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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.
Multiplication in {\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{2^m}, Gaussian normal basis, elliptic curve cryptography.

D. Hankerson, A. Menezes, F. Hu, M. Long, R. Dahab and J. L?pez, "Software Multiplication Using Gaussian Normal Bases," in IEEE Transactions on Computers, vol. 55, no. , pp. 974-984, 2006.
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