Generalized negligible morphisms and their tensor ideals

We introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $${\mathcal {C}}$$ C over a local ring R. If the maximal ideal of R is generated by a single element, we show that any thick ideal of $${\mathcal {C}}$$ C admits an explicitly given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.

The Walsh Transform of a Class of Boolean Functions

The Walsh transform is an important tool to investigate cryptographic properties of Boolean functions. This paper is devoted to study the Walsh transform of a class of Boolean functions defined as [see formula in PDF], by making use of the known conclusions of Walsh transform and the properties of trace function, and the conclusion is obtained by generalizing an existing result.

An Attribute-Based Access Control for IoT Using Blockchain and Smart Contracts

With opportunities brought by the Internet of Things (IoT), it is quite a challenge to maintain concurrency and privacy when a huge number of resource-constrained distributed devices are involved. Blockchain have become popular for its benefits, including decentralization, persistence, immutability, auditability, and consensus. Great attention has been received by the IoT based on the construction of distributed file systems worldwide. A new generation of IoT-based distributed file systems has been proposed with the integration of Blockchain technology, such as the Swarm and Interplanetary File System. By using IoT, new technical challenges, such as Credibility, Harmonization, large-volume data, heterogeneity, and constrained resources are arising. To ensure data security in IoT, centralized access control technologies do not provide credibility. In this work, we propose an attribute-based access control model for the IoT. The access control lists are not required for each device by the system. It enhances access management in terms of effectiveness. Moreover, we use blockchain technology for recording the attribute, avoiding data tempering, and eliminating a single point of failure at edge computing devices. IoT devices control the user’s environment as well as his or her private data collection; therefore, the exposure of the user’s personal data to non-trusted private and public servers may result in privacy leakage. To automate the system, smart contracts are used for data accessing, whereas Proof of Authority is used for enhancing the system’s performance and optimizing gas consumption. Through smart contracts, ciphertext can be stored on a blockchain by the data owner. Data can only be decrypted in a valid access period, whereas in blockchains, the trace function is achieved by the storage of invocation and the creation of smart contracts. Scalability issues can also be resolved by using the multichain blockchain. Eventually, it is concluded from the simulation results that the proposed system is efficient for IoT.

Isoscattering strings of concatenating graphs and networks

We identify and investigate isoscattering strings of concatenating quantum graphs possessing n units and 2n infinite external leads. We give an insight into the principles of designing large graphs and networks for which the isoscattering properties are preserved for $$n \rightarrow \infty $$ n → ∞ . The theoretical predictions are confirmed experimentally using $$n=2$$ n = 2 units, four-leads microwave networks. In an experimental and mathematical approach our work goes beyond prior results by demonstrating that using a trace function one can address the unsettled until now problem of whether scattering properties of open complex graphs and networks with many external leads are uniquely connected to their shapes. The application of the trace function reduces the number of required entries to the $$2n \times 2n $$ 2 n × 2 n scattering matrices $${\hat{S}}$$ S ^ of the systems to 2n diagonal elements, while the old measures of isoscattering require all $$(2n)^2$$ ( 2 n ) 2 entries. The studied problem generalizes a famous question of Mark Kac “Can one hear the shape of a drum?”, originally posed in the case of isospectral dissipationless systems, to the case of infinite strings of open graphs and networks.

Projective objects and the modified trace in factorisable finite tensor categories

For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:(1)${\mathcal{C}}$ always contains a simple projective object;(2)if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$-action;(3)the action of the Grothendieck ring of ${\mathcal{C}}$ on the span of internal characters of projective objects can be diagonalised;(4)the linearised Grothendieck ring of ${\mathcal{C}}$ is semisimple if and only if ${\mathcal{C}}$ is semisimple.Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in ${\mathcal{C}}$ carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of $S$-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular $S$-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.

Power trace functions over finite fields

We count the number of solutions to a power trace function equal to a constant and use this to find the probability of a successful attack on an authentication code proposed by Ding et al. (2005) [C. Ding, A. Salomaa, P. Solé and X. Tian, Three constructions of authentication/secrecy codes, J. Pure Appl. Algebra 196 (2005) 149–168].

Further results on permutation polynomials via linear translators

In this paper, by using linear translators, we characterize a new class of permutation polynomials of the form [Formula: see text], which has a more general form than [Formula: see text]. Then, we present the compositional inverses of such permutation polynomials. Furthermore, by specifying the functions [Formula: see text] and [Formula: see text], we can get some new permutation polynomials of the forms [Formula: see text] and [Formula: see text] where [Formula: see text] is the trace function.

Single Image Dehazing using Transmission Estimation

Image dehazing is a system which can reduce poor consequence of haze on pictures which increase competence of the picture/tape meting out method in the indistinct climate. The present study, presents a simple image dehazing technique. System deletes the initial transmittal, exactly depends on hidden area division which in turn reduces the initial transmittal. First, we build raw transmittal in Gaussian pyramid using selective transmittal extraction prior. Second, a patch based on Laplacian pyramid which gives the value of nonlinear re-trace function point by point. Third, is universal Laplacian map which states the segmentation-based re-traced image. All the three segmentation guide the performer for transmittal refinement. Finally, the dehazing output gets back from the refined transmittal and atmospheric scattered technique.

DEPENDENCE OF THE STRUCTURAL PROPERTIES OF THE DUAL BASIS ON THE TYPE OF THE CHARACTERISTIC POLYNOMIAL

The relevance of the topic is connected with the solution of problems of processing of recurrent sequences. The method used to determine the initial phase of a pseudorandom sequence constructed by the characteristic polynomial allows its processing directly in the receiving process. This approach does not require mandatory accumulation of the entire sequence, which significantly increases the efficiency of the system, which uses a dual basis as a method of decoding recurrent sequences. The method of decoding using the dual basis of the Galois field is an effective means of protection against errors in a digital data transmission system, where the codes of BCH, Reed-Solomon and other varieties of recurrent sequences can be used as noise resistant codes. At the same time, the calculation of the elements of the basis dual to the left power basis generates a number of problems related to mathematical calculations over the elements of a finite field. In this paper, the methods of finding the elements of the dual basis are considered, the variants of simplified procedures for their calculation are proposed, and the task is to study the structural properties of the dual basis, which depend on the type of the characteristic polynomial. Research subject. The work is devoted to the study of binary Galois fields, in particular, the properties of the left power and dual basis are affected. Method. On the example of M-sequences, the procedure for finding the elements of the dual basis through the trace function is considered in detail using the mathematical apparatus developed by O. S. Korovitsky. Core results. The analysis of the given properties of the dual basis allowed to reveal new structural dependences between its elements. Practical relevance. The revealed structural dependences between the elements of the dual basis of the Galois field make it possible to significantly simplify the implementation of the procedure for the formation of its elements.