# American Institute of Mathematical Sciences

ISSN:
1930-5346

eISSN:
1930-5338

All Issues

## Advances in Mathematics of Communications

2016 , Volume 10 , Issue 4

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2016, 10(4): 683-694 doi: 10.3934/amc.2016034 +[Abstract](758) +[PDF](354.8KB)
Abstract:
We give the structure of constacyclic codes over some chain rings. We also provide conditions on the equivalence between constacyclic codes and cyclic codes over finite chain rings.As a special case,we consider the structure of $(\alpha + \beta p)$-constacyclic codes of length $p^s$ over $GR(p^e,r)$.
2016, 10(4): 695-706 doi: 10.3934/amc.2016035 +[Abstract](630) +[PDF](359.9KB)
Abstract:
It is shown that the residue code of a self-dual $\mathbb{Z}_4$-code of length $24k$ (resp. $24k+8$) and minimum Lee weight $8k+4 \text{ or }8k+2$ (resp. $8k+8 \text{ or }8k+6$) is a binary extremal doubly even self-dual code for every positive integer $k$. A number of new self-dual $\mathbb{Z}_4$-codes of length $24$ and minimum Lee weight $10$ are constructed using the above characterization. These codes are Type I $\mathbb{Z}_4$-codes having the largest minimum Lee weight and the largest Euclidean weight among all Type I $\mathbb{Z}_4$-codes of that length. In addition, new extremal Type II $\mathbb{Z}_4$-codes of length $56$ are found.
2016, 10(4): 707-723 doi: 10.3934/amc.2016036 +[Abstract](485) +[PDF](385.9KB)
Abstract:
Cyclic codes have wide applications in data storage systems and communication systems. Employing binary two-prime Whiteman generalized cyclotomic sequences of order 6, we construct several classes of cyclic codes over the finite field $\mathrm{GF}(q)$ and give their generator polynomials. And we also calculate the minimum distance of some cyclic codes and give lower bounds on the minimum distance for some other cyclic codes.
2016, 10(4): 725-741 doi: 10.3934/amc.2016037 +[Abstract](701) +[PDF](388.3KB)
Abstract:
Plateaued functions have been introduced by Zheng and Zhang in 1999 as good candidates for designing cryptographic functions since they possess many desirable cryptographic characteristics. Plateaued functions bring together various nonlinear characteristics and include two important classes of Boolean functions defined in even dimension: the well-known bent functions ($0$-plateaued functions) and the semi-bent functions ($2$-plateaued functions). Bent functions have been extensively investigated since 1976. Very recently, the study of semi-bent functions has attracted a lot of attention in symmetric cryptography. Many intensive progresses in the design of such functions have been made especially in recent years. The paper is devoted to the construction of semi-bent functions on the finite field $\mathbb{F}_{2^n}$ ($n=2m$) in the line of a recent work of S. Mesnager [IEEE Transactions on Information Theory, Vol 57, No 11, 2011]. We extend Mesnager's results and present a new construction of infinite classes of binary semi-bent functions in polynomial trace. The extension is achieved by inserting mappings $h$ on $\mathbb{F}_{2^n}$ which can be expressed as $h(0) = 0$ and $h(uy) = h_1(u)h_2(y)$ with $u$ ranging over the circle $U$ of unity of $\mathbb{F}_{2^n}$, $y \in \mathbb{F}_{2^m}^{*}$ and $uy \in \mathbb{F}_{2^n}^{*}$, where $h_1$ is a isomorphism on $U$ and $h_2$ is an arbitrary mapping on $\mathbb{F}_{2^m}^{*}$. We then characterize the semi-bentness property of the extended family in terms of classical binary exponential sums and binary polynomials.
2016, 10(4): 743-752 doi: 10.3934/amc.2016038 +[Abstract](553) +[PDF](341.3KB)
Abstract:
We extend the construction of $p^2$-periodic binary threshold sequences derived from Fermat quotients to the $d$-ary case where $d$ is an odd prime divisor of $p-1$, and then by defining cyclotomic classes modulo $p^{2}$, we present exact values of the linear complexity under the condition of $d^{p-1}\not \equiv 1 \pmod {p^2}$. Also, we extend the results to the Euler quotients modulo $p^{r}$ with odd prime $p$ and $r \geq 2$. The linear complexity is very close to the period and is of desired value for cryptographic purpose. The results extend the linear complexity of the corresponding $d$-ary sequences when $d$ is a primitive root modulo $p^2$ in earlier work. Finally, partial results for the linear complexity of the sequences when $d^{p-1} \equiv 1 \pmod {p^2}$ is given.
2016, 10(4): 753-764 doi: 10.3934/amc.2016039 +[Abstract](629) +[PDF](358.4KB)
Abstract:
Aspects of the properties, enumeration and construction of points on diagonal and Hermitian hypersurfaces have been considered extensively in the literature and are further considered here. The zeta function of diagonal hypersurfaces is given as a direct result of the work of Wolfmann. Recursive construction techniques for the set of rational points of Hermitian hypersurfaces are of interest. The relationship of these techniques here to the construction of codes on hypersurfaces is briefly noted.
2016, 10(4): 765-795 doi: 10.3934/amc.2016040 +[Abstract](629) +[PDF](556.9KB)
Abstract:
The aim of this text is to construct and to enumerate self-dual $\theta$-cyclic and $\theta$-negacyclic codes over $\mathbb{F}_{p^2}$ where $p$ is a prime number and $\theta$ is the Frobenius automorphism.
2016, 10(4): 797-809 doi: 10.3934/amc.2016041 +[Abstract](614) +[PDF](366.0KB)
Abstract:
