Efficient multiplications in F(5)5n and F(7)7n

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. (2008) [11 and Montgomery (2005) [12]. The timing results of implementations of the Karatsuba type formulas and the proposed formula for multiplication in F(5)5.89 are given.


Efficient multiplication in double-struck F sign3ℓm, m ≥ 1 and 5 ≤ ℓ ≤ 18
Cenk, Murat; Özbudak, Ferruh (2008-06-14)
Using a method based on Chinese Remainder Theorem for polynomial multiplication and suitable reductions, we obtain an efficient multiplication method for finite fields of characteristic 3. Large finite fields of characteristic 3 are important for pairing based cryptography [3]. For 5 <= l <= 18, we show that our method gives canonical multiplication formulae over F-3lm for any m >= 1 with the best multiplicative complexity improving the bounds in [6]. We give explicit formula in the case F-36.97.
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 ...
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.
Faster characteristic three polynomial multiplication and its application to NTRU Prime decapsulation
Yeniaras, Esra; Cenk, Murat (2022-01-01)
Efficient computation of polynomial multiplication over characteristic three fields is required for post-quantum cryptographic applications which gain importance upon the recent advances in quantum computers. In this paper, we propose three new polynomial multiplication algorithms over F-3 and show that they are more efficient than the current state-of-the-art algorithms. We first examine through the well-known multiplication algorithms in F-3[x] including the Karatsuba 2-way and 3-way split formulas along ...
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, “Efficient multiplications in F(5)5n and F(7)7n,” JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, pp. 177–183, 2011, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/32584.