Synchronization of M-sequences based on fast Нadamard transform

Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

We consider Holder continuous circulant (2 × 2) matrix functions \({{\bf G}^1_2}\) defined on the fractal boundary Γ of a Jordan domain Ω in \({\mathbb{R}^{2n}}\). The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitian Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitian Dirac operators. In Brackx et al. (Bull Braz Math Soc 40(3): 395–416, 2009) a Hermitian Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from which a Hilbert transform was derived in Brackx et al. (J Math Anal Appl 344: 1068–1078, 2008) for the case of domains with smooth boundary. However, crucial parts of the method are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitian Hilbert transform in that case. As a consequence, we give necessary and sufficient conditions for the Hermitian monogenicity of a circulant matrix function \({{\bf G}^1_2}\) in the interior and exterior of Ω, in terms of its boundary value \({{\bf g}^1_2={\bf G}^1_2|_\Gamma}\), extending in this way also results of Abreu Blaya et al. (Bound. Value Probl. 2008: 2008) (article ID 425256), (article ID 385874), where Γ is required to be Ahlfors–David regular.

On the construction of irreducible and primitive polynomials from [formula omitted] to [formula omitted

MDS matrices are used in the design of diffusion layers in many block ciphers and hash functions due to their optimal branch number. But MDS matrices, in general, have costly implementations. So in search for efficiently implementable MDS matrices, there have been many proposals. In particular, circulant, Hadamard, and recursive MDS matrices from companion matrices have been widely studied. In a recent work, recursive MDS matrices from sparse DSI matrices are studied, which are of interest due to their low fixed cost in hardware implementation. In this paper, we present results on the exhaustive search for (recursive) MDS matrices over GL(4, F2). Specifically, circulant MDS matrices of order 4, 5, 6, 7, 8; Hadamard MDS matrices of order 4, 8; recursive MDS matrices from companion matrices of order 4; recursive MDS matrices from sparse DSI matrices of order 4, 5, 6, 7, 8 are considered. It is to be noted that the exhaustive search is impractical with a naive approach. We first use some linear algebra tools to restrict the search to a smaller domain and then apply some space-time trade-off techniques to get the solutions. From the set of solutions in the restricted domain, one can easily generate all the solutions in the full domain. From the experimental results, we can see the (non) existence of (involutory) MDS matrices for the choices mentioned above. In particular, over GL(4, F2), we provide companion matrices of order 4 that yield involutory MDS matrices, circulant MDS matrices of order 8, and establish the nonexistence of involutory circulant MDS matrices of order 6, 8, circulant MDS matrices of order 7, sparse DSI matrices of order 4 that yield involutory MDS matrices, and sparse DSI matrices of order 5, 6, 7, 8 that yield MDS matrices. To the best of our knowledge, these results were not known before. For the choices mentioned above, if such MDS matrices exist, we provide base sets of MDS matrices, from which all the MDS matrices with the least cost (with respect to d-XOR and s-XOR counts) can be obtained. We also take this opportunity to present some results on the search for sparse DSI matrices over finite fields that yield MDS matrices. We establish that there is no sparse DSI matrix S of order 8 over F28 such that S8 is MDS.

Irreducible polynomials play an important role in design of Forward Error Correction (FEC) codes for data transmission with integrity and automatic correction of data, as for example, Low-Density Parity Check codes. The usage of irreducible polynomials enables construction of non-prime-order finite fields. Most of the irreducible polynomials belong to binary Galois field. The important analytical concept is optimisation of irreducible polynomials for use in FECs in nonbinary Galois (NBG) field, leading to the development of an algorithm for LDPC that can work with nonbinary Galois fields. According to studies, the Tanner graph for ‘nonbinary Low Density Parity Check’ codes might get sparser as the field’s dimensions rise, ensuring that they do much better than their binary counterparts. A detailed discussion of representation of nonbinary irreducible polynomial and the computations involved have been illustrated. The concept has been tried for NB-LDPC codes. To prove the notion, computational complexity is found from different parameters such as performance of error correction capability, complexity cost and simulation time taken. Such detail study makes the NBG fields-based FEC very suitable for high-speed data transmission with self-error correction.

