Galois Fields

In this paper, we develop the theory for constructing DNA cyclic codes of odd length over $R=\Z_4[u]/\langle u^2-1 \rangle$ based on the deletion distance. Firstly, we relate DNA pairs with a special 16 elements of ring $R$. Cyclic codes of odd length over $R$ satisfy the reverse constraint and the reverse-complement constraint are discussed in this paper. We also study the $GC$-content of these codes and their deletion distance. The paper concludes with some examples of cyclic DNA codes with $GC$-content and their respective deletion distance.

El proyecto que se presenta es una herramienta grafica didáctica utilizando el complemento Guide de Matlab. Para esto hemos dividido la interfaz en dos partes: la primera que será de comparación visual, donde la persona aplica los diferentes operadores morfológicos conocidos (dilatación, erosión, cierre y apertura), a una imagen con su respectivo elemento estructurante, y observara de manera visual el efecto que produce el operador morfológico en la imagen resultante y la parte de aplicación, en la cual podremos obtener la cantidad de objetos que presenta la imagen, con sus respectivas áreas y perímetros de cada forma o estructura que muestre la imagen.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is constant equal to the inverse 1/ √ d, with d the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality d + 1 have been derived for prime power dimensions d = p m using the tools of abstract algebra (Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions the cardinality is much less. The bases can be reinterpreted as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. The MUB states are related to additive characters of Galois fields (in odd characteristic p) and of Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for physical states and find them related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in our quest of minimal uncertainty in quantum information primitives.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

NOTICE June 2025: A few centuries have passed since Leibniz in 1695. What is Beyond the Fractional Derivative? Consult Sec. 9.4, 9.3 of "Solution to 9 Notable Problems.." It raises entirely new questions in both areas: Galois Fields (and K-Algebras) and Differential Operators. Someone (FAST) will have to surpass the other challenges posed there, resulting in a generalization. Find apps for these novel non-traditional, non-fractional operators.
ABSTRACT:
We extend the Theory of Differentially δ-Uniform Functions in its various confines, now its influence is multiplied; we provide a special collection of K-Algebras. This solves hot challenges, e.g.: 1). We generalize for the second time Dillon's switching method in the (general) differentially uniform context. 2). With a singular style, we develop the analysis of Differential Equations exclusively for Galois fields (Section 9.4: an in-depth discussion, plus questions that require immediate action); on which there is hardly any research done and in a completely different sense, where the investigated functions have co-domain that is never the structure F_pn. Besides, we generalize the elegant BDC of Steinbach-Posthoff, and construct fascinating Chain Complexes. We introduce the F_pn-Schrödinger equation (and the newest Black-Scholes). 3). We obtain the novel Fundamental Theorem of Calculus associated with a derivative order type other than the famous Fractional. Further, we conduct the corresponding Algebraic Attack on the functions that currently offer the best resistance, and identify the survivors.

In this paper we study the structure and properties of additive right and left polycyclic codes induced by a binary vector a in Fn 2 . We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if C is a right polycyclic code induced by a vector a ∈ Fn 2 , then the Hermitian dual of C is a sequential code induced by a. As an application of these codes, we present examples of additive right polycyclic codes over F4 with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over F4. 1. Preliminaries A linear code of length n over a finite field F is a subspace of F. An Additive code of length n over a finite field F is a subgroup of F. Additive codes over the finite field F4 = { 0, 1, α, α } where α + α + 1 = 0 were introduced in [3] because of their applications in quantum computing. Define t...

Linear code defined by a sparse parity check matrix It is empirically known that (2,k)-regular LDPC codes exhibit good decoding performance. Non-binary low-density parity-check (LDPC) codes Linear code defined by a sparse parity check matrix It is empirically known that (2,k)-regular LDPC codes exhibit good decoding performance.

