Results on the multiplication in finite fields of characteristic three using modified polynomial representation and normal elements in binary fields

Download

index.pdf

Date

2013

Author

Özel, Canan

Metadata

Show full item record

Item Usage Stats

69
views

27
downloads

In this thesis, we study on the multiplication in finite fields of characteristic three. We use Charlier and Hermite polynomials to represent elements in F_3^n for obtaining alternative representations to the standart polynomial representation. We give multiplication methods in these representations to multiply elements in F_3^n. We compute the multiplication and reduction complexities in each representation and compare the complexity results with the ones in the standart polynomial representation. Charlier and Hermite polynomial representations enable us to find irreducible binomials. We show that in some cases, there is a set of irreducible binomials in each representation to do modular reduction with lower addition complexity than the one in the standart polynomial representation. We give a multiplier architecture in Charlier polynomial representation of finite fields F_3^n, where n=2 (mod 3). Then, we examine cubing and inversion operations in this representation. We investigate the matrix-vector product method for multiplication of the field elements in Hermite polynomial representation and we generalize the reduction complexity by using the reduction matrix in this matrix-vector product method. Finally, we focus on the optimal normal basis construction in binary fields and find a connection between optimal normal basis elements and Hermite polynomials in these fields.

Subject Keywords

Polynomials., Finite fields (Algebra)., Characteristic functions.

URI

http://etd.lib.metu.edu.tr/upload/12615899/index.pdf
https://hdl.handle.net/11511/22583

Collections

Graduate School of Applied Mathematics, Thesis

Suggestions

OpenMETU
Core

On the arithmetic operations over finite fields of characteristic three with low complexity AKLEYLEK, SEDAT; Özbudak, Ferruh; Özel, Claire Susanna (2014-03-15) In this paper, the Hermite polynomial representation is adapted as a new way to represent certain finite fields of characteristic three. We give the multiplication method to multiply two elements of F-3n in the Hermite polynomial representation with subquadratic computational complexity by using a divide-and-conquer idea. We show that in some cases there is a set of irreducible binomials in the Hermite polynomial representation to obtain modular reduction with a lower addition complexity than the standard p...
A geometric approach to absolute irreducibility of polynomials Koyuncu, Fatih; Özbudak, Ferruh; Department of Mathematics (2004) This thesis is a contribution to determine the absolute irreducibility of polynomials via their Newton polytopes. For any field F; a polynomial f in F[x1, x2,..., xk] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F; i.e. irreducible over every algebraic extension of F. We present some new results giving integrally indecomposable classes of polytopes. Conse...
On the exact number of solutions of certain linearized equations Özbudak, Ferruh (2014-11-01) In this note we have revisited some of the results of Trachtenberg (On the cross-correlation functions of maximal linear sequences, Ph.D. thesis, University of Southern California, Los Angeles, 1970), which are directly related with the number of solutions of some special linearized polynomials over finite fields. In some cases we give improvements. Also, we give some results on the exact number of solutions of certain linearized equations depending on the coefficients of that equation.
On multiplication in finite fields Cenk, Murat; Özbudak, Ferruh (2010-04-01) We present a method for multiplication in finite fields which gives multiplication algorithms with improved or best known bilinear complexities for certain finite fields. Our method generalizes some earlier methods and combines them with the recently introduced complexity notion (M) over cap (q)(l), which denotes the minimum number of multiplications needed in F-q in order to obtain the coefficients of the product of two arbitrary l-term polynomials modulo x(l) in F-q[x]. We study our method for the finite ...
A specific type of permutation and complete permutation polynomials over finite fields Ongan, Pinar; GÜLMEZ TEMÜR, BURCU (World Scientific Pub Co Pte Lt, 2020-04-01) In this paper, we study polynomials of the form f(x) = x (qn-1/q-1+1) + bx is an element of F-qn[x], where n = 5 and list all permutation polynomials (PPs) and complete permutation polynomials (CPPs) of this form. This type of polynomials were studied by Bassalygo and Zinoviev for the cases n = 2 and n = 3, Wu, Li, Helleseth and Zhang for the case n = 4, p not equal 2, Bassalygo and Zinoviev answered the question for the case n = 4, p= 2 and finally by Bartoli et al. for the case n = 6. Here, we determine a...

Citation Formats

C. Özel, “Results on the multiplication in finite fields of characteristic three using modified polynomial representation and normal elements in binary fields,” Ph.D. - Doctoral Program, Middle East Technical University, 2013.