Multiplicative Inverse

Let p be a prime number. We show that there is a one-to-one correspondence between the set of strongly nilpotent braces and the set of nilpotent pre-Lie rings of cardinality p n , for sufficiently large p. Moreover, there is an injective mapping from the set of left nilpotent pre-Lie rings into the set of left nilpotent braces of cardinality p n for n+1 < p. As an application, by using well known results about the correspondence between braces and Hopf-Galois extensions we use pre-Lie algebras to describe Hopf-Galois extensions.

We define combinatorial representations of finite skew braces and use this idea to produce a database of skew braces of small size. This database is then used to explore different concepts of the theory of skew braces such as ideals, series of ideals, prime and semiprime ideals, Baer and Wedderburn radicals and solvability. The paper contains several questions.

In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent F p -braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence between finite F p -braces and pre-Lie algebras proposed by Wolfgang Rump in 2014. As an application example, a classification of all right nilpotent F p -braces generated by one element of cardinality p 4 is obtained. It is also shown that the sum of a finite number of left nilpotent ideals in a left brace is a left nilpotent ideal, therefore every finite brace contains the largest left nilpotent ideal. The motivation for this paper is the following assertion, made by Wolfgang Rump on page 141 of for finite right braces: Suppose that G is the adjoint group of a brace A. The 1-cocycle G → A would then lead to a complete Right RSA struture of g via Lazard's correspondence. We provide a formal proof of this correspondence as it appears none have been published. This correspondence means we can use pre-Lie algebras to characterise finite braces of cardinality p n , and construct examples of braces using purely algebraic methods, instead of the more typical group theory-based methods or by computer or hand calculations. This can be used to characterise the structure of any finite brace, since it was shown in that every finite brace is completely determined by its adjoint group and braces which are its Sylow's subgroups. In , page 135, Rump developed a connection between left nilpotent R-braces and pre-Lie algebras over the field of real numbers. In the case of pre-Lie algebras over finite fields, this method can be applied to obtain braces from pre-Lie algebras for sufficiently large p using Lazard's correspondence. However it is not immediately clear how to obtain a pre-Lie algebra from every brace. It is also not clear if every brace will be an image of some pre-Lie algebra under Lazard's corresponence. Therefore it is not immediately clear how to attach a pre-Lie algebra in a reversible way to every brace, although it is clear how to assign pre-Lie algebras to braces which were already obtained from pre-Lie algebras using Lazard's correspondence in the different direction. Here we show how to attach to every finite right nilpotent F p -brace a finite nilpotent pre-Lie algebra over the field F p . We develop a simple algebraic formula for this passage from braces to pre-Lie algebras. For the passage the other way, from finite pre-Lie algebras to finite braces, we can use the method from page 135 . We then show that this correspondence is one-to-one. Moreover, the passages from braces to pre-Lie algebras and from pre-Lie algebras to braces are reversible by each other. Therefore our formulas, which at first glance do not resemble Lazard's correspondence, in fact correspond to Lazard's correspondence applied to multiplicative groups of braces and therefore agree

A low latency architecture to compute the multiplicative inverse and division in a finite field GF(2 ) is presented. Compared to other proposals with the same complexity, this circuit has lower latency and can be used in error-correction or cryptography to increase system throughput. This architecture takes advantage of the simplicity to computing powers (2 ) of an element in the Galois Field. The inverse of an element is computed in two stages: power calculation and multiplication. A division can be performed using only one more multiplication in the inversion circuit.

The substitution box (S-box) component is the heart of the Advanced Encryption Standard (AES) algorithm. The S-box values are generated from the multiplicative inverse of Galois finite field GF(2 8) with an affine transform. There are many techniques of gaining the multiplicative inverse values were proposed. Most of the hardware implementations of S-box were using look-up tables (LUTs) (memory-based) to store the values which employ the largest area in design. In this paper, a software method of producing the multiplicative inverse values, which is the generator of S-box values and the possibilities of implementing the methods in hardware applications will be discussed. The method is using the log and antilog values. The method is modified to create a memory-less value generator in AES hardware-based implementation. The implementation is proposed to embed on limited memory, small-sized FPGA.

