2 - Galois Fields

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

Circulant matrices form an important class of matrices that have been continuously studied due to their nice algebraic structures and wide applications. In this study, we focus specifically on negacirculant matrices, which are known as extensions of the classical circulant matrices. The algebraic structures of the rings of n × n n\times n negacirculant matrices over finite fields and over finite commutative chain rings are presented. Subsequently, the algebraic structures and enumeration of the unit groups of such matrix rings are established. Additionally, the number of non-singular n × n n\times n negacirculant matrices with prescribed determinant is given in some cases. Conjectures and open problems are proposed as well as a brief discussion in the case where the underlying ring is a finite commutative ring with identity is also presented.

Linear dynamical systems over finite rings

On the ℓ-DLIPs of codes over finite commutative rings

Coded Distributed Matrix Multiplication (CDMM) is a distributed matrix multiplication (DMM) for large-scale matrices through a coding scheme such that any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> worker nodes among all <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> worker nodes can recover the final product, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> corresponds to the length of the code and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> is called the recovery threshold. The state-of-art CDMM schemes, such as EP codes for Single DMM and GCSA codes for batch DMM, are defined over a Galois field GF(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i>) of size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ≥ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>. These are inefficient for small Galois fields such as GF(2) and the integer residue ring Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i></sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> due to the lack of invertible elements for interpolation. DMM over Z<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">_p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> (such as Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">64</sup> ) is well-motivated in practice due to their direct compatibility with hardware. In this work, we construct efficient CDMM over the Galois ring GR(p<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i>) which is an extension ring over Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i></sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> of degree <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i>, particularly, GR(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p, d</i>) = GF(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup>) is the Galois field and GR(<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i></sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup>, 1) = Z<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">_p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup>. We first give a general CDMM framework for the batch of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> matrix multiplications via the famous RMFE (Cascudo et al. Crypto’18). Compared with GCSA, our construction has a smaller recovery threshold by a factor of 1/<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>. Next, we optimize EP codes via batch preprocessing of the input matrices. We give two types of Single CDMM, which can achieve almost the same performance as EP codes over a Galois field with size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ≥ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>. Finally, we present the experimental analysis of our CDMM on Galois rings.

A variety of Lie rings is called just-non-Cross if it itself is not Cross, but each of its proper subvarieties is Cross, i.e. is generated by a finite ring. In this paper, we completely describe the solvable just-non-Cross varieties both of Lie rings and of Lie R-algebras where R is a finite commutative ring with identity and, in particular, where R is a finite field. We find algorithms which allow us to determine whether a given identity defines a Cross variety of Lie algebras; also, using the multiplication and addition tables of a finite Lie algebra, we find algorithms for extracting its identities.Bibliography: 10 titles.

We introduce three new constructions of systematic authentication codes over finite fields and Galois rings. Our first construction uses resilient functions over finite fields and provides optimal impersonation and substitution probabilities. Our two other constructions are defined over Galois rings: one is based on resilient maps attaining optimal probabilities as well, while the other is based on maps with maximum Fourier transforms. For the special case of characteristic $$p^2$$p2, the maps used on our third construction are bent. Furthermore, we give a generalised construction for the case of characteristic $$p^s$$ps, with $$s \ge 2$$s?2. The second and third codes over Galois rings, restricted to the particular case of Galois fields, are different than the first code introduced in this paper: the corresponding source and tag spaces differ, and the encoding maps classes are pairwise different.

The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enumerators of linear codes over the ring of integers modulo 4. We also look at Gray maps from Galois rings to the finite field of prime size. We first give an application of the distance preserving Gray maps to obtain good ternary codes, and then we give an inductive algebraic construction of a distance preserving Gray map from Galois rings with the homogeneous distance to the field of prime size with the Hamming distance as well as a combinatorial construction. By using the Gray maps, we obtain some results about the weight enumerators of linear codes over the ring of dual numbers. In the last part of this work, we consider the permutation invariance of binary codes and the connection with linearity over certain rings. Among the rings considered, we have different variations and extensions of the ring of dual numbers. In this context we consider the Reed-Muller codes and answer questions about permutation invariance of Reed-Muller codes under certain permutation groups and the linearity of Reed-Muller codes over these aforementioned rings.