This paper presents a new area efficient, bit-level pipelined, linear systolic array architecture performing exponentiation over large Finite Field GF(2 m ). It is based on one multiplier and implements the square-and-multiply algorithm. The architecture is regular, expendable to any field order and programmable with respect to field-generating polynomial P(x). It allows the input elements to enter a linear systolic array in the same order and the system only requires one pipelined control signal. The operations are overlapped at higher system frequency in order to reduce the total delay of the exponentiation computation. A systematic approach is applied for implementing a digit-serial architecture in order to reduce power consumption. The resulting circuit is highly regular and programmable with respect to P(x). An analysis of the performance comparison is described as function of the digit-size. A comparison is made with the bit serial architecture based on the performance improve...

It is shown that at the present stage of the evolution of the universe, cosmological acceleration is an inevitable kinematical consequence of quantum theory in semiclassical approximation. Quantum theory does not involve such classical concepts as Minkowski or de Sitter spaces. In classical theory, when choosing Minkowski space, a vacuum catastrophe occurs, while when choosing de Sitter space, the value of the cosmological constant can be arbitrary. On the contrary, in quantum theory there are no uncertainties in view of the following: 1) the de Sitter algebra is the most general ten-dimensional Lie algebra; 2) the Poincare algebra is a special degenerate case of the de Sitter algebra in the limit R → ∞ where R is the contraction parameter for the transition from the de Sitter to the Poincare algebra and R has nothing to do with the radius of de Sitter space; 3) R is fundamental to the same extent as c and h: c is the contraction parameter for the transition from the Poincare to the Galilean algebra and h is the contraction parameter for the transition from quantum to classical theory; 4) as a consequence, the question (why the quantities (c, h, R) have the values which they actually have) does not arise. The solution to the problem of cosmological acceleration follows from results on irreducible representations of the de Sitter algebra. This solution is free of uncertainties and does not involve dark energy, quintessence and other exotic mechanisms the physical meaning of which is a mystery.

Several mathematical frameworks of phase-locking are developed. The classical Huyghens approach is generalized to include all harmonic and subharmonic resonances and is found to be connected to 1/f noise and prime number theory. Two types of quantum phase-locking operators are defined, one acting on the rational numbers, the other on the elements of a Galois field. In both cases we analyse in detail the phase properties and find that they are related to the Riemann zeta function and to incomplete Gauss sums, respectively.

, J. Hansen uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in P 2 . In this paper, we generalize Hansen's results from P 2 to P m ; we also show that the hypotheses of [8] may be weakened. The proof is succinct and follows by combining the Cayley-Bacharach Theorem and the bounds on evaluation codes obtained in .

It is shown that in a quantum theory over a Galois field, the famous Dirac's result about antiparticles is generalized such that a particle and its antiparticle are already combined at the level of irreducible representations of the symmetry algebra without assuming the existence of a local covariant equation. We argue that the very existence of antiparticles is a strong indication that nature is described by a finite field rather than by complex numbers.

– This paper discusses methods for efficient implementation of algorithms for determination of Fixed-polarity Galois field (FPGF) expressions for q-valued logic functions (p- prime). Calculations of coefficients in FPGF expressions can be most efficiently performed in terms of space and time, if conversion of an expression for a given polarity into the expression for another polarity is done along the so-called extended dual polarity route. However, there are many possible extended dual polarity routes which differ in the associated processing rules. The complexity of whatever software or hardware resources required for calculations of FPGF expressions depends on the number of different processing rules required along a route. In this paper, we introduce the notion of homogeneous extended dual polarity routes and show that calculations along these routes can be performed with a reduced number of processing rules required. Due to that, both hardware and software implementation of the...

This paper proposes to combine classical approaches to the optimization of arithmetic expressions by choosing po- larity of variables with the linearization of switching func- tions (LSF). Experimental results presented in the paper, show that in classical approaches, transition from Positive-polarity arithmetic expressions to Fixed-polarity arithmetic expres- sions may provide for 30-40% of reduced number of coeffi- cients. However, when combined with LSF another 20% of reduction in the non-zero coefficients count can be achieved.

