An alternative class of irreducible polynomials for optimal extension fields

Authors:
Yin Li;Gong-Liang Chen;Jian-Hua Li
Affiliations:
School of Information Security Engineering, Shanghai Jiaotong University, Shanghai, China;School of Information Security Engineering, Shanghai Jiaotong University, Shanghai, China;Department of Electronic Engineering, Shanghai Jiaotong University, Shanghai, China
Venue:
Designs, Codes and Cryptography
Year:
2011

Citing 10
Cited 0

Use of elliptic curves in cryptography

Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms

The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Modern computer algebra

Modern computer algebra
Parallel Multiplication in GF(2^k) usingPolynomial Residue Arithmetic

Designs, Codes and Cryptography
Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms

CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Modular Multiplication in GF(pk) Using Lagrange Representation

INDOCRYPT '02 Proceedings of the Third International Conference on Cryptology: Progress in Cryptology
Fast Implementation of Elliptic Curve Arithmetic in GF(pn)

PKC '00 Proceedings of the Third International Workshop on Practice and Theory in Public Key Cryptography: Public Key Cryptography
Practical fast polynomial multiplication

SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
Arithmetic Operations in Finite Fields of Medium Prime Characteristic Using the Lagrange Representation

IEEE Transactions on Computers
Reduction Optimal Trinomials for Efficient Software Implementation of the ηT Pairing

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

Quantified Score

Hi-index	0.00

Visualization

Abstract

Optimal extension fields (OEF) are a class of finite fields used to achieve efficient field arithmetic, especially required by elliptic curve cryptosystems (ECC). In software environment, OEFs are preferable to other methods in performance and memory requirement. However, the irreducible binomials required by OEFs are quite rare. Sometimes irreducible trinomials are alternative choices when irreducible binomials do not exist. Unfortunately, trinomials require more operations for field multiplication and thereby affect the efficiency of OEF. To solve this problem, we propose a new type of irreducible polynomials that are more abundant and still efficient for field multiplication. The proposed polynomial takes the advantage of polynomial residue arithmetic to achieve high performance for field multiplication which costs O(m 3/2) operations in $${\mathbb{F}_p}$$ . Extensive simulation results demonstrate that the proposed polynomials roughly outperform irreducible binomials by 20% in some finite fields of medium prime characteristic. So this work presents an interesting alternative for OEFs.