Properties of a new complex number-theoretic z-transform (CNT z-transform) over a finite ring are presented here and related to the usual z-transform. Using the Chinese remainder theorem, it is convenient to use finite rings that are isomorphic to the direct sum of finite or Galois fields of the form GF(q2) where q is a Mersenne prime. Many properties of the usual z-transform are preserved in the CNT z-transform. This transform is used in the present paper to design both recursive and nonrecursive FIR filters on a finite ring. The advantages of the FIR filter on a finite ring are the following: 1) the absence of a roundoff error build up in the computation of either the recursive or nonrecursive realization of the filter; 2) when the FIR filter is recursive, the question of stability does not arise as long as the magnitudes of the impulse response and the input sequence do not exceed their design values; 3) for the frequency sample representation of the FIR filter an absolute error bound on the impulse response function can be obtained in terms of the power spectrum. The time required to compute a nonrecursive FIR filter on the Galois field GF(q2), where q is a Mersenne prime, is competitive with the similar nonrecursive realization on the usual complex number field, using the FFT algorithm.

Matrix product codes with rosenbloom-tsfasman metric

Given a finite (associative, unital) ring R, let K(R) denote the set of polynomials in R[x] that send each element of R to 0 under evaluation. We study K(R) and its elements. We conjecture that K(R) is a two-sided ideal of R[x] for any finite ring R, and prove the conjecture for several classes of finite rings (including commutative rings, semisimple rings, local rings, and all finite rings of odd order). We also examine a connection to sets of integer-valued polynomials.

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.

Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or else a lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. Lattice isomorphisms of finite one-generated rings with identity are studied. We elucidate the general structure of finite one-generated rings with identity and also give necessary and sufficient conditions for a finite ring decomposable into a direct sum of Galois rings to be generated by one element. Conditions are found under which the projective image of a ring decomposable into a direct sum of finite fields is a one-generated ring. We look at lattice isomorphisms of one-generated rings decomposable into direct sums of Galois rings of different types. Three main types of Galois rings are distinguished: finite fields, rings generated by idempotents, and rings of the form GR(pn,m), where m > 1 and n > 1. We specify sufficient conditions for the projective image of a one-generated ring decomposable into a sum of Galois rings and a nil ideal to be generated by one element.

A celebrated theorem of Wedderburn says that a finite associative division ring is a finite (commutative) field. An elegant proof of this, due to E. Artin [3, p. 72], reduces the problem to showing the reduced (or generic) norm has a nontrivial zero, and then applies the theorem (conjectured by Artin and proved by Chevalley) that a polynomial with coefficients in a finite field and zero constant term has a nontrivial zero in that field if its degree is less than the number of variables. The Wedderburn theorem was generalized by A. A. Albert to finite strictly power-associative division rings of characteristic # 2 in [1, p. 301 ] and [2, p. i1 ] (see also [5 ]). The proof reduced the powerassociative case to the associative case by using the properties of Jordan division algebras. It is the purpose of this note to show that Artin's method carries over directly to the power-associative case using the generic norm introduced by N. Jacobson [4]. By avoiding Jordan algebras one is able to extend Albert's result to the characteristic 2 case. The usual definition of a division ring requires that left and right multiplications by a nonzero element be bijective. If the ring is finite it is necessarily an algebra, over a finite field 4, which is algebraic (even finite-dimensional) and without zero divisors. Also, as we shall see later, the algebra necessarily has a unit element. We wish to consider something slightly more general. For our purposes we say an algebraic power-associative algebra is a division algebra if it is unital and no element xO 0 is a zero divisor in the subalgebra (D[x] it generates (since 4<[x] is finite-dimensional, this is equivalent to saying x is invertible in b [x]). Our conditions are less restrictive, since the Jordan algebra of a quadratic form can be a division algebra in our sense with zero divisors. With this definition we have

In this paper, we study generalized Reed-Solomon codes (GRS codes) over commutative and noncommutative rings, we show that the classical Welch-Berlekamp and Guruswami-Sudan decoding algorithms still hold in this context, and we investigate their complexities. Under some hypothesis, the study of noncommutative GRS codes over finite rings leads to the fact that GRS codes over commutative rings have better parameters than their noncommutative counterparts. Also, GRS codes over finite fields have better parameters than their commutative rings counterparts. But we also show that given a unique decoding algorithm for a GRS code over a finite field, there exists a unique decoding algorithm for a GRS code over a truncated power series ring with a better asymptotic complexity. Moreover, we generalize a lifting decoding scheme to obtain new unique and list decoding algorithms designed to work when the base ring is, for example, a Galois ring or a truncated power series ring or the ring of square matrices over the latter ring.