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Based on Toeplitz matrix-vector products and coordinate transformation techniques, we present a new scheme for subquadratic space complexity parallel multiplication in GF(2^{n}) using the shifted polynomial basis. Both the space complexity and the asymptotic gate delay of the proposed multiplier are better than those of the best existing subquadratic space complexity parallel multipliers. For example, with n being a power of 2, the space complexity is about 8 percent better, while the asymptotic gate delay is about 33 percent better, respectively. Another advantage of the proposed matrix-vector product approach is that it can also be used to design subquadratic space complexity polynomial, dual, weakly dual, and triangular basis parallel multipliers. To the best of our knowledge, this is the first time that subquadratic space complexity parallel multipliers are proposed for dual, weakly dual, and triangular bases. A recursive design algorithm is also proposed for efficient construction of the proposed subquadratic space complexity multipliers. This design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature.
Finite field, subquadratic space complexity multiplier, coordinate transformation, shifted polynomial basis, Toeplitz matrix.

M. A. Hasan and H. Fan, "A New Approach to Subquadratic Space Complexity Parallel Multipliers for Extended Binary Fields," in IEEE Transactions on Computers, vol. 56, no. , pp. 224-233, 2007.
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