Montgomery Multiplication in GF(2^k
Designs, Codes and Cryptography
IEEE Transactions on Computers
Efficient Arithmetic on Koblitz Curves
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
An Efficient Optimal Normal Basis Type II Multiplier
IEEE Transactions on Computers
A Modified Massey-Omura Parallel Multiplier for a Class of Finite Fields
IEEE Transactions on Computers
Finite Field Multiplier Using Redundant Representation
IEEE Transactions on Computers
Efficient Software Implementation for Finite Field Multiplication in Normal Basis
ICICS '01 Proceedings of the Third International Conference on Information and Communications Security
High-Speed Software Multiplication in F2m
INDOCRYPT '00 Proceedings of the First International Conference on Progress in Cryptology
Elliptic Scalar Multiplication Using Point Halving
ASIACRYPT '99 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
Software Implementation of Elliptic Curve Cryptography over Binary Fields
CHES '00 Proceedings of the Second International Workshop on Cryptographic Hardware and Embedded Systems
Low Complexity Multiplication in a Finite Field Using Ring Representation
IEEE Transactions on Computers
Guide to Elliptic Curve Cryptography
Guide to Elliptic Curve Cryptography
Fast Normal Basis Multiplication Using General Purpose Processors
IEEE Transactions on Computers
Hardware and Software Normal Basis Arithmetic for Pairing-Based Cryptography in Characteristic Three
IEEE Transactions on Computers
Polynomial and Normal Bases for Finite Fields
Journal of Cryptology
On the Number of Trace-One Elements in Polynomial Bases for $$\mathbb{F}_{2^n}$$
Designs, Codes and Cryptography
Efficient Algorithms and Architectures for Field Multiplication Using Gaussian Normal Bases
IEEE Transactions on Computers
Efficient Algorithms and Architectures for Field Multiplication Using Gaussian Normal Bases
IEEE Transactions on Computers
Journal of VLSI Signal Processing Systems
Software Implementation of Arithmetic in
WAIFI '07 Proceedings of the 1st international workshop on Arithmetic of Finite Fields
Transactions on computational science XI
Gauss periods as constructions of low complexity normal bases
Designs, Codes and Cryptography
Quantum binary field inversion: improved circuit depth via choice of basis representation
Quantum Information & Computation
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
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Fast algorithms for multiplication in finite fields are required for several cryptographic applications, in particular for implementing elliptic curve operations over binary fields {\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{2^m}. In this paper, we present new software algorithms for efficient multiplication over {\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{2^m} that use a Gaussian normal basis representation. Two approaches are presented, direct normal basis multiplication and a method that exploits a mapping to a ring where fast polynomial-based techniques can be employed. Our analysis, including experimental results on an Intel Pentium family processor, shows that the new algorithms are faster and can use memory more efficiently than previous methods. Despite significant improvements, we conclude that the penalty in multiplication is still sufficiently large to discourage the use of normal bases in software implementations of elliptic curve systems.