This paper introduces the notion of a strongly prime ideal, and shows that the largest solvable ideal in a finite brace equals the intersection of all strongly prime ideals in this brace. This is used to generalise some well known results from ring theory into the context of braces and pre-Lie algebras. Several open questions are also posed.

Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.

For the implementation of error-correcting codes, cryptographic algorithms, and the construction of homomorphic methods for privacy-preserving, there is a need for methods of performing operations on elements GF(2m) that have low computational complexity. This paper analyzes the existing methods of performing operations on the elements GF(2m) and proposes a new method based on the use of a sparse table of elements of this field. The object of research is the processes of operations in information security systems. The subject of research is methods and algorithms for performing operations on elements GF(2m). The purpose of this research is to develop and improve methods and algorithms for performing operations on elements GF(2m) to reduce their computational complexity. Empirical methods and methods of mathematical and software modeling are used in the research. Existing and proposed algorithms are implemented using the C# programming language in the Visual Studio 2015 development environment. Experimental research of existing and developed algorithms was carried out according to the proposed method, which allows to level the influence of additional parameters on the results of the research. The conducted research on methods for performing operations on the elements GF(2m) shows the expediency of using a sparse table of field elements. This approach makes it possible to reduce the amount of RAM required for the software and hardware implementation of the developed method compared to the classical tabular method, which requires storage of a full table of correspondence of the polynomial and index representation of the field elements. In addition, the proposed method gives an increase in speed of more than 4 times for the operations of calculating the multiplicative inverse element and exponentiation. As a result, the proposed method allows to reduce the computational complexity of error-correcting codes, cryptographic algorithms, and the homomorphic methods for privacy-preserving.

This manuscript deals with the theorem on diminution of the Extended Euclidean Algorithm for finding the multiplicative inverse of non-zero elemental polynomials of Galois field  with respect to a monic irreducible polynomial  over , where  is a prime and is any positive integer. This method became successful in finding the multiplicative inverse of all those non-zero polynomials for which the Extended Euclidean Algorithm fails. We find the inverse of all 342 non-zero elemental polynomials of  using an irreducible polynomial  We also used Cayley Hamilton’s theorem for finding the multiplicative inverse of the non-zero elemental polynomials of a finite field.

Standardization of decimal floating-point formats by IEEE in IEEE 754-2008 Standards fuelled the interest on decimal floating-point architectures among the global research community. Although decimal arithmetic architecture research attracted computer scientists for the last two decades, the major thrust was observed past the year 2008. Multiple proposals have been witnessed for decimal arithmetic units, mostly adders/subtractors, and multipliers. Very few designs have been proposed in the division domain. This article proposes decimal division hardware based on sutras from Vedic Mathematics, the ancient mathematics system. We present a Reduced Magnitude Divisor Generator which converts each digit of the actual divisor into a reduced digit set [-5, 5] using a unique combination/modification of the Vedic Sutras. The divisor digit magnitude reduction also minimizes the product set of multiplication as the single-digit multiplier belongs to the reduced digit set [0, 5] barring the sign...

In this paper, we first discussed multiplicative metric mapping by giving some topological properties of the relevant multiplicative metric space. As an interesting result of our discussions, we observed that the set of positive real numbers R_+ is a complete multiplicative metric space with respect to the multiplicative absolute value function. Furthermore, we introduced concept of multiplicative contraction mapping and proved some fixed point theorems of such mappings on complete multiplicative metric spaces.

In this work, we determine the general solution of the quinquevigintic functional equation and also investigate its stability of this equation in the setting of matrix normed spaces and the framework of matrix non-Archimedean fuzzy normed spaces by using the fixed point method.

We prove a weak form of the Krull-Schmidt Theorem concerning the behavior of direct-product decompositions of $G$-groups, biuniform abelian $G$-groups, $G$-semidirect products and the $G$-set $Hom(H,A)$. Here $G$ and $A$ are groups and $H$ is a $G$-group. Our main result is the following. Let $P$ be any group. Let $H_1,\ldots,H_n, H_1&#39;,\ldots,H_t&#39;$ be $n+t$ biuniform abelian normal subgroups of $P$. Suppose that the products $H_1,\ldots,H_n, H_1&#39;,\ldots,H_t&#39;$ are direct, that is, $H_1 \ldots H_n = H_1 \times \ldots \times H_n$ and $H_1&#39;\ldots H_t&#39; =H_1&#39;\times\ldots\times H_t&#39;$. Then the normal subgroups $H_1\times\ldots\timesH_n$ and $H_1&#39;\times\ldots\times H_t&#39;$ of $P$ are $P$-isomorphic if and only if $n = t$ and there exist two permutations $\sigma$ and $\tau$ of $\{1,2,\ldots,n\}$ such that $[H_i]_m = [H&#39;_\sigma(i)]_m$ and $[H_i]_e = [H&#39;_\tau(i)]_e$ for every $i = 1,2,\ldots,n$.

