Affine Polynomial Research Articles

A note on the reducibility of binary affine polynomials

Stickelberger---Swan Theorem is an important tool for determining parity of the number of irreducible factors of a given polynomial. Based on this theorem, we prove in this note that every affine polynomial A(x) over $${\mathbb{F}_2}$$ with degree >1, where A(x) = L(x) + 1 and $${L(x)=\sum_{i=0}^{n}{x^{2^i}}}$$ is a linearized polynomial over $${\mathbb{F}_2}$$ , is reducible except x 2 + x + 1 and x 4 + x + 1. We also give some explicit factors of some special affine pentanomials over $${\mathbb{F}_2}$$ .

Cryptanalysis of an Extension of the Hill Cipher

B. Thilaka and K. Rajalakshmi [2] propose extensions of Hill Cipher using affine transformation and polynomial transformation to make it more secure. In this article, we show that these extensions of Hill Cipher are prone to cryptanalytic attacks. We have proposed known plaintext attack for cryptanalizing ciphers based on the modified Hill Cipher system. We illustrate the method with the help of examples and show that the suggested modifications in Hill Cipher do not make it significantly stronger.

AN EXTENSION OF HILL CIPHER USING GENERALISED INVERSES AND mth RESIDUE MODULO n

Hill Cipher is a symmetric polyalphabetic block cipher that enciphers a string of letters into another string of the same length using the linear transformation y=xK. In deciphering, the determinant value must be less than 26 and relatively prime to 26 so that the matrix K of the linear transformation is invertible in modulo 26. The Affine Hill cipher extends the concept of Hill cipher by using the non-linear transformation y=xK+b. In this paper, we extend this concept to encrypt a plaintext of blocksize m to a ciphertext of blocksize n≥m using (a) affine transformation and (b) polynomial transformation to make this cipher more secure. Here the matrix K of the transformation need not be a square matrix. To enable decryption, we state the conditions to be satisfied by K which are as follows.Case (a): (i) For affine transformation, the generalised inverse K + of the matrix K corresponding to the transformation should satisfy the equation KK +=I in modulo p where p is a chosen prime p>26. For m=n, K + is the usual inverse of the matrix K.Case(b): (i) For polynomial transformation, the generalised inverse K + should satisfy the above condition, (ii) If r is the degree of the polynomial, then choose those values of s≤r such that the sth root of modulo p exists for all elements in Z p . In other words, choose those values of s that are relatively prime to Φ(p).

불감시간을 갖는 Affine 시스템의 안정도 해석과 제어기 설계

The Nyquist robust stability margin is proposed as a measure of robust stability for systems with Affine TFM(Transfer Function Matrix) parametric uncertainty. The parametric uncertainty is modeled through a Affine TFM MIMO (Multi-Input Multi-Output) description with dead-time, and the unstructured uncertainty through a bounded perturbation of Affine polynomials. Gershgorin's theorem and concepts of diagonal dominance and GB(Gershgorin Bands) are extended to include model uncertainty. Multiloop PI/PID controllers can be tuned by using a modified version of the Ziegler-Nichols (ZN) relations. Consequently, this paper provides sufficient conditions for the robustness of Affine TFM MIMO uncertain systems with dead-time based on Rosenbrock's DNA. Simulation examples show the performance and efficiency of the proposed multiloop design method for Affine uncertain systems with dead-time.

Improved hybrid algorithm for finding roots of error-locator polynomials

In this paper we suggest a hybrid method for finding roots of error locator polynomials. We first review a fast version of the Chien search algorithm based on the decomposition of the error locator polynomial into a sum of multiples of affine polynomials. We then propose to combine it with modified analytical methods for solution of polynomials of small degree in radicals. We suggest, in particular, two efficient decompositions, whose combination with analytical algorithms yields a significant reduction in time-complexity, as proved by means of simulation. Copyright © 2003 AEI.

Affine Transformations, Polynomials, and Proportionality

Affine Transformations, Polynomials, and Proportionality

Fast algorithm for computing the roots of error locator polynomials up to degree 11 in Reed-Solomon decoders

The central problem in the implementation of a Reed-Solomon code is finding the roots of the error locator polynomial. In 1967, Berlekamp et al. found an algorithm for finding the roots of an affine polynomial in GF(2/sup m/) that can be used to solve this problem. In this paper, it is shown that this Berlekamp-Rumsey-Solomon (1967) algorithm, together with the Chien (1964) search method, makes possible a fast decoding algorithm in the standard-basis representation that is naturally suitable in a software implementation. Finally, simulation results for this fast algorithm are given.

