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<p><b>Abstract</b>—This paper presents two new systolic arrays to realize Euclid's algorithm for computing inverses and divisions in finite fields <it>GF</it>(2<super><it>m</it></super>) with the standard basis representation. One of these two schemes is parallel-in parallel-out, and the other is serial-in serial-out. The former employs <it>O</it>(<it>m</it><super>2</super>) area complexity to provide the maximum throughput in the sense of producing one result every clock cycle, while the latter achieves a throughput of one result per <it>m</it> clock cycles using <it>O</it>(<it>m</it>· log<sub>2</sub><it>m</it>) area complexity. Both of the proposed architectures are highly regular and, thus, well suited to VLSI implementation. As compared to existing related systolic architectures with the same throughput performance, the proposed parallel-in parallel-out scheme reduces the hardware complexity (and, thus, the area-time product) by a factor of <it>O</it>(<it>m</it>) and the proposed serial-in serial-out scheme by a factor of <it>O</it>(<it>m</it>/log<sub>2</sub><it>m</it>).</p>
Finite field division, finite field inversion, parallel-in parallel-out architecture, serial-in serial-out architecture, standard basis, systolic array, VLSI.

J. Guo and C. Wang, "Systolic Array Implementation of Euclid's Algorithm for Inversion and Division in GF (2m)," in IEEE Transactions on Computers, vol. 47, no. , pp. 1161-1167, 1998.
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