Mathematics

Research

14 pages, 328 KiB

Open AccessFeature PaperArticle

by Víctor Álvarez, José Andrés Armario, María Dolores Frau, Félix Gudiel, María Belén Güemes and Amparo Osuna

Mathematics 2021, 9(8), 857; https://doi.org/10.3390/math9080857 - 14 Apr 2021

Cited by 2 | Viewed by 1405

Abstract

Since Horadam and de Launey introduced the cocyclic framework on combinatorial designs in the 1990s, it has revealed itself as a powerful technique for looking for (cocyclic) Hadamard matrices. Ten years later, the series of papers by Kotsireas, Koukouvinos and Seberry about Hadamard [...] Read more.

Since Horadam and de Launey introduced the cocyclic framework on combinatorial designs in the 1990s, it has revealed itself as a powerful technique for looking for (cocyclic) Hadamard matrices. Ten years later, the series of papers by Kotsireas, Koukouvinos and Seberry about Hadamard matrices with one or two circulant cores introduced a different structured approach to the Hadamard conjecture. This paper is built on both strengths, so that Hadamard matrices with cocyclic cores are introduced and studied. They are proved to strictly include usual Hadamard matrices with one and two circulant cores, and therefore provide a wiser uniform approach to a structured Hadamard conjecture. Full article

(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)

24 pages, 3920 KiB

Open AccessArticle

A Multi-Criteria Computer Package-Based Energy Management System for a Grid-Connected AC Nanogrid

by Carlos Roncero-Clemente, Eugenio Roanes-Lozano and Fermín Barrero-González

Mathematics 2021, 9(5), 487; https://doi.org/10.3390/math9050487 - 27 Feb 2021

Cited by 4 | Viewed by 1646

Abstract

The electric system scenario has been changing during the last years moving to a distributed system with a high penetration of renewables. Due to the unpredictable behavior of some renewables sources, the development of the energy management system is considered crucial to guarantee [...] Read more.

The electric system scenario has been changing during the last years moving to a distributed system with a high penetration of renewables. Due to the unpredictable behavior of some renewables sources, the development of the energy management system is considered crucial to guarantee the reliability and stability of the system. At the same time, increasing the lifespan of the energy storage system is one of the most important points to take into account. In this sense, a software package implemented in the computer algebra system Maple is proposed in this work to control a grid-connected nanogrid with hybrid energy storage system (composed by batteries and supercapacitors). The energy management system considers several rules as the state of charge of the energy storage system, the photovoltaic power generation and the load profile, the nanogrid power trend and the energy prices. The improved performance of the nanogrid is proven by simulations in MATLAB/Simulink. Full article

(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)

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15 pages, 478 KiB

Open AccessArticle

High-Speed Implementation of PRESENT on AVR Microcontroller

by Hyeokdong Kwon, Young Beom Kim, Seog Chung Seo and Hwajeong Seo

Mathematics 2021, 9(4), 374; https://doi.org/10.3390/math9040374 - 13 Feb 2021

Cited by 7 | Viewed by 2306

Abstract

We propose the compact PRESENT on embedded processors. To obtain high-performance, PRESENT operations, including an add-round-key, a substitute layer and permutation layer operations are efficiently implemented on target embedded processors. Novel PRESENT implementations support the Electronic Code Book (ECB) and Counter (CTR). The [...] Read more.

We propose the compact PRESENT on embedded processors. To obtain high-performance, PRESENT operations, including an add-round-key, a substitute layer and permutation layer operations are efficiently implemented on target embedded processors. Novel PRESENT implementations support the Electronic Code Book (ECB) and Counter (CTR). The implementation of CTR is improved by using the pre-computation for one substitute layer, two diffusion layer, and two add-round-key operations. Finally, compact PRESENT on target microcontrollers achieved 504.2, 488.2, 488.7, and 491.6 clock cycles per byte for PRESENT-ECB, 16-bit PRESENT-CTR (RAM-based implementation), 16-bit PRESENT-CTR (ROM-based implementation), and 32-bit PRESENT-CTR (ROM-based implementation) modes of operation, respectively. Compared with former implementation, the execution timing is improved by 62.6%, 63.8%, 63.7%, and 63.5% for PRESENT-ECB, 16-bit PRESENT-CTR (RAM based implementation), 16-bit PRESENT-CTR (ROM-based implementation), and 32-bit PRESENT-CTR (ROM-based implementation) modes of operation, respectively. Full article

