\) the trace function from \(\mathbb {F}_{q}\) onto \(\mathbb {F}_{p}\) . In this paper, for a positive integer t, let \(D\subset \mathbb {F}^{t}_{q}\) and \(D=\{(x_{1},x_{2}) \in (\mathbb {F}_{q}^{*})^{2} : \mathrm {Tr}_{m}(x_{1}+x_{2})=0\}\) , we define a p-ary linear code \(\mathcal {C}_{D}\) by $$\begin{aligned} \mathcal {C}_{D}=\left\{ \mathbf {c}(a_{1},a_{2}) : (a_{1},a_{2})\in \mathbb {F}^{2}_{q}\right\} , \end{aligned}$$ where $$\begin{aligned} \mathbf {c}(a_{1},a_{2})=\left( \mathrm {Tr}_{m}\left( a_{1}x^{2}_{1}+a_{2}x^{2}_{2}\right) \right) _{(x_{1},x_{2})\in D}. \end{aligned}$$ We compute the weight enumerators of the punctured codes \(\mathcal {C}_{D}\) . PubDate: 2018-01-01 DOI: 10.1007/s00200-017-0329-8 Issue No:Vol. 29, No. 1 (2018)

Authors:Mridul Nandi; Tapas Pandit Pages: 77 - 102 Abstract: Predicate encryption (PE), a generalization of attribute-based encryption (ABE), is a versatile tool for providing access control over data. The underlying predicate for a PE is parametrized by an index, called system parameter or simply system-index. A system-index, in general, consists of component(s) from \(\mathbb {N}\) . Yamada et al. in PKC 2011 proposed a verifiability-based conversion from CPA to CCA-secure ABE. This conversion was generalized by Yamada et al. in PKC 2012 from ABE to PE. In the later conversion, the authors considered the system-index to be a single component. In practice, there are many schemes, e.g., functional encryption for general relations and hierarchical-inner product (HIP) encryption schemes of Okamoto-Takashima in CRYPTO 2010, CANS 2011 and EUROCRYPT 2012, where system-indices consist of more than a single component. Therefore, for these schemes, the conversion of Yamada et al. (in PKC, 2012) is out of scope. In this paper, we revisit the CPA to CCA conversion for PE and propose a new conversion based on verifiability. The proposed conversion works irrespective of the number of components in the system-indices. It generalizes the existing conversion of Yamada et al. (in PKC, 2011) from ABE to PE. The PE schemes which are realized by the conversion of Yamada et al. (2011) are also realized by our conversion. Therefore, the conversion of ours has more scope than the conversion proposed in 2012. We show that all the aforementioned CPA-secure schemes for general relations and HIP relation are easily converted to the corresponding CCA-secure schemes by our conversion. Further, we show a generic conversion from CPA to CCA-secure functional encryption for regular languages which captures the existing PE schemes for regular languages. PubDate: 2018-01-01 DOI: 10.1007/s00200-017-0330-2 Issue No:Vol. 29, No. 1 (2018)

Authors:Qian Liu; Yujuan Sun; WeiGuo Zhang Abstract: Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. In this paper, for an integer s satisfying \(s=\frac{q^n-1}{2}+q^r\) , we give six classes of permutation polynomials of the form \((ax^{q^m}-bx+\delta )^s+L(x)\) over \(\mathbb {F}_{q^n}\) , and for s satisfying \(s(p^m-1)\equiv p^m-1\ (mod\ p^n-1)\) or \(s(p^{{\frac{k}{2}}m}-1)\equiv p^{km}-1 (mod\ p^n-1)\) , we propose three classes of permutation polynomials of the form \((aTr_m^n(x)+\delta )^s+L(x)\) over \(\mathbb {F}_{p^n}\) , respectively. PubDate: 2018-01-31 DOI: 10.1007/s00200-018-0350-6

Authors:Javad Doliskani Abstract: For an elliptic curve E over a finite field \(\mathbb {F}_q\) , where q is a prime power, we propose new algorithms for testing the supersingularity of E. Our algorithms are based on the polynomial identity testing problem for the p-th division polynomial of E. In particular, an efficient algorithm using points of high order on E is given. PubDate: 2018-01-16 DOI: 10.1007/s00200-018-0349-z

