We show that the multiplication operation c = a b r^(-1) in the field
GF(2^{k}) can be implemented significantly faster in software than the
standard multiplication, where r is a special fixed element of the field. This
operation is the finite field analogue of the Montgomery multiplication for
modular multiplication of integers. We give the bit-level and word-level
algorithms for computing the product, perform a thorough performance analysis,
and compare the algorithm to the standard multiplication algorithm in
GF(2^{k}). The Montgomery multiplication can be used to obtain fast
software implementations of the discrete exponentiation operation, and is
particularly suitable for cryptographic applications where k is large.