The theory of polynomials over finite fields is important for investigating the algebraic structure of finite fields as well as for many applications. Above all, irreducible polynomials—the prime elements of the polynomial ring over a finite field—are indispensable for constructing finite fields and computing with the elements of a finite field. Section 1 introduces the notion of the order of a polynomial. An important fact is the connection between minimal polynomials of primitive elements (so-called primitive polynomials) and polynomials of the highest possible order for a given degree. Results about irreducible polynomials going beyond those discussed in the previous chapters are presented in Section 2. The next section is devoted to constructive aspects of irreducibility and deals also with the problem of calculating the minimal polynomial of an element in an extension field. Certain special types of polynomials are discussed in the last two sections. Linearized polynomials are singled out by the property that all the exponents occurring in them are powers of the characteristic. The remarkable theory of these polynomials enables us, in particular, to give an alternative proof of the normal basis theorem. Binomials and trinomials — that is, two-term and three-term polynomials—form another class of polynomials for which special results of considerable interest can be established. We remark that another useful collection of polynomials— namely, that of cyclotomic polynomials—was already considered in Chapter 2, Section 4, and that some additional information on cyclotomic polynomials is contained in Section 2 of the present chapter.

The optimal detection for coded system requires the use of a maximum a posteriori (MAP) detection. A list sphere detector (LSD) can be used to approximate the MAP detector. Depending on the used list size, LSD provides a tradeoff between the performance and the computational complexity. The LSD output candidate list is used to calculate the approximation of the probability log-likelihood ratio (LLR) of each transmitted bit. The list should be large enough and it should include at least one candidate for both possible bits for good approximation. The use of a small list size causes inaccurate and, especially, very large LLRs that prevent the decoder from correcting the falsely detected signals and, thus, degrades performance. We study the effect of the LLR clipping to the performance and complexity of the LSD algorithm. We show that by limiting the dynamic range of the LLR the required LSD list size can be decreased, and, thus, the complexity of the algorithms is decreased. The optimal dynamic range values for LLR clipping are determined and the effect of the clipping to the complexity of the LSD algorithms is analyzed.

Fast forward and inverse transforms for multiple-valued input binary output PLI logic are derived. New matrix functions are introduced which allow feasible transformation of arbitrary q-valued input binary output functions. This allows representation of the function in the algebra of GF(2), where q/spl ges/2. The representation of MVB functions in various functional bases having always fast algorithms is the foundation for the efficient synthesis of PLI expansions of MVB functions. The computational complexity of such algorithms is essentially reduced when compared to standard matrix multiplication and inversion based methods of the earlier work in PLI logic.

Some relations between the irreducible polynomials over a finite field and its quadratic extension

The S-Box and Inverse S-Box in the tradition advance encryption standard is fixed and it is made by the composite field arithmatic to find the multiplicative inverse in the finite field GF (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ). But in traditional AES the use of irreducible polynomial m(x) = x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> + x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> + x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> + x+1 to find out multiplicative inverse, is known to the attacker. So if we can make use of different irreducible polynomial every time to the finite field of GF (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ) and send this the receiver combined with the secret key, then every time a new irreducible polynomial is used and a random S-Box is generated. Hence, the security of the algorithm is enhanced. In this paper, we devise an algorithm, which make use of different irreducible polynomial to finite field of GF (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ) to make random S-Box and Inverse S-Box.

In this chapter, we study several aspects of matrices over finite fields. We begin with some basic results on the rank and order of matrices and then apply some of these results to discuss an interesting theoretical topic, namely matrix representations of an extension field \({\text{GF}}(q^{n} )\) over \({\text{GF}}(q)\). Following this, we provide results on the numbers of matrices over \({\text{GF}}(q)\) of various important special types – such as symmetric, orthogonal and circulant matrices – that will be needed later to study basis representations of extension fields. Last but not least, we also consider the Discrete Fourier Transform, which is not only a very useful tool for some results given in the present chapter, but has many interesting further applications.

Linear recursive sequences generated from irreducible polynomials over GF(q), q=P/sup m/, a prime power, are investigated. These sequences are expressed as vector linear recursive sequences whose component sequences may be obtained directly from an irreducible polynomial over GF(p). The relationship between these components is investigated. A method of generating the primitive and nonprimitive irreducible polynomials is also discussed. >

Implementation aspects of list sphere decoder algorithms for MIMO-OFDM systems

In this paper, we present a method to compute all the irreducible and primitive polynomials of degree m over a finite field. We also describe two concrete implementations of our method with respective time complexities O(m2 + m log m) and O(m2 + log m). These implementations, using in parallel different devices introduced to operate in these fields [1], [7], allows us to reduce the time complexity of our method below that of the best previously known methods [3]. Our algorithm is especially well-suited for applications using large finite fields.

In this paper proposed new one-way matrix function and a new algorithm for the transfer the private key encryption to open communication channels based on the generalized PP, Galois and Fibonacci matrices is discussed. The purpose of this article is to describe the features, principles of a generalized primitive polynomial that extends the classic term primitive polynomial.