On multiplication in finite fields

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 fields F(q)n, where 2 <= n <= 18 and q = 2, 3,4 and we improve or reach the currently best known bilinear complexities. We also give some applications in cryptography. (C) 2010 Published by Elsevier Inc.


Classification of function fields with class number three
BİLHAN, Mehpare; Buyruk, Dilek; Özbudak, Ferruh (2015-11-01)
We give the full list of all algebraic function fields over a finite field with class number three up to isomorphism. Our list consists of explicit equations of algebraic function fields which are mutually non-isomorphic over the full constant field.
On the Representation of Finite Fields
Akleylek, Sedat; Özbudak, Ferhun; Department of Cryptography (2010)
The representation of field elements has a great impact on the performance of the finite field arithmetic. In this thesis, we give modified version of redundant representation which works for any finite fields of arbitrary characteristics to design arithmetic circuits with small complexity. Using our modified redundant representation, we improve many of the complexity values. We then propose new representations as an alternative way to represent finite fields of characteristic two by using Charlier and Herm...
Additive polynomials and primitive roots over finite fields
Özbudak, Ferruh (2001-01-01)
We prove existence of primitive roots with a prescribed nonzero image using the arithmetic of algebraic function fields for a class of polynomials over sufficiently large finite fields.
Efficient multiplications in F(5)5n and F(7)7n
Cenk, Murat; Özbudak, Ferruh (2011-08-15)
Efficient multiplications in finite fields of characteristics 5 and 7 are used for computing the Eta pairing over divisor class groups of the hyperelliptic curves Lee et al. (2008) [1]. In this paper, using the recent methods for multiplication in finite fields, the explicit formulas for multiplication in F(5)5n and F(7)7n are obtained with 10 multiplications in F(5)n for F(5)5n and 15 multiplications in F(7)n for F(7)7n improving the results in Cenk and Ozbudak (2008) [4], Cenk et al. (2009) [5], Lee et al...
Free storage basis conversion over extension field
Harold, Ndangang Yampa; Akyıldız, Ersan; Department of Cryptography (2014)
The representation of elements over finite fields play a great impact on the performance of finite field arithmetic. So if efficient representation of finite field elements exists and conversion between these representations is known, then it becomes easy to perform computation in a more efficient way. In this thesis, we shall provide a free storage basis conversion in the extension field F_(q^p) of F_q between Normal basis and Polynomial basis and vice versa. The particularity of this thesis is that, our t...
Citation Formats
M. Cenk and F. Özbudak, “On multiplication in finite fields,” JOURNAL OF COMPLEXITY, pp. 172–186, 2010, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/32570.