Era uma noite de inverno e os termómetros não precisavam anunciar. Todos já sentiam, é certo que uns muito mais que os outros, a frieza daquela noite. Se nós, nas ruas, falássemos, mesmo que baixinho, uma densa fumaça branca sairia de nossas bocas e narinas. Se nos tocássemos, fora do cobertor, a ponta de nossos dedos se enrugariam imediatamente. Se nos beijássemos, nossas línguas se secariam, nossas bocas se rachariam. Se, por algum deslize, escorregássemos para o chão, fora do papelão úmido, congelaríamos. No entanto, tudo isso pouco importava, não poderiam impedir o toque carinhoso, pois arriscaríamos o infortúnio de qualquer modo, implicitamente já havíamos prometido aquecer um ao outro em todas as noites frias, seja com os olhos, com as mãos, com a boca ou simplesmente com o amor. Tocando seu peito, olhávamos fixamente um para o outro quando, do outro lado da rua escura, ouvimos o barulho de um grande portão se abrindo. Nenhuma luz se acendeu no casarão. Aquietamo-nos de imediato nas sombras e, de lá, cautelosamente saiu um homem. Era alto, bem agasalhado, imaginamos que entre seus trinta e quarenta anos, e carregava consigo uma caixa que, por seu volume, dificultava suas tentativas de fechar a passagem que acabara de fazer uso. Observamos de longe enquanto ele se esforçava para fazê-lo sem depositar a caixa no chão. Não conseguiu. O constante barulho resultou na aparição de uma pequena garotinha, que descendo depressa e descalça as escadas, acendeu uma luz. Estava eufórica, usava um pijama colorido e tinha um semblante assustado. Encarando o homem, disse: "Pai, fica! Não precisa ir agora". Ele pouco se importou. Ficaram se entreolhando por alguns instantes e, pelo tempo que se seguiu, acreditamos que tenham trocado outras palavras, mas a distância -maldita seja ela nessas horas -nos excluía do restante da conversa. Passado o instante, ele voltou-se novamente para a rua, sem ao menos respondê-la, sequer olhou para trás. "Pai, para! Não vai embora!", repetia ela chorando. Da Mulher de Ló à Orfeu e Eurídice, a história nos ensinou que olhar para trás em momentos de grande decisão pode desencadear arrependimentos, maldições e desgraças, muitas vezes irreversíveis. Por exemplo, no primeiro caso, Ló foi avisado por dois anjos que deveria abandonar, junto de sua mulher e filhos, a cidade de Sodoma que, infestada pelo pecado, seria completamente destruída pelo Deus de Abraão. Para que todos eles se salvassem do castigo divino, a única condição imposta pelos seres celestiais era de que ninguém, enquanto fugissem da cidade, olhasse para trás. Contudo, a Mulher de Ló, sem conter o seu ímpeto, desobedeceu a ordem de seus guardiões e, fazendo-o, transformou-se em uma estátua de sal. Nota-se que, no segundo caso, não menos verídico, à tragédia inicia-se com a morte de Eurídice. Orfeu, inconformado com a perda de sua amada, dirige-se ao mundo dos mortos para buscá-la, desejando convencer o próprio deus da morte, Hades, a libertá-la. Tocando sua harpa, o herói consegue fazê-lo, mas com a única condição de que, até chegar ao mundo dos vivos, nenhum dos dois deveria olhar para trás. Orfeu concordou e, tomando Eurídice, caminharam em di-

This paper introduces a new block encryption algorithm designed for the cryptographic protection of data. The paper introduces and explains a newly devised exponentiation modulo (EM) transform method, utilized to obtain the S-block, an essential element within the presented algorithm. A method of optimizing the choice of keys and increasing the efficiency of calculation was also used. It is proposed that incorporating characteristics of cryptographic primitives functioning within the Galois field into the algorithm can lead to favorable outcomes. To increase the encryption algorithm's speed, non-positional polynomial notation systems and a working base index table are used. The paper discusses the implementation of an encryption algorithm in C++ and examines the statistical characteristics of the resulting ciphertexts. For experimental testing of statistical safety, a set of statistical tests by National Institute of Standards and Technology (NIST) and D. Knuth was used. Furthermore, the resulting S-box was examined using linear, differential, and algebraic cryptanalysis techniques. In the future, this proposed S-box will be implemented in the encryption algorithm being developed for the preliminary encryption of confidential data.