(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)

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13 pages, 286 KiB

Open AccessArticle

A Discussion of a Cryptographical Scheme Based in

F

-Critical Sets of a Latin Square

by Laura M. Johnson and Stephanie Perkins

Mathematics 2021, 9(3), 285; https://doi.org/10.3390/math9030285 - 31 Jan 2021

Viewed by 1494

Abstract

This communication provides a discussion of a scheme originally proposed by Falcón in a paper entitled “Latin squares associated to principal autotopisms of long cycles. Applications in cryptography”. Falcón outlines the protocol for a cryptographical scheme that uses the

F

-critical sets associated [...] Read more.

This communication provides a discussion of a scheme originally proposed by Falcón in a paper entitled “Latin squares associated to principal autotopisms of long cycles. Applications in cryptography”. Falcón outlines the protocol for a cryptographical scheme that uses the

F

-critical sets associated with a particular Latin square to generate access levels for participants of the scheme. Accompanying the scheme is an example, which applies the protocol to a particular Latin square of order six. Exploration of the example itself, revealed some interesting observations about both the structure of the Latin square itself and the autotopisms associated with the Latin square. These observations give rise to necessary conditions for the generation of the

F

-critical sets associated with certain autotopisms of the given Latin square. The communication culminates with a table which outlines the various access levels for the given Latin square in accordance with the scheme detailed by Falcón. Full article

(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)

8 pages, 254 KiB

Open AccessArticle

Boolean Functions and Permanents of Sylvester Hadamard Matrices

by José Andrés Armario

Mathematics 2021, 9(2), 177; https://doi.org/10.3390/math9020177 - 17 Jan 2021

Cited by 6 | Viewed by 2045

Abstract

One of the fastest known general techniques for computing permanents is Ryser’s formula. On this note, we show that this formula over Sylvester Hadamard matrices of order

2^{m}

,

H_{m}

, can be carried out by enumerating m-variable Boolean functions [...] Read more.

One of the fastest known general techniques for computing permanents is Ryser’s formula. On this note, we show that this formula over Sylvester Hadamard matrices of order

2^{m}

,

H_{m}

, can be carried out by enumerating m-variable Boolean functions with an arbitrary Walsh spectrum. As a consequence, the quotient

p e r (H_{m}) / 2^{2^{m}}

might be a measure of the “density” of m-variable Boolean functions with high nonlinearity. Full article

(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)

20 pages, 858 KiB

Open AccessFeature PaperArticle

Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

by Raúl M. Falcón, Víctor Álvarez, María Dolores Frau, Félix Gudiel and María Belén Güemes

Mathematics 2021, 9(2), 113; https://doi.org/10.3390/math9020113 - 6 Jan 2021

Cited by 1 | Viewed by 1425

Abstract

The classical design of cocyclic Hadamard matrices has recently been generalized by means of both the notions of the cocycle of Hadamard matrices over Latin rectangles and the pseudococycle of Hadamard matrices over quasigroups. This paper delves into this topic by introducing the [...] Read more.

The classical design of cocyclic Hadamard matrices has recently been generalized by means of both the notions of the cocycle of Hadamard matrices over Latin rectangles and the pseudococycle of Hadamard matrices over quasigroups. This paper delves into this topic by introducing the concept of the pseudococycle of a partial Hadamard matrix over a Latin rectangle, whose fundamentals are comprehensively studied and illustrated. Full article

(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)

22 pages, 569 KiB

Open AccessArticle

Efficient Implementation of ARX-Based Block Ciphers on 8-Bit AVR Microcontrollers

by YoungBeom Kim, Hyeokdong Kwon, SangWoo An, Hwajeong Seo and Seog Chung Seo

Mathematics 2020, 8(10), 1837; https://doi.org/10.3390/math8101837 - 19 Oct 2020

Cited by 9 | Viewed by 2828

Abstract

As the development of Internet of Things (IoT), the data exchanged through the network has significantly increased. To secure the sensitive data with user’s personal information, it is necessary to encrypt the transmitted data. Since resource-constrained wireless devices are typically used for IoT [...] Read more.

