Search Results (15)

A method is proposed that reduces the computation of the reduced digital convolution operation to a set of independent convolutions computed in Galois fields. The reduced digital convolution is understood as a modified convolution operation whose result is a function taking discrete values in the same discrete scale as the original functions. The method is based on the use of partial convolutions, reduced to computing a modulo integer $q_0$, which is the product of several prime numbers: $q_0=p_1{p}_2\dots p_n$. It is shown that it is appropriate to use the expansion of the number $q_0$, to $q=p_0{p}_1{p}_2\dots p_n$, where $p_0$ is an additional prime number, to compute the reduced digital convolution. This corresponds to the use of additional digits in the number system used to convert to partial convolutions. The inverse procedure, i.e., reducing the result of calculations modulo q to the result corresponding to calculations modulo $q_0$, is provided by the formula that used only integers proved in this paper. The possibilities of practical application of the obtained results are discussed. Full article

The main objective of this article is to classify all finite singleton local rings, which are associative rings characterized by a unique maximal ideal and a distinguished basis consisting of a single element. These rings are associated with four positive integer invariants $p,n,s$, and $t,$ where p is a prime number. In particular, we aim to classify these rings and count them up to isomorphism while maintaining the same set of invariants. We have found interesting cases of finite singleton local rings with orders of $p^6$ and $p^7$ that hold substantial importance in the field of coding theory. Full article

Suppose R is a finite chain ring with invaraints $p,n,r,k,k^{\prime },m.$ Suppose G is also the subset of all $\phi \in$ Aut$\left(R\right),$ the automorphism group of $R,$ such that $\phi \left({\pi }^{k^{\prime }}\right)={\pi }^{k^{\prime }},$ where $\pi$ is a generator of the maximal ideal of $R.$ It was found that G is a group that is, in some sense, the set of all symmetries of $\left\{{\pi }^{k^{\prime }}\right\}.$ The main purpose of this article is to describe the structure of $G.$ The subgroup G helps us understand the structure of Aut$\left(R\right)$ in the general case which in turn provides immediate results in classifying chain rings. Full article

Let R be a finite commutative chain ring with invariants $p,n,r,k,m.$ The purpose of this article is to study j-diagrams for the one group $H=1+J\left(R\right)$ of $R,$ where $J\left(R\right)=\left(\pi \right)$ is Jacobson radical of $R.$ In particular, we prove the existence and uniqueness of j-diagrams for such one group. These j-diagrams help us to solve several problems related to chain rings such as the structure of their unit groups and a group of all symmetries of $\left\{{\pi }^{k^{\prime }}\right\},$ where $k^{\prime }\mid k$. The invariants $p,n,r,k,m$ and the Eisenstein polynomial by which R is constructed over its Galois subring determine fully the j-diagram for $H.$ Full article

A local ring is an associative ring with unique maximal ideal. We associate with each Artinian local ring with singleton basis four invariants (positive integers) $p,n,s,t.$ The purpose of this article is to describe the structure of such rings and classify them (up to isomorphism) with the same invariants. Every local ring with singleton basis can be constructed over its coefficient subring by a certain polynomial called the associated polynomial. These polynomials play significant role in the enumeration. Full article

Let R be a Galois ring, $GR(p^n,r)$, of characteristic $p^n$ and of order $p^{nr}$. In this article, we study cyclic codes of arbitrary length, N, over R. We use discrete Fourier transform (DFT) to determine a unique representation of cyclic codes of length, N, in terms of that of length, $p^s$, where $s=v_p\left(N\right)$ and $v_p$ are the p-adic valuation. As a result, Hamming distance and dual codes are obtained. In addition, we compute the exact number of distinct cyclic codes over R when $n=2$. Full article

We provide three new authentication schemes without secrecy. The first two on finite fields and Galois rings, using Gray map for this link. The third construction is based on Galois rings. The main achievement in this work is to obtain optimal impersonation and substitution probabilities in the schemes. Additionally, in the first and second scheme, we simplify the source space and obtain a better relationship between the size of the message space and the key space than the one given in a recent paper. Finally, we provide a third scheme on Galois rings. Full article

A finite ring with an identity is a chain ring if its lattice of left ideals forms a unique chain. Let R be a finite chain ring with invaraints $p,n,r,k,k^{\prime },m.$ If $n=1,$ the automorphism group $Aut\left(R\right)$ of R is known. The main purpose of this article is to study the structure of $Aut\left(R\right)$ when $n>1.$ First, we prove that $Aut\left(R\right)$ is determined by the automorphism group of a certain commutative chain subring. Then we use this fact to find the automorphism group of R when $p\nmid k.$ In addition, $Aut\left(R\right)$ is investigated under a more general condition; that is, R is very pure and p need not divide $k.$ Based on the j-diagram introduced by Ayoub, we were able to give the automorphism group in terms of a particular group of matrices. The structure of the automorphism group of a finite chain ring depends essentially on its invaraints and the associated j-diagram. Full article