Low-density parity-check (LDPC) codes are widely recognized for their excellent forward error correction, near-Shannon-limit performance, and support for high data rates with effective hardware parallelization. Their convolutional counterpart, LDPC convolutional codes (LDPC-CCs), offer additional advantages such as variable codeword lengths, unlimited paritycheck matrices, and simpler encoding and decoding. These features make LDPC-CCs particularly suitable for practical implementations with varying channel conditions and data frame sizes. This paper investigates the performance of LDPC-CCs using the extrinsic information transfer (EXIT) chart, a graphical tool for analyzing iterative decoding. EXIT charts visualize mutual information exchange and help predict convergence behavior, estimate performance thresholds, and optimize code design. Starting with the EXIT chart principles for LDPC codes, we derived the mutual information functions for variable and check nodes in regular and irregular LDPC-CC tanner graphs. This involved adapting existing EXIT functions to the periodic parity-check matrix of LDPC-CCs. We compare regular and irregular LDPC-CC constructions, examining the impact of degree distributions and the number of periods in the parity-check matrix on convergence behavior. Our simulations show that irregular LDPC-CCs consistently outperform regular ones, and the EXIT chart analysis confirms that LDPC-CCs demonstrate superior bit error rate (BER) performance compared to equivalent LDPC block codes.

It is shown that in a quantum theory over a Galois field, the famous Dirac's result about antiparticles is generalized such that a particle and its antiparticle are already combined at the level of irreducible representations of the symmetry algebra without assuming the existence of a local covariant equation. We argue that the very existence of antiparticles is a strong indication that nature is described by a finite field rather than by complex numbers.

de Sitter symmetry on quantum level implies that operators describing a given system satisfy commutation relations of the de Sitter algebra. This approach gives a new perspective on fundamental notions of quantum theory. We discuss applications of the approach to the cosmological constant problem, gravity, and particle theory.

In this paper, a parallel, power-efficient and scalable word-based crypto architecture is proposed that performs the operations required for scalar point multiplication including add, multiplication and inversion operations on GF(2 m ) operands. The proposed architecture distinguishes itself from exiting architectures, including our prior architecture, by the fact that its resource usage and power-consumption is based on the input data. Hence, such architecture might be used for various operand sizes without modifying or reconfiguring the underlying hardware. The architecture has also the ability to perform several different operations in parallel when each operation requires a small key size which significantly increases the overall performance and throughput of the system. In the absence of parallel requests, the remaining unused modules will be turned off in order to save power. The experimental results show significant improvement in the timing, throughput and energy performances with a slight overhead in the circuit area.

By extending the notion of minimum rank distance, this paper introduces two new relative code parameters of a linear code C 1 of length n over a field extension F q m and its subcode C 2 C 1 . One is called the relative dimension/intersection profile (RDIP), and the other is called the relative generalized rank weight (RGRW). We clarify their basic properties and the relation between the RGRW and the minimum rank distance. As applications of the RDIP and the RGRW, the security performance and the error correction capability of secure network coding, guaranteed independently of the underlying network code, are analyzed and clarified. We propose a construction of secure network coding scheme, and analyze its security performance and error correction capability as an example of applications of the RDIP and the RGRW. Silva and Kschischang showed the existence of a secure network coding in which no part of the secret message is revealed to the adversary even if any dim C 1 -1 links are wiretapped, which is guaranteed over any underlying network code. However, the explicit construction of such a scheme remained an open problem. Our new construction is just one instance of secure network coding that solves this open problem.

A new cellular structure for a versatile Reed-Solomon (RS) decoder is introduced based on time domain decoding algorithm. The time domain decoding algorithm is restructured to be suitable for introducing the cellular structure. The main advantages of this structure are its versatility and very simple cellular structure. By versatile decoder we mean a decoder that can be programmed to decode any (n, k) RS code defined in Galois field 2 m with a fixed block length n and a fixed symbol size m. This decoder can correct both errors and erasures for any message length k. The introduced decoder is cellular and has a very simple structure and hence it is suitable for VLSI designs. Index Terms-Cellular structure, comparison of RS decoding structures, time domain RS decoder, versatile RS decoder, VLSI.