The Fast Inverse Square Root algorithm has been used in 3D games of past for lighting and reflection calculations, because it offers up to four times performance gains. This paper presents a hardware implementation of the same algorithm on an FPGA board by designing the complete architecture and successfully mapping it on Xilinx Spartan 3E after thorough functional verification. The results show that this implementation provides a very efficient single-precision floating point inverse square root calculator with a practically accurate result being made available after just 12 short clock cycles. This performance measure is far superior to the software counterpart of the algorithm, and is not processor dependent like rsqrtss of x86 SSE instruction set. Results of this work can aid FPGA based vector processors or graphic processing units with 3D rendering. The hardware design can also form part of a larger floating point arithmetic unit for dedicated reciprocal square root calculations.

The Fast Inverse Square Root algorithm has been used in 3D games of past for lighting and reflection calculations, because it offers up to four times performance gains. This paper presents a hardware implementation of the same algorithm on an FPGA board by designing the complete architecture and successfully mapping it on Xilinx Spartan 3E after thorough functional verification. The results show that this implementation provides a very efficient single-precision floating point inverse square root calculator with a practically accurate result being made available after just 12 short clock cycles. This performance measure is far superior to the software counterpart of the algorithm, and is not processor dependent like rsqrtss of x86 SSE instruction set. Results of this work can aid FPGA based vector processors or graphic processing units with 3D rendering. The hardware design can also form part of a larger floating point arithmetic unit for dedicated reciprocal square root calculations.

General results on multiplicative lattices found recently by Facchini, Finocchiaro and Janelidze have been studied in the particular case of groups by Facchini, de Giovanni and Trombetti. In this paper we prove that these results hold not only for the multiplicative lattices of all normal subgroups of a group, but also for much more general multiplicative lattices. Therefore they can be applied to other algebraic structures, for instance to braces.

It is shown that over an arbitrary field there exists a nil algebra R whose adjoint group R o is not an Engel group. This answers a question by Amberg and Sysak from 1997 [5] and answers related questions from [3, 44]. The case of an uncountable field also answers a recent question by Zelmanov. In [37], Rump introduced braces and radical chains A n+1 = A • A n and A (n+1) = A (n) • A of a brace A. We show that the adjoint group A o of a finite right brace is a nilpotent group if and only if A (n) = 0 for some n. We also show that the adjoint group of A o of a finite left brace A is a nilpotent group if and only if A n = 0 for some n. Moreover, if A is a finite brace whose adjoint group A o is nilpotent then A is the direct sum of braces whose cardinatities are powers of prime numbers. Notice that A o is sometimes called the multiplicative group of a brace A (for example in [13]). We also introduce a chain of ideals A [n] of a left brace A and then use it to investigate braces which satisfy A n = 0 and A (m) = 0 for some m, n (Theorems 2, 3). In Section 2 we describe connections between our results and braided groups and the non-degenerate involutive set-theoretic Yang-Baxter equation. It is worth noticing that by a result by Gateva-Ivanova [17] braces are in oneto-one correspondence with braided groups with involutive braided operators.

The substitution box (S-box) component is the heart of the Advanced Encryption Standard (AES) algorithm. The S-box values are generated from the multiplicative inverse of Galois finite field GF(2 8) with an affine transform. There are many techniques of gaining the multiplicative inverse values were proposed. Most of the hardware implementations of S-box were using look-up tables (LUTs) (memory-based) to store the values which employ the largest area in design. In this paper, a software method of producing the multiplicative inverse values, which is the generator of S-box values and the possibilities of implementing the methods in hardware applications will be discussed. The method is using the log and antilog values. The method is modified to create a memory-less value generator in AES hardware-based implementation. The implementation is proposed to embed on limited memory, small-sized FPGA.

