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A Fast Algorithm for Multiplicative Inversion in GF(2m) Using Normal Basis
May 2001 (vol. 50 no. 5)
pp. 394-398

Abstract—A fast algorithm for multiplicative inversion in $GF(2^m)$ using normal basis is proposed. It is an improvement on those proposed by Itoh and Tsujii and by Chang et al., which are based on Fermat's Theorem and require $O(\log m)$ multiplications. The number of multiplications is reduced by decomposing $m-1$ into several factors and a small remainder.

[1] C.C. Wang,T.K. Truong,H.M. Shao,L.J. Deutsch,J.K. Omura, and I.S. Reed,"VLSI Architectures for Computing Multiplications and Inverses inGF(2m)," IEEE Trans. Computers, vol. 34, no. 8, pp. 709-716, Aug. 1985.
[2] T. Itoh and S. Tsujii, “A Fast Algorithm for Computing Multiplicative Inverses in$GF(2^m)$Using Normal Basis,” Information and Computing, vol. 78, pp. 171-177, 1988.
[3] T. Itoh and S. Tsujii, “A Fast Algorithm for Computing Multiplicative Inverses in Finite Fields Using Normal Basis,” IEICE Trans. (A)., vol. J70-A, no. 11, pp. 1637-1645, Nov. 1989 (in Japanese).
[4] G-L. Feng,"A VLSI Architecture for Fast Iinversion inGF(2m)," IEEE Trans. Computers, vol. 38, no. 10, pp. 1,383-1,386, Oct. 1989.
[5] T. Chang, E. Lu, Y. Lee, Y. Leu, and H. Shyu, “Two Algorithms for Computing Multiplicative Inverses in$GF(2^m)$Using Normal Basis,” accepted by Information Processing Letters.
[6] Stefano Nocentini, "The Planning Ritual," Datamation, vol. 31, p. 128, Apr.15, 1985.
[7] Y. Asano, T. Itoh, and S. Tsujii, “Generalized Fast Algorithm for Computing Multiplicative Inverses in$GF(2^m)$,” Electronics Letters, vol. 25, no. 10, pp. 664-665, May 1989.
[8] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes. New York: North-Holland, 1977.
[9] J.L. Massey and J.K. Omura, “Computational Method and Apparatus for Finite Field Arithmetic,” US Patent Application, submitted 1981.

Index Terms:
Finite field, finite field inversion, Fermat's theorem, normal basis.
Citation:
Naofumi Takagi, Jun-ichi Yoshiki, Kazuyoshi Takagi, "A Fast Algorithm for Multiplicative Inversion in GF(2m) Using Normal Basis," IEEE Transactions on Computers, vol. 50, no. 5, pp. 394-398, May 2001, doi:10.1109/12.926155
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