Codes over the ring of integers modulo 4 have been studied by many researchers. Negacyclic codes such that the length of the code is odd have been characterized over the alphabet 4 , and furthermore, have been generalized to the case of the alphabet being a finite commutative chain ring. In this paper, we investigate negacyclic codes of length 2 over Galois rings. The structure of negacyclic codes of length 2 over the Galois rings GR(2), as well as that of their duals, are completely obtained. The Hamming distances of negacyclic codes over GR(2) in general, and over 2 in particular are studied. Among other more general results, the Hamming distances of all negacyclic codes over 2 of length 4 8 and 16 are given. The weight distributions of such negacyclic codes are also discussed.

Cuando se trabaja en proyectos de diseño, utilizando FPGA,  es necesario tomar decisiones respecto de cómo abordar cálculos complejos, considerando los valores ya sea en Punto Fijo o en Punto Flotante. Entonces el diseñador puede plantearse preguntas como ¿cuál aritmética se debe usar? ¿qué aritmética es mejor para el problema que se desea resolver? ¿qué criterios se debe aplicar para tomar la decisión? ¿da lo mismo cuál se use?
Estas preguntas se abordan en esta presentación, al definir funciones matemáticas, diseñar su implementación en FPGA, y establecer criterios de comparación.

This paper presents a thorough exploration of Cryptoprocessors in the context of the Internet of Things (IoT). The focus is on hardware support, cryptographic processors, and energy-efficient designs to fortify the security of IoT ecosystems. Leveraging insights from recent research papers, the survey examines various aspects of Cryptoprocessor implementations, ranging from lightweight countermeasures to advanced latticebased post-quantum cryptography. The proposed paper amalgamates findings from studies on hardware architectures, energyefficient designs, and innovative cryptographic algorithms to provide a holistic overview of state-of-the-art advancements in IoT security. The synthesis of this survey's exploration aims to contribute to the ongoing discourse on securing IoT devices through robust hardware-backed cryptographic solutions.

A new approach to ensuring information security is proposed for using in the zone of direct radio visibility. The approach is based on a direct analogy between radio and optical radiation. In the latter case, information protection can be ensured by identifying the point or direction from which the signal comes; all other light sources are cut off by the observer. It is shown that a similar method of information protection can also be implemented in the radio range, which corresponds to modern trends in the development of radio holography. The simplest scheme for detecting the direction to a transmitter, identified as “friend” is presented. Its operability is demonstrated by means of simulation modeling. Ways for the further development of this approach are determined, based on the fact that modern technologies make it possible to implement distributed reception of a radio signal (to use a relatively large number of receivers), which makes it possible to implement direct analogues of such optical elements as a lens, holograms.

This paper analyzes the transformative role of algebraic geometry in the development of robust error-correcting codes, essential for maintaining data integrity in digital communication and storage systems. By leveraging the mathematical structures of algebraic curves over finite fields, algebraic-geometric (AG) codes have emerged as powerful alternatives to classical codes, such as Reed-Solomon and BCH. We explore how key geometric properties-such as genus and rational points-contribute to the construction of codes with enhanced error-correction capabilities, outperforming traditional methods in terms of minimum distance and code length. Additionally, recent advancements in decoding algorithms for AG codes are discussed, addressing challenges related to computational complexity. Through a detailed comparative analysis, the paper highlights the practical advantages of AG codes in modern communication systems, demonstrating their potential for improving data reliability in high-noise environments. Finally, we c

Since the debut of the Advanced Encryption Standard (AES), it has been thoroughly studied by hardware designers with the goal of reducing the area and delay of the hardware implementation of this cryptosystem. This paper proposes an implementation of the AES mix columns operation. In this paper, a compact architecture for the AES mix columns operation and its inverse is presented. The hardware implementation is compared with previous work done in this area. We show that our design has a lower gate count than other designs that implement both the forward and the inverse mix columns operation.