We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang-Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces.

Most public key crypto systems use finite field modulo arithmetic. This modulo arithmetic is applied on real numbers, binary values and polynomial functions. The computation cost is based on how it works with minimum use of scarce resources like processor and memory We have implemented the determination of the multiplicative in-verse of a polynomial over GF(2(superscript p)) with minimum computational cost. The ”Extended Euclidean Algorithm” (EEA) has been demonstrated to work very well manually for integers and polynomials. However polynomial manipulation cannot be computerized directly. We have implemented the same by using simple bit wise shift and XOR operations. In small applications like smart cards, mo-bile devices and other small memory devices, this method works very well. To the best of our knowledge, the pro-posed algorithm seems to be the first, efficient and cost effective implementation of determining the multiplicative inverse of polynomials over GF(2(superscript p)) ...

We link the recent theory of L-algebras to previous notions of Universal Algebra and Categorical Algebra concerning subtractive varieties, commutators, multiplicative lattices, and their spectra. We show that the category of L-algebras is subtractive and normal in the sense of Zurab Janelidze, but neither the category of L-algebras nor that of pre-L-algebras are Mal'tsev categories, hence in particular they are not semi-abelian. Therefore L-algebras are a rather peculiar example of an algebraic structure.

This paper describes the hardware implementation methodologies of fixed point binary division algorithms. The implementations have been extended for the execution of the reciprocal of the binary numbers. Radix-2 (binary) implementations of digit recurrence and multiplicative based methods have been considered for comparison. Functionality of the algorithms have been verified in Verilog hardware description language (HDL) and synthesized in Xilinx ISE 8.2i targeting the device xc4vlx15-12sf363 of Virtex4 family. Implementation was done for both signed and unsigned number systems, having bit width of operands vary as an exponential function of , where =2 to 5. Performance parameters have been calculated in terms of clock frequency, FPGA slice utilization, latency and power consumption. Implementation results indicate that multiplicative based algorithm is superior in terms of latency, while digit recurrence algorithms are consuming low power along-with less area overhead.

Let $M$ be a lattice module over a $C$-lattice $L$. Let $Spec^{s}(M)$ be the collection of all second elements of $M$. In this paper, we consider a topology on $Spec^{s}(M)$, called the second classical Zariski topology as a generalization of concepts in modules and investigate the interplay between the algebraic properties of a lattice module $M$ and the topological properties of $Spec^{s}(M)$. We investigate this topological space from the point of view of spectral spaces. We show that $Spec^{s}(M)$ is always $T_{0}-$space and each finite irreducible closed subset of $Spec^{s}(M)$ has a generic point.

To Mirian Andrés Gómez in memoriam Resumen. En este artículo consideramos cadenas finitas de elementos en retículos distributivos acotados L, que pueden ser representadas como morfismos Chn → L con dominio una n-cadena estándar con n + 2 elementos. Consideramos también cocadenas, esto es, morfismos de retículos L → Chn asociados a cadenas de ideales primos de L. Una cadena pertenece a una cocadena de la misma longitud si la correspondiente composición de morfismos Chn → L → Chn es la identidad. Definimos cadenas enlazadas mediante una relación ecuacional entre cadenas de la misma longitud, estableciendo algunas propiedades de dicha relación. Finalmente, probamos que una cadena pertenece a una cocadena sí y sólo si no están enlazadas. Este resultado significa que la dimensión de Krull de retículos distributivos acotados puede definirse en términos de cadenas enlazadas de elementos.

Computation of multiplicative inverses in finite fields GF (p) and GF (2 n) is the most time consuming operation in elliptic curve cryptography especially when affine coordinates are used. Since the existing algorithms based on extended Euclidean algorithm do not permit a fast software implementation, projective coordinates, which eliminate almost all of the inversion operations from the curve arithmetic, are preferred. In this paper, we demonstrated that affine coordinates implementation provides a comparable speed to that of projective coordinates with careful hardware realization of existing algorithms for calculating inverses in both fields without utilizing special moduli or irreducible polynomials. We presented two inversion algorithms for binary extension and prime fields, which are slightly modified versions of the Montgomery inversion algorithm. The similarity of the two algorithms allows the design of a single unified hardware architecture that performs the computation of inversion in both fields. We also proposed a hardware structure where the field elements are represented using a multi-word format. This feature allows a scalable architecture able to operate in a broad range of precision, which has certain advantages in cryptographic applications. In addition, we included statistical comparison of four inversion algorithms in order to help choose the best one amongst them for implementation onto hardware.