As the development of Internet of Things (IoT), the data exchanged through the network has significantly increased. To secure the sensitive data with user’s personal information, it is necessary to encrypt the transmitted data. Since resource-constrained wireless devices are typically used for IoT services, it is required to optimize the performance of cryptographic algorithms which are computation-intensive tasks. In this paper, we present efficient implementations of ARX-based Korean Block Ciphers (HIGHT and LEA) with CounTeR (CTR) mode of operation, and CTR_DRBG, one of the most widely used DRBGs (Deterministic Random Bit Generators), on 8-bit AVR Microcontrollers (MCUs). Since 8-bit AVR MCUs are widely used for various types of IoT devices, we select it as the target platform in this paper. We present an efficient implementation of HIGHT and LEA by making full use of the property of CTR mode, where the nonce value is fixed, and only the counter value changes during the encryption. On our implementation, the cost of additional function calls occurred by the generation of look-up table can be reduced. With respect to CTR_DRBG, we identified several parts that do not need to be computed. Thus, precomputing those parts in offline and using them online can result in performance improvements for CTR_DRBG. Furthermore, we applied several optimization techniques by making full use of target devices’ characteristics with AVR assembly codes on 8-bit AVR MCUs. Our proposed table generation way can reduce the cost for building a precomputation table by around 6.7% and 9.1% in the case of LEA and HIGHT, respectively. Proposed implementations of LEA and HIGHT with CTR mode on 8-bit AVR MCUs provide 6.3% and 3.8% of improved performance, compared with the previous best results, respectively. Our implementations are the fastest compared to previous LEA and HIGHT implementations on 8-bit AVR MCUs. In addition, the proposed CTR_DRBG implementations on AVR provide better performance by 37.2% and 8.7% when the underlying block cipher is LEA and HIGHT, respectively. Full article

(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)

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14 pages, 312 KiB

Open AccessArticle

Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication

by Yansheng Wu and Yoonjin Lee

Mathematics 2020, 8(9), 1495; https://doi.org/10.3390/math8091495 - 3 Sep 2020

Cited by 4 | Viewed by 2583

Abstract

It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the [...] Read more.

It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length

n \geq 7

. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes. Full article

(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)

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14 pages, 661 KiB

Open AccessArticle

Comparison of Entropy and Dictionary Based Text Compression in English, German, French, Italian, Czech, Hungarian, Finnish, and Croatian

by Matea Ignatoski, Jonatan Lerga, Ljubiša Stanković and Miloš Daković

Mathematics 2020, 8(7), 1059; https://doi.org/10.3390/math8071059 - 1 Jul 2020

Cited by 10 | Viewed by 5146

Abstract

The rapid growth in the amount of data in the digital world leads to the need for data compression, and so forth, reducing the number of bits needed to represent a text file, an image, audio, or video content. Compressing data saves storage [...] Read more.

The rapid growth in the amount of data in the digital world leads to the need for data compression, and so forth, reducing the number of bits needed to represent a text file, an image, audio, or video content. Compressing data saves storage capacity and speeds up data transmission. In this paper, we focus on the text compression and provide a comparison of algorithms (in particular, entropy-based arithmetic and dictionary-based Lempel–Ziv–Welch (LZW) methods) for text compression in different languages (Croatian, Finnish, Hungarian, Czech, Italian, French, German, and English). The main goal is to answer a question: ”How does the language of a text affect the compression ratio?” The results indicated that the compression ratio is affected by the size of the language alphabet, and size or type of the text. For example, The European Green Deal was compressed by 75.79%, 76.17%, 77.33%, 76.84%, 73.25%, 74.63%, 75.14%, and 74.51% using the LZW algorithm, and by 72.54%, 71.47%, 72.87%, 73.43%, 69.62%, 69.94%, 72.42% and 72% using the arithmetic algorithm for the English, German, French, Italian, Czech, Hungarian, Finnish, and Croatian versions, respectively. Full article

(This article belongs to the Special Issue Codes, Designs, Cryptography and Optimization)

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