In this contribution, we propose two successive interference cancellation (SIC) schemes for a fast frequency hopping (FFH) multiple access (MA) system using M-ary frequency shift keying (MFSK) and invoking multiuser detection (MUD). One of the proposed schemes invokes clipped combining, while the other scheme employs both product combining and clipped combining. The SIC schemes are adapted from a scheme proposed by U.-C. Fiebig in 1996. The basic principle of the SIC schemes is that detection is carried out in multiple stages and during each stage, only the most reliable symbols are detected. In subsequent stages, the interference contributed by the already detected symbols may be removed. The performance of the proposed schemes is evaluated and compared to that of Fiebig's scheme, when the FFH-MFSK system operates in a Nakagamim fading MA channel. The simulation results demonstrate that the proposed schemes attain a better bit error rate performance than Fiebig's scheme.

We show how to construct large families of quantum stabilizer and subsystem codes from algebro-geometric toric codes extending the known construction of subsystem codes from cyclic codes and extending the construction of stabilizer codes from toric codes in an earlier work by one of the authors. Since algebro-geometric toric codes are higher dimensional extensions of cyclic codes, we obtain this way a new and rich source of quantum stabilizer and subsystem codes. The fact that these codes are produced from polytopes enable us to produce large families of codes by varying the shape of the polytopes. In fact, by varying the shape of the polytopes defining the toric varieties, we show that we obtain families of quantum codes whose parameters improve steadily and with several of these codes being either Quantum MDS codes or Near Quantum MDS codes.

We prove that the lexicographic, degree lexicographic and the degree reverse lexicographic orders for monomials in R n = K[X 1 ,. .. , X n ] are uniquely determined by their induced orderings, (i.e. their restrictions to R n,i = K[X 1 ,. .. ,X i ,. .. , X n ]), when n 4. We also show that for any n 4 there are monomial orders that are not uniquely determined by their induced orderings, and provide examples of these orders for each n. the ustars conference proceedings 1 (2014), #R00

A new algorithm is shown that converts disjoint cube representation of Boolean functions into fixed-polarity Generalized Reed-Muller Expansions (GRME). Since the known fast algorithm Ihal generates the GRME based on the factorization or thc Reed-Muller transform matrix always starts from the truth table (minterms) of Boolean function, then the described method has the advantages due to smaller required computer memory. Moreover, for the Boolean functions described by only fcw disjoint cubes the method is much more efficient than the fast algorithm.

An important part of based on elliptical curves cryptographic data protection is multipliers of Galois fields. For based on elliptical curves digital signatures, not only prime but also extended Galois fields GF(pm) are used. The article provides a theoretical justification for the use of extended Galois fields GF(dm) with characteristics d > 2, and a criterion for determining the best field is presented. With the use of the proposed criterion, the best fields, which are advisable to use in data protection, are determined.
Cores (VHDL descriptions of digital units) are considered as structural part of based on FPGA devices. In the article methods for cryptoprocessors cores creating were analyzed. The article describes the generator of VHDL descriptions of extended Galois field multipliers with big characteristic (up to 2998). The use of mathematical packages for calculations to improve the quality of information security is also considered.
The Galois field multipliers generator creates the VHDL description of multipliers schemes, describes connections of their parts and generates VHDL descriptions of these parts as result of Quine-McCluskey Boolean functions minimization method. However, the execution time of the algorithm increases with increasing amount of input data. Accordingly, generating field multipliers with large characteristic can take frерom a few seconds to several tens of seconds.
It's important to simplify the design and minimize logic gates number in a field programmable gate array (FPGA) because it will speed up the operation of multipliers. The generator creates multipliers according to the three variants.
The efficiency of using multipliers for fields with different characteristics was compared in article.
The expediency of using extended Galois fields GF(dm) with characteristics d > 2 in data protection tools is analyzed, a criterion for comparing data protection tools based on such Galois fields is determined, and the best fields according to the selected criterion when implemented according to a certain algorithm are determined.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The article describes the function of the BCH code decoder for correcting of double errors in main memory. Some basic algebraic concepts needed to understand the structure of the BCH code, which is used to secure the main memory of a "Multiprocessor Computing System", are explained and the method of decoding this code is shown as an example.