Random numbers are widely employed in cryptography and security applications. If the generation process is weak, the whole chain of security can be compromised: these weaknesses could be exploited by an attacker to retrieve the information, breaking even the most robust implementation of a cipher. Due to their intrinsic close relationship with analogue parameters of the circuit, True Random Number Generators are usually tailored on specific silicon technology and are not easily scalable on programmable hardware, without affecting their entropy. On the other hand, programmable hardware and programmable System on Chip are gaining large adoption rate, also in security critical application, where high quality random number generation is mandatory. The work presented herein describes the design and the validation of a digital True Random Number Generator for cryptographically secure applications on Field Programmable Gate Array. After a preliminary study of literature and standards specifying requirements for random number generation, the design flow is illustrated, from specifications definition to the synthesis phase. Several solutions have been studied to assess their performances on a Field Programmable Gate Array device, with the aim to select the highest performance architecture. The proposed designs have been tested and validated, employing official test suites released by NIST standardization body, assessing the independence from the place and route and the randomness degree of the generated output. An architecture derived from the Fibonacci-Galois Ring Oscillator has been selected and synthesized on Intel Stratix IV, supporting throughput up to 400 Mbps. The achieved entropy in the best configuration is greater than 0.995. Full article

A finite ring with an identity whose lattice of ideals forms a unique chain is called a finite chain ring. Let R be a commutative chain ring with invariants $p,n,r,k,m.$ It is known that R is an Eisenstein extension of degree k of a Galois ring $S=GR(p^n,r).$ If $p-1$ does not divide k, the structure of the unit group $U\left(R\right)$ is known. The case $(p-1)\mid k$ was partially considered by M. Luis (1991) by providing counterexamples demonstrated that the results of Ayoub failed to capture the direct decomposition of $U\left(R\right).$ In this article, we manage to determine the structure of $U\left(R\right)$ when $(p-1)\mid k$ by fixing Ayoub’s approach. We also sharpen our results by introducing a system of generators for the unit group and enumerating the generators of the same order. Full article

In this study, we explore maximum distance separable (MDS) self-dual codes over Galois rings $GR(p^m,r)$ with $p\equiv -1(mod4)$ and odd r. Using the building-up construction, we construct MDS self-dual codes of length four and eight over $GR(p^m,3)$ with ( $p=3$ and $m=2,3,4,5,6$ ), ( $p=7$ and $m=2,3$ ), ( $p=11$ and $m=2$ ), ( $p=19$ and $m=2$ ), ( $p=23$ and $m=2$ ), and ( $p=31$ and $m=2$ ). In the building-up construction, it is important to determine the existence of a square matrix U such that $U{U}^T=-I$ , which is called an antiorthogonal matrix. We prove that there is no $2\times 2$ antiorthogonal matrix over $GR(2^m,r)$ with $m\ge 2$ and odd r. Full article

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension $\mathbf{R}/{\mathrm{Z}}_{p^r}$ , where $\mathbf{R}=\mathrm{GR}(p^r,n).$ We present a criterion on the normal basis for $\mathbf{R}/{\mathrm{Z}}_{p^r}$ and reduce this problem to one of finite field extension $\overline{\mathbf{R}}/{\overline{\mathrm{Z}}}_{p^r}={\mathbb{F}}_q/{\mathbb{F}}_p(q=p^n)$ by Theorem 1. We determine all optimal normal bases for Galois ring extension. Full article

The aim of the present paper is twofold. First, to give the main ideas behind quantum computing and quantum information, a field based on quantum-mechanical phenomena. Therefore, a short review is devoted to (i) quantum bits or qubits (and more generally qudits), the analogues of the usual bits 0 and 1 of the classical information theory, and to (ii) two characteristics of quantum mechanics, namely, linearity, which manifests itself through the superposition of qubits and the action of unitary operators on qubits, and entanglement of certain multi-qubit states, a resource that is specific to quantum mechanics. A, second, focus is on some mathematical problems related to the so-called mutually unbiased bases used in quantum computing and quantum information processing. In this direction, the construction of mutually unbiased bases is presented via two distinct approaches: one based on the group SU(2) and the other on Galois fields and Galois rings. Full article

Two new systematic authentication codes based on the Gray map over a Galois ring are introduced. The first introduced code attains optimal impersonation and substitution probabilities. The second code improves space sizes, but it does not attain optimal probabilities. Additionally, it is conditioned to the existence of a special class of bent maps on Galois rings. Full article

Fractional repetition (FR) codes are a class of distributed storage codes that replicate and distribute information data over several nodes for easy repair, as well as efficient reconstruction. In this paper, we propose three new constructions of FR codes based on relative difference sets (RDSs) with $\lambda =1$ . Specifically, we propose new $(q^2-1,q,q)$ FR codes using cyclic RDS with parameters $(q+1,q-1,q,1)$ constructed from q-ary m-sequences of period $q^2-1$ for a prime power q, $(p^2,p,p)$ FR codes using non-cyclic RDS with parameters $(p,p,p,1)$ for an odd prime p or $p=4$ and $(4^l,2^l,2^l)$ FR codes using non-cyclic RDS with parameters $(2^l,2^l,2^l,1)$ constructed from the Galois ring for a positive integer l. They are differentiated from the existing FR codes with respect to the constructable code parameters. It turns out that the proposed FR codes are (near) optimal for some parameters in terms of the FR capacity bound. Especially, $(8,3,3)$ and $(9,3,3)$ FR codes are optimal, that is, they meet the FR capacity bound for all k. To support various code parameters, we modify the proposed $(q^2-1,q,q)$ FR codes using decimation by a factor of the code length $q^2-1$ , which also gives us new good FR codes. Full article