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Finite Field Multiplier Using Redundant Representation
November 2002 (vol. 51 no. 11)
pp. 1306-1316

Abstract—This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our architectures is that they provide area-time trade-offs which enable us to implement the multipliers in a partial-parallel/hybrid fashion. This hybrid architecture has great significance in its VLSI implementation in very large fields. The squaring operation using the redundant representation is simply a permutation of the coordinates. It is shown