In this article, we investigate the construction of lightweight MDS matrices over the matrix polynomial residue ring. According to distributions of the minimum polynomial, distributions of XOR count and equivalence classes of MDS matrices, we propose an algorithm, which not only can construct lightest MDS matrices, but also is evidently more efficient than previous methods. Moreover, we investi...

A Computational Theory of Laurent Polynomial Rings and Multidimensional FIR Systems by HYUNG-JU PARK Doctor of Philosophy in Mathematics University of California at Berkeley A Laurent polynomial ring is a natural ground ring for the study of FIR (Finite Impulse Response) systems in signal processing since an FIR lter bank can be seen as a matrix over this ring, and the notion of perfect reconst...

Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I [x] for some ideals I of R. All considered rings are associative but not necessarily have identities. Köthe’s conjecture states that a ring without nil ideals has no one-sided nil ideals. It is equivalent [4] to the assertion that polynomial rings over nil rings are Jaco...

In this paper we shall primarily study polynomial integrability of the differential system ẋ = −y + Pn(x, y), ẏ = x + Qn(x, y), n = 2, 3, where Pn and Qn are homogeneous polynomials of degree n. By taking various yet very elementary ways, we not only straightforwardly find the necessary and sufficient integrability conditions but also explicitly present the corresponding polynomial first integr...

My primary area of research has been algebraic geometry and its interaction with the theory of linear systems of partial differential equations (PDEs). This interaction is provided through the study of D-modules; i.e., modules over the ring of algebraic differential operators. The computational aspects of the subject interest me most. The use of Gröbner bases yields relatively effective algorit...

Via explicit diagonalization of the chiral SU (N) 2 fusion matrices, we discuss the possibility of representing the fusion ring of the chiral SU (N) models, at level K = 2, by a polynomial ring in a single variable when N is odd and by a polynomial ring in two-variable when N is even.

In earlier eighties of XX century A.A.Nechaev has obtained the criterion of full period of a Galois polynomial over primary residue ring Z2n . Also he has obtained necessary conditions of maximal period of the Galois polynomial over Z2n in terms of coefficients of this polynomial. Further A.S.Kuzmin has obtained analogous results for the case of Galois polynomial over primary residue ring of od...

We develop a rigorous algebraic background for representations of triangle groups in linear groups over algebras arising from factor rings of multivariate polynomial rings. This is then used to substantially improve the existing upper bounds on the order of epimorphic images of triangle groups with a given injectivity radius and, analogously, the size of the associated hypermaps of a given type...