Authors:Yun Gao; Jian Gao; Tingting Wu; Fang-Wei Fu Pages: 457 - 467 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: 2017-12-01 DOI: 10.1007/s00200-017-0315-1 Issue No:Vol. 28, No. 6 (2017)

Authors:Zohreh Rajabi; Kazem Khashyarmanesh Pages: 469 - 495 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: 2017-12-01 DOI: 10.1007/s00200-017-0317-z Issue No:Vol. 28, No. 6 (2017)

Authors:Krzysztof Ziemiański Pages: 497 - 525 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: 2017-12-01 DOI: 10.1007/s00200-017-0316-0 Issue No:Vol. 28, No. 6 (2017)

Authors:Gerardo Vega Pages: 527 - 533 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: 2017-12-01 DOI: 10.1007/s00200-017-0318-y Issue No:Vol. 28, No. 6 (2017)

Authors:Xiaoni Du; Yunqi Wan Pages: 535 - 547 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: 2017-12-01 DOI: 10.1007/s00200-017-0319-x Issue No:Vol. 28, No. 6 (2017)

Authors:Michele Rossi; Lea Terracini Pages: 351 - 368 Abstract: The main object of the present paper is a numerical criterion determining when a Weil divisor of a \({\mathbb {Q}}\) –factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of \({\mathbb {Z}}\) –linear interpretation of Gale duality and secondary fan as developed in several previous papers of us. As a byproduct we get a computation of the Cartier index of a Weil divisor and a numerical characterization of weak \({\mathbb {Q}}\) –Fano, \({\mathbb {Q}}\) –Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studied. PubDate: 2017-11-01 DOI: 10.1007/s00200-016-0308-5 Issue No:Vol. 28, No. 5 (2017)

Authors:W. Fish Pages: 369 - 386 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: 2017-11-01 DOI: 10.1007/s00200-016-0310-y Issue No:Vol. 28, No. 5 (2017)

Authors:Miriam Abdón; Robert Rolland Pages: 387 - 408 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: 2017-11-01 DOI: 10.1007/s00200-016-0311-x Issue No:Vol. 28, No. 5 (2017)

Authors:T. Aaron Gulliver; Masaaki Harada Pages: 409 - 424 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: 2017-11-01 DOI: 10.1007/s00200-017-0312-4 Issue No:Vol. 28, No. 5 (2017)

Authors:Ulrich Oberst Pages: 437 - 456 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: 2017-11-01 DOI: 10.1007/s00200-017-0314-2 Issue No:Vol. 28, No. 5 (2017)

Authors:Amaro Barreal; Capi Corrales Rodrigáñez; Camilla Hollanti Abstract: Algebraic space–time coding—a powerful technique developed in the context of multiple-input multiple-output (MIMO) wireless communications—has profited tremendously from tools from Class Field Theory and, more concretely, the theory of central simple algebras and their orders. During the last decade, the study of space–time codes for practical applications, and more recently for future generation (5G \(+\) ) wireless systems, has provided a practical motivation for the consideration of many interesting mathematical problems. One such problem is the explicit computation of orders of central simple algebras with small discriminants. In this article, we consider the most interesting asymmetric MIMO channel setups and, for each treated case, we provide explicit pairs of fields and a corresponding non-norm element giving rise to a cyclic division algebra whose natural order has the minimum possible discriminant. PubDate: 2017-12-14 DOI: 10.1007/s00200-017-0348-5

Authors:Gerardo Vega Abstract: We generalize and simplify the results of Sharma and Bakshi (Finite Fields Appl 18(1):144–159 2012) on the weight-distributions of irreducible cyclic codes of prime-power length. PubDate: 2017-11-29 DOI: 10.1007/s00200-017-0347-6

Authors:Pamela E. Harris; Erik Insko; Anthony Simpson Abstract: The multiplicity of a weight \(\mu \) in an irreducible representation of a simple Lie algebra \(\mathfrak {g}\) with highest weight \(\lambda \) can be computed via the use of Kostant’s weight multiplicity formula. This formula is an alternating sum over the Weyl group and involves the computation of a partition function. In this paper we consider a q-analog of Kostant’s weight multiplicity and present a SageMath program to compute q-multiplicities for the simple Lie algebras. PubDate: 2017-11-01 DOI: 10.1007/s00200-017-0346-7