Given a finite bijective non-degenerate set-theoretic solution (X, r) of the Yang-Baxter equation we characterize when its structure monoid M (X, r) is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions (X, r) and show, for example, that this property is equivalent to the solution associated to the structure monoid M (X, r) (respectively structure group G(X, r)) being a multipermuation solution and that G = G(X, r) is solvable of derived length not exceeding the multipermutation level of (X, r) enlarged by one, generalizing results of Gateva-Ivanova and Cameron obtained in the involutive case. Moreover, we also prove that if X is finite and G = G(X, r) is nilpotent, then the torsion part of the group G is finite, it coincides with the commutator subgroup [G, G] + of the additive structure of the skew left brace G and G/[G, G] + is a trivial left brace. G = G(X, r) = gr(X | xy = λ x (y)ρ y (x) for all x, y ∈ X)

Reverse conversion is an important exercise in achieving the properties of Residue Number System (RNS). Current algorithms available for reverse conversion exhibits greater computational overhead in terms of speed and area. In this paper, we have developed a new algorithm for reverse conversion for two-moduli set and three-moduli set that are very simple and with fewer multiplicative inverse operations than there are in the traditional algorithms like the Chinese Remainder Theorem (CRT) and Mixed Radix Conversion (MRC).

There are many results on stability of various forms of functional equations available in the theory of functional equations. The intention of this paper is to introduce an advanced and a new multi-dimensional reciprocal-quadratic functional equation involving $p&gt;1$ variables. It is interesting to note that it has two different solutions, namely, quadratic and multiplicative inverse quadratic functions. We solve its various stability problems in the setting of non-zero real numbers and non-Archimedean fields via fixed point approach.

We focus on the fragment TFA of λ-calculus which is known to contain only terms which normalize in polynomial time. Inside TFA we translated BEA, a well known, imperative and fast algorithm which calculates the multiplicative inverse of binary finite fields. The translation suggests how to categorize the operations of BEA in sets which drive the design of a variant that we called DCEA. On several common architectures we show that these two algorithms have comparable performances, while on UltraSPARC and ARM architectures the variant we synthesized from a purely functional source can go considerably faster than BEA.

In this paper, we first discussed multiplicative metric mapping by giving some topological properties of the relevant multiplicative metric space. As an interesting result of our discussions, we observed that the set of positive real numbers R_+ is a complete multiplicative metric space with respect to the multiplicative absolute value function. Furthermore, we introduced concept of multiplicative contraction mapping and proved some fixed point theorems of such mappings on complete multiplicative metric spaces.

We present a k-bit encoding of the k-bit binary integers based on a discrete logarithm representation. The representation supports a discrete logarithm number system (DLS) that allows integer multiplication to be reduced to addition and integer exponentiation to be reduced to multiplication. We introduce right-to-left bit serial conversion, deconversion and unified conversion/deconversion algorithms between binary and DLS. The conversion algorithms utilize O(k) additions, do not require use of a multiplier, and are applicable at least up to 128 bit integers. We illustrate the use of the representation in determining a novel and efficient integer power modulo 2 k operation 2 k y x , and compare hardware performance with a current state-of-the-art method. Furthermore, we describe properties of the conversion mappings that allow compact table lookup structures to be employed for direct conversion-to and deconversion-from the DLS encoding. Our lookup architecture allows 16-bit conversion and deconversion mappings to be realized with table sizes of order 2-8 Kbytes, which is up to a 64x size reduction of the 128 Kbytes of an arbitrary 16-bits-in 16-bits-out function table. Performance and area results are given that demonstrate effectiveness of the table lookup architecture. The lookup methodology extends to other 16-bit integer functions such as multiplicative inverse and squaring operations.