Resumen El proyecto que se presenta es una herramienta grafica didáctica utilizando el complemento Guide de Matlab. Para esto hemos dividido la interfaz en dos partes: la primera que será de comparación visual, donde la persona aplica los diferentes operadores morfológicos conocidos (dilatación, erosión, cierre y apertura), a una imagen con su respectivo elemento estructurante, y observara de manera visual el efecto que produce el operador morfológico en la imagen resultante y la parte de aplicación, en la cual podremos obtener la cantidad de objetos que presenta la imagen, con sus respectivas áreas y perímetros de cada forma o estructura que muestre la imagen.

We argue that the main reason of crisis in quantum theory is that nature, which is fundamentally discrete and even finite, is described by classical mathematics involving the notions of infinitely small, continuity etc. Moreover, since classical mathematics has its own foundational problems which cannot be resolved (as follows, in particular, from Gödel's incompleteness theorems), the ultimate physical theory cannot be based on that mathematics. In the first part of the work we discuss inconsistencies in standard quantum theory and reformulate the theory such that it can be naturally generalized to a formulation based on finite mathematics. It is shown that: a) as a consequence of inconsistent definition of standard position operator, predictions of the theory contradict the data on observations of stars; b) the cosmological acceleration and gravity can be treated simply as kinematical manifestations of quantum de Sitter symmetry, i.e. the cosmological constant problem does not exist, and for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed. In the second part we first prove that classical mathematics is a special degenerate case of finite mathematics in the formal limit when the characteristic p of the field or ring in the latter goes to infinity. This implies that mathematics describing nature at the most fundamental level involves only a finite number of numbers while the notions of limit and infinitely small/large and the notions constructed from them (e.g. continuity, derivative and integral) are needed only in calculations describing nature approximately. In a quantum theory based on finite mathematics, the de Sitter gravitational constant depends on p and disappears in the formal limit p → ∞, i.e. gravity is a consequence of finiteness of nature. The application to particle theory gives that the notion of a particle and its antiparticle is only approximate and, as a consequence: a) the electric charge and the baryon and lepton quantum numbers can be only approximately conserved; b) particles which in standard theory are treated as neutral (i.e. coinciding with their antiparticles) cannot be elementary. We argue that only Dirac singletons can be true elementary particles and discuss a conjecture that classical time t manifests itself as a consequence of the fact that p changes, i.e. p and not t is the true evolution parameter.

We consider massless elementary particles in quantum theory based on a Galois field (GFQT). In our recent paper hep-th/0206078 where the GFQT has been discussed in detail, it has been shown that the theory possesses a new symmetry between particles and antipar-ticles, which has no analog in the standard approach. In the present paper we prove that the symmetry is compatible with all the operators describing such particles. As a consequence, the existence of mass-less neutral elementary particles in the GFQT is incompatible with the usual relation between spin and statistics (in particular, this implies that in the GFQT, the photon and the graviton can be only composite particles). We argue that the same statement is also valid for neutral particles with arbitrary masses.

In our previous papers it has been shown that quantum theory based on a Galois field (GFQT) possesses a new symmetry between particles and antiparticles, and for massless particles this symmetry (called the AB one) is compatible with all representation operators of the symmetry algebra. In the present paper, it is shown that the AB symmetry is compatible with all representation operators of the symmetry algebra in the general case. As a consequence of simple arithmetical considerations , this symmetry is compatible with the vacuum condition, only for particles with half-integer spin (in usual units). If the AB symmetry is combined with the spin-statistics theorem, one arrives at the following conclusions: in quantum theory based on a Galois field i) any neutral particle can be only composite but not elementary (this property has been proved for the massless case in hep-th/0207192); ii) any interaction can involve only an even number of creation/annihilation operators.

Standard mathematics involves such notions as infinitely small/large, continuity and standard division. This mathematics is usually treated as fundamental while finite mathematics is treated as inferior. Standard mathematics has foundational problems (as follows, for example, from Gödel's incompleteness theorems) but it is usually believed that this is less important than the fact that it describes many experimental data with high accuracy. We argue that the situation is the opposite: standard mathematics is only a degenerate case of finite one in the formal limit when the characteristic of the ring or field used in finite mathematics goes to infinity. Therefore foundational problems in standard mathematics are not fundamental.

