Description of unitary representations of the group of infinite 𝑝-adic integer matrices

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

Two-Module Weight-Based Sum Code in Residue Ring Modulo M=4

The paper describes research results of features of error detection in data vectors by sum codes. The task is relevant in this setting, first of all, for the use of sum codes in the implementation of the checkable discrete systems and the technical means for the diagnosis of their components. Methods for sum codes constructing are described. A brief overview in the field of methods for sum codes constructing is provided. The article highlights codes for which the values of all data bits are taken into account once by the operations of summing their values or the values of the weight coefficients of the bits during the formation of the check vector. The paper also highlights codes that are formed when the data vectors are initially divided into subsets, in particular, into two subsets. An extension of the sum code class obtained by isolating two independent parts in the data vectors, as well as weighting the bits of the data vectors at the stage of code construction, is proposed. The paper provides a generalized algorithm for two-module weighted codes construction, and describes their features obtained by weighing with non-ones weight coefficients for one of data bits in each of the subvectors, according to which the total weight is calculated. Particular attention is paid to the two-module weight-based sum code, for which the total weight of the data vector in the residue ring modulo M = 4 is determined. It is shown that the purpose of the inequality between the bits of the data vector in some cases gives improvements in the error detection characteristics compared to the well-known two-module codes. Some modifications of the proposed two-module weighted codes are described. A method for calculating the total number of undetectable errors in the two-module sum codes in the residue ring modulo M = 4 with one weighted bit in each of the subsets is proposed. Detailed characteristics of error detection by the considered codes both by the multiplicities of undetectable errors and by their types (unidirectional, symmetrical and asymmetrical errors) are given. The proposed codes are compared with known codes. A method for the synthesis of two-module sum encoders on a standard element base of the single signals adders is proposed. The classification of two-module sum codes is presented.

Optimization of the FIR Filter Structure in Finite Residue Field Algebra

This paper introduces a method for optimizing non-recursive filtering algorithms. A mathematical model of a non-recursive digital filter is proposed and a performance estimation is given. A method for optimizing the structural implementation of the modular digital filter is described. The essence of the optimization is that by using the property of the residue ring and the properties of the symmetric impulse response of the filter, it is possible to obtain a filter having almost a half the length of the impulse response compared to the traditional modular filter. A difference equation is given by calculating the output sample of modules p1 … pn in the modified modular digital filter. The performance of the modular filters was compared with the performance of positional non-recursive filters implemented on a digital signal processor. An example of the estimation of the hardware costs is shown to be required for implementing a modular digital filter with a modified structure. This paper substantiates the expediency of applying the natural redundancy of finite field algebra codes on the example of the possibility to reduce hardware costs by a factor of two. It is demonstrated that the accuracy of data processing in the modular digital filter is higher than the accuracy achieved with the implementation of filters on digital processors. The accuracy advantage of the proposed approach is shown experimentally by the construction of the frequency response of the non-recursive low-pass filters.

On groups containing the additive group of the residue ring or the vector space

Groups which are most frequently used as key addition groups in iterative block ciphers include the regular permutation representation $\begin{array}{} \displaystyle V_{n}^{+} \end{array}$ of the group of vector key addition, the regular permutation representation $\begin{array}{} \displaystyle \mathbb{Z}_{{2^n}}^{+} \end{array}$ of the additive group of the residue ring, and the regular permutation representation $\begin{array}{} \displaystyle \mathbb{Z}_{{2^n} + 1}^ \odot \end{array}$ of the multiplicative group of a prime field (in the case where 2n + 1 is a prime number). In this work we consider the extension of the group Gn generated by $\begin{array}{} \displaystyle V_{n}^{+} \end{array}$ and $\begin{array}{} \displaystyle \mathbb{Z}_{{2^n}}^{+} \end{array}$ by means of transformations and groups which naturally arise in cryptographic applications. Examples of such transformations and groups are the groups $\begin{array}{} \displaystyle \mathbb{Z}_{{2^d}}^{+} \times V_{n - d}^{+} ~\text{and}~ V_{n - d}^{+}\times \mathbb{Z}_{{2^d}}^{+} \end{array}$ and pseudoinversion over the field GF(2n) or over the Galois ring GR(2md, 2m).

On the number of subsets of the residue ring such that the difference of any pair of elements is not invertible

The paper is concerned with subsets I of the residue group Zd in which the difference of any two elements is not relatively prime to d. The class of such subsets is denoted by U(d), the class of sets from U(d) of cardinality r is denoted by U(d, r). The present paper gives formulas for evaluation or estimation of |U(d)| and |U(d, r)|.