One the challenging in hardware performance is to designing a high speed calculating unit. The higher of calculations speeds in a computer system will be pointed out in terms of performance. As a result, designing a high speed calculating unit is of utmost importance. In this paper, we start design whit this knowledge that one multiplier made of several adder and one divider made of several sub tractor. Therefore, if the fast adder or fast multiplier designed, performance will be improved. In this design, a circuit is designed in a manner that without a need for transforming numbers or letters from the given bases into binary bases, the multiplication of two numbers for the same base is done in that base. Reduction in the number of conversions in the calculating unit, causes reduction in the consumption power and an increase in the operating speed of the system. In this design, a very small Data Oriented Memory is used to save numerical and character data.

Summary. We formalize in the Mizar System [3], [4], definitions and basic propositions about primary ideals of a commutative ring along with Chapter 4 of [1] and Chapter III of [8]. Additionally other necessary basic ideal operations such as compatibilities taking radical and intersection of finite number of ideals are formalized as well in order to prove theorems relating primary ideals. These basic operations are mainly quoted from Chapter 1 of [1] and compiled as preliminaries in the first half of the article.

In this paper we present a single-chip FPGA full encryptor/decryptor core design of the AES algorithm. Our design performs all of them, encryption, decryption and key scheduling processes. High performance timing figures are obtained through the use of a pipelined architecture. Moreover, several modifications to the conventional AES algorithm's formulations have been introduced, thus allowing us to obtain a significant reduction in the total number of computations and the path delay associated to them. Particularly, for the implementation of the most costly step of AES, multiplicative inverse in GF(2 8), two approaches were considered. The first approach uses pre-computed values stored in a lookup table giving fast execution times of the algorithm at the price of memory requirements. Our second approach computes multiplicative inverse by using composite field techniques, yielding a reduction in the memory requirements at the cost of an increment in the execution time. The obtained results indicate that both designs are competitive with the fastest complete AES single-chip FGPA core implementations reported to date. Our first approach requires up to 11.8% less CLB slices, 21.5% less BRAMs and yields up to 18.5% higher throughput than the fastest comparable implementation reported in literature.

Let p be a prime number. We show that there is a one-to-one correspondence between the set of strongly nilpotent braces and the set of nilpotent pre-Lie rings of cardinality p n , for sufficiently large p. Moreover, there is an injective mapping from the set of left nilpotent pre-Lie rings into the set of left nilpotent braces of cardinality p n for n + 1 < p. As an application, by using well known results about the correspondence between braces and Hopf-Galois extensions we use pre-Lie algebras to describe Hopf-Galois extensions. We also translate some results from brace theory to pre-Lie algebras. For the passage from pre-Lie rings to braces we use the same method as described in [41].

We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang-Baxter equation are explored using the structure of skew left braces. Contents 14 5. Braces with cyclic multiplicative group 20 6. Indecomposable solutions 23 Acknowledgements 25 References 25

We define combinatorial representations of finite skew braces and use this idea to produce a database of skew braces of small size. This database is then used to explore different concepts of the theory of skew braces such as ideals, series of ideals, prime and semiprime ideals, Baer and Wedderburn radicals and solvability. The paper contains several questions. Contents 13 Funding 13 References 14

Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.

This paper presents a procedure for calculating multiplicative inverses modulo 2 m , based on a novel mathematical approach. The procedure is suitable for software implementation on a general-purpose processor. When counting the total number of word-level processor multiplications, the computational effort involved in calculating a multiplicative inverse is 2/3 that of a single multiplication of m-bit values, in addition to a few word-level multiplications. For standard processor word sizes, the number of these additional multiplications does not exceed 12. This introduces a clear advantage of the proposed method when compared to other known methods presented in the literature.

Let K be a ring and let G be a totally ordered group whose elements act as automorphisms on K. We denote by K*G the skew group ring over K. The prime radical P (K*G) is a homogeneous ideal of K*G ([5], Theorem 1.2). From this, P (K*G) = S (K) *G, where S (K) is a G-ideal of K. In this paper we shall apply the same method as in [2] and [3] to study the prime radical of K*G. By this way, we shall obtain similar results to the one above for a sequence of ideals of K*G, beginning with the Noether radical of K*G. Furthermore, we shall give a characterization of S(K). Finally, we shall compare S(K) with the prime radical P(K).