This work analyses the output sequence from a cryptographic non-linear generator, the so-called shrinking generator. This sequence, known as the shrunken sequence, can be built by interleaving a unique PN-sequence whose characteristic polynomial serves as basis for the shrunken sequence's characteristic polynomial. In addition, the shrunken sequence can be also generated from a linear model based on cellular automata. The cellular automata here proposed generate a family of sequences with the same properties, period and characteristic polynomial, as those of the shrunken sequence. Moreover, such sequences appear several times along the cellular automata shifted a fixed number. The use of discrete logarithms allows the computation of such a number. The linearity of these cellular automata can be advantageously employed to launch a cryptanalysis against the shrinking generator and recover its output sequence.
2016, 10(4): 811-823 doi: 10.3934/amc.2016042 +[Abstract](574) +[PDF](409.8KB)
Abstract:
Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. It is proved that optimal $(v, \{3, 5\}, 1, (1/2, 1/2))$-OOCs exist for some complete congruence classes of $v$. In this paper, for $Q\in \{(2/3,1/3), (3/4,1/4)\}$, by using skew starters, it is also proved that optimal $(v, \{3,5\}, 1, Q)$-OOCs exist for some complete congruence classes of $v$.
2016, 10(4): 825-845 doi: 10.3934/amc.2016043 +[Abstract](636) +[PDF](499.2KB)
Abstract:
We construct new complementary sequence sets of size $4$, using a graphical description. We explain how the construction can be seen as a special case of a less explicit array construction by Parker and Riera and, at the same time, a generalization of another construction by the same authors. Some generalizations of the construction are also given, which are not in the construction of Parker and Riera. Lower bounds and upper bounds of the number of sequences in the constructions are analyzed.
2016, 10(4): 847-850 doi: 10.3934/amc.2016044 +[Abstract](612) +[PDF](264.1KB)
Abstract:
Let $[n,k]_q$ be a projective two-weight linear code over ${\rm GF}(q)^n$. In this correspondence, 9 codes are constructed in which $k=3$.
2016, 10(4): 851-860 doi: 10.3934/amc.2016045 +[Abstract](637) +[PDF](371.8KB)
Abstract:
We specialize Möller's algorithm to the computation of Gröbner bases related to lattices. We give the complexity analysis of our algorithm. Then we provide experiments showing that our algorithm is more efficient than Buchberger's algorithm for computing the associated Gröbner bases. Furthermore we show that the binomial ideal associated to the lattice can be constructed from a set of binomials associated with a set of generators of the corresponding label code. This result is presented in a general way by means of three ideal constructions associated with group codes that constitute the same ideal. This generalizes earlier results for specific cases of group codes such as linear codes, codes over ${\mathbb Z}_m$ and label codes of lattices.
2016, 10(4): 861-870 doi: 10.3934/amc.2016046 +[Abstract](560) +[PDF](352.7KB)
Abstract:
In this paper we use the nonrepresentable ring $E_{p}^{(m)}$ to introduce public key cryptosystems in noncommutative settings and based on the Semigroup Action Problem and the Decomposition Problem respectively.
2016, 10(4): 871-893 doi: 10.3934/amc.2016047 +[Abstract](663) +[PDF](434.4KB)
Abstract:
In this paper we use group theoretic tools to obtain random variables which violate linear rank inequalities, that is inequalities which always hold on ranks of subspaces. We consider ten of the 24 (non-Shannon type) generators of linear rank inequalities in five variables and look at them as group inequalities. We prove that for primes $p,q$, groups of order $pq$ always satisfy these ten group inequalities. We give partial results for groups of order $p^2q$, and find that the symmetric group $S_4$ is the smallest group to yield violations for two among the ten group inequalities.
2016, 10(4): 895-919 doi: 10.3934/amc.2016048 +[Abstract](597) +[PDF](576.9KB)
Abstract:
Correlation immune functions were introduced to protect some shift register based stream ciphers against correlation attacks. For cryptographic applications, relaxing the concept of correlation immunity has been highlighted and proved to be more appropriate in several cryptographic situations. Various weakened notions of correlation immunity and resiliency have been widely introduced for cryptographic functions, but those notions are difficult to handle.
As a variation, we focus on the notion of $\varphi$-correlation immunity which is closely related to (fast) correlation attacks on stream ciphers based on nonlinear combiner model. In particular, we exhibit new connections between $\varphi$-correlation immunity and $\epsilon$-almost resiliency, which are two distinct approaches for characterizing relaxed resiliency. We also extend the concept of $\varphi$-correlation immunity introduced by Carlet et al. in 2006 for Boolean functions to vectorial functions and study the main cryptographic parameters of $\varphi$-correlation immune functions. Moreover, we provide new primary constructions of $\varphi$-resilient functions with good designed immunity profile. Specially, we propose a new recursive method to construct $\varphi$-resilient functions with high nonlinearity, high algebraic degree, and monotone increasing immunity profile.
2016, 10(4): 921-929 doi: 10.3934/amc.2016049 +[Abstract](650) +[PDF](286.5KB)
Abstract:
Polycyclic codes are ideals in quotients of polynomial rings by a principal ideal. Special cases are cyclic and constacyclic codes. A MacWilliams relation between such a code and its annihilator ideal is derived. An infinite family of binary self-dual codes that are also formally self-dual in the classical sense is exhibited. We show that right polycyclic codes are left polycyclic codes with different (explicit) associate vectors and characterize the case when a code is both left and right polycyclic for the same associate polynomial. A similar study is led for sequential codes.

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