It is argued that the main reason of crisis in quantum theory is that nature, which is fundamentally discrete, is described by continuous mathematics. Moreover, no ultimate physical theory can be based on continuous mathematics because, as follows from Gödel's incompleteness theorems, any mathematics involving the set of all natural numbers has its own foundational problems which cannot be resolved. In the first part of the paper inconsistencies in standard approach to quantum theory are discussed and the theory is reformulated such that it can be naturally generalized to a formulation based on discrete and finite mathematics. Then the cosmological acceleration and gravity can be treated simply as kinematical manifestations of de Sitter symmetry on quantum level (i.e. for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed). In the second part of the paper motivation, ideas and main results of a quantum theory over a Galois field (GFQT) are described. In contrast to standard quantum theory, GFQT is based on a solid mathematics and therefore can be treated as a candidate for ultimate quantum theory. The presentation is non-technical and should be understandable by a wide audience of physicists and mathematicians.

We consider a system of two free bodies in de Sitter invariant quantum mechanics. De Sitter invariance is understood such that a set of operators describing a system satisfies commutation relations of the de Sitter algebra. Our approach does not involve quantum field theory, de Sitter space and its geometry (metric and connection). At very large distances the standard relative distance operator describes a well known cosmological acceleration. In particular, the cosmological constant problem does not exist and there is no need to involve dark energy or other fields for solving this problem. At the same time, for systems of macroscopic bodies this operator does not have correct properties at lesser distances and should be modified. We propose a modification which has correct properties, reproduces Newton's gravity and the precession of Mercury's perihelion if the width of the de Sitter momentum distribution δ for a macroscopic body is inversely proportional to its mass m. We argue that fundamental quantum theory should be based on a Galois field with a large characteristic p which is a fundamental constant characterizing laws of physics in our Universe. Then one can give a natural explanation that δ = const R/(mG) where R is the radius of the Universe (such that Λ = 3/R 2 is the cosmological constant) and G is a quantity defining Newton's gravity. A very rough estimation gives G ≈ R/(m N lnp) where m N is the nucleon mass. If R is of order 10 26 m then lnp is of order 10 80 and therefore p is of order exp(10 80). In the formal limit p → ∞ gravity disappears, i.e. in our approach gravity is a consequence of finiteness of nature.

It is shown that in a quantum theory based on a Galois field, the famous Dirac's result about antiparticles is generalized such that a particle and its antiparticle are already combined at the level of irreducible representations of the symmetry algebra without assuming the existence of a local covariant equation. We argue that the very existence of antiparticles is a strong indication that nature is described by a finite field rather than by complex numbers.

We consider massless elementary particles in a quantum theory based on a Galois field (GFQT). We previously showed that the theory has a new symmetry between particles and antiparticles, which has no analogue in the standard approach. We now prove that the symmetry is compatible with all operators describing massless particles. Consequently, massless elementary particles can have only the half-integer spin (in conventional units), and the existence of massless neutral elementary particles is incompatible with the spin-statistics theorem. In particular, this implies that the photon and the graviton in the GFQT can only be composite particles.

Systems of free particles in a quantum theory based on a Galois field (GFQT) are discussed in detail. In this approach infinities cannot exist, the cosmological constant problem does not arise and one irreducible representation of the symmetry algebra necessarily describes a particle and its antiparticle simultaneously. As a consequence, well known results of the standard theory (spin-statistics theorem; a particle and its antiparticle have the same masses and spins but opposite charges etc.) can be proved without involving local covariant equations. The spin-statistics theorem is simply a requirement that quantum theory should be based on complex numbers. Some new features of GFQT are as follows: i) elementary particles cannot be neutral; ii) the Dirac vacuum energy problem has a natural solution and the vacuum energy (which in the standard theory is infinite and negative) equals zero as it should be; iii) the charge operator has correct properties only for massless particles with the spins 0 and 1/2. In the AdS version of the theory there exists a dilemma that either the notion of particles and antiparticles is absolute and then only particles with a half-integer spin can be elementary or the notion is valid only when energies are not asymptotically large and then supersymmetry is possible.