Applications of Frequency Dependent Wavelet Shrinkage to Analyzing Quality of Image Registration

A fundamental problem in medical imaging is to transport an atlas associated with a template image onto new data. This is sometimes achieved by registering (warping) the template to the data and using the induced (spatial) deformation field to impose an atlas on the data.There are a number of alignment techniques which bring two images in register; however, few studies have been done to quantitatively compare various warping methods. In this paper we compare different wavelet space thresholding schemes (uniform, spatially, and frequency adaptive, Bayesian) and develop two approaches for quantitative evaluation of the performance of three nonaffine warping techniques and one affine polynomial (12 parameter) registration on groups of data. The results of the affine and nonaffine warps are evaluated, both on structural magnetic resonance imaging (MRI) data and on the corresponding functional positron emission tomography (PET) data using a quantitative approach based on the discrete wavelet transform. Using i...

Random spherical uncertainty in estimation and robustness

A theorem is formulated that gives an exact probability distribution for a linear function of a random vector uniformly distributed over a ball in n-dimensional space. This mathematical result is illustrated via applications to a number of important problems of estimation and robustness under spherical uncertainty. These include parameter estimation, characterization of attainability sets of dynamical systems, and robust stability of affine polynomial families.

Computational studies on generation and control in magnetron sputtering plasma

We devise and analyze vertex-based schemes on polyhedral meshes to approximate advection–reaction equations. Error estimates of order O(h3/2) are established in the discrete inf–sup stability norm which includes the mesh-dependent weighted advective derivative. The two key ingredients are a local polyhedral reconstruction map leaving affine polynomials invariant, and a local design of stabilization whereby gradient jumps are only penalized across some subfaces in the interior of each mesh cell. Numerical results are presented on three-dimensional polyhedral meshes.

An improved technique for plotting robust root locus

Abstract This paper deals with the problem of plotting the robust root loci of a feedback control system which includes affine parametric uncertainties in the coefficients of the plant transfer function. First, based on a geometrical interpretation of the value set of a family of affine parametric uncertain polynomials, an efficient algorithm is proposed for checking whether the set includes the origin of the complex plane. This zero‐inclusion test algorithm is then applied along with a pivoting procedure to trace out the boundaries of the cross sections of the robust root loci. Since the proposed technique of generating a robust root locus needs not to construct all exposed edges of the value set in testing the zero inclusion condition, and needs not to use a brute‐force two‐dimensional gridding approach in plotting the boundary of a cross section of the robust root loci, its advantage of saving computational cost is obviously significant.

On Linear Fractional Representations of Systems with Parametric Uncertainty

This paper introduces a new procedure to obtain low order linear Pactional representations of systems with parametric uncertainty. It is based on recursive sum and product decompositions of multidimensional polynomial matrices that reduce the problem to trivial representations of affine polynomial matrices. This approach is illustrated with a simple example.

Calculation of Zero Dynamics for Affine MIMO-Systems Using the Ritt Algorithm

This paper considers the problem of calculating the zero dynamics by using the Ritt algorithm of differential algebra. A result is given which shows how the Ritt algorithm captures the structure of the zero dynamics for affine polynomial MIMO systems when the system has a vector relative degree. We also discuss how the algorithm can be used when the system is not polynomial or it has no vector relative degree.

On the Degree of Multivariate Bernstein Polynomial Operators

Let σ be a d-dimensional simplex with vertices v 0, ..., v d and B n (ƒ,·) denote the nth degree Bernstein polynomial of a continuous function ƒ on σ. Dahmen and Micchelli ( Stud. Sci. Hungar. 23 (1988), 265-287) proved that B n (ƒ,·) ≥ B n+1 (ƒ,·), n ∈ N, for any convex function ƒ on σ, and it is clear that a necessary and sufficient condition for the inequality to become an identity for all n ∈ N is that ƒ is an affine polynomial. Let σ m be the mth simplicial subdivision of σ (which will be defined precisely later). By using a degree-raising formula, the result of Dahmen and Micchelli can be extended to B mn (ƒ,·), ≥ B mn+1 (ƒ,·), n ∈ N, for any which is convex on every cell of σ m . The objective of this paper is to derive conditions under which this inequality becomes an identity.

The Divisors of $x^{2m} + x$ of Constant Derivatives and Degree $2^{m - 2}$

This paper provides a new proof of the fact that the polynomials of degree $2^{m-2}$ over the Galois field $GF(2^m) (m \geq 2)$, which are fully reducible and admit no multiple factor (i.e., which divide x^{2^m} + x$) and whose derivatives are constant are affine polynomials. The author determines explicitly these polynomials, which are related to a problem in coding theory that is recounted.

Large Step Path-Following Methods for Linear Programming, Part II: Potential Reduction Method

The algorithm proposed in this paper is the affine polynomial potential reduction method, with new procedures for updating the lower bounds for an optimal solution of the linear programming problem. A method is developed for updating the lower bounds by large steps, with strict control over the duality gaps associated with each iterate. Two algorithms are obtained by this approach: the first one has complexity $O(nL)$ iterations, and a very simple updating procedure; the second one updates the lower bounds at points very near the central trajectory and achieves a complexity of $O(\sqrt{n} L)$ iterations.

Affine polynomial identity rings and their generalizations

Affine polynomial identity rings and their generalizations