Fast Bit Parallel-Shifted Polynomial Basis Multipliers in $GF(2^{n})$

A new nonpipelined bit-parallel-shifted polynomial basis multiplier for GF(2<sup>n</sup>) is presented. For some irreducible trinomials, the space complexity of the multiplier matches the best results available in the literature, and its gate delay is equal to T <sub>A</sub>+lceillog<sub>2</sub>nrceilT<sub>X</sub>, where T<sub>A </sub> and T<sub>X</sub> are the delay of one two-input and and xor gates, respectively. To the best of our knowledge, this is the first time that the gate delay bound T<sub>A</sub>+lceillog<sub>2</sub>nrceilT<sub>X</sub> is reached. For some irreducible pentanomials, its gate delay is equal to T<sub>A </sub>+(1+lceillog<sub>2</sub>nrceil)T<sub>X</sub>. NIST has recommended five binary fields for the elliptic curve digital signature algorithm applications: GF(2<sup>163</sup>), GF(2<sup>233</sup>), GF(2 <sup>283</sup>), GF(2<sup>409</sup>), and GF(2<sup>571</sup>), but no irreducible trinomials exist for three degrees, viz., 163, 283 and 571. For the three corresponding binary fields, we show that the gate delay of the proposed multiplier is T<sub>A</sub>+(1+lceillog<sub>2</sub>nrceil)T<sub>X</sub>. This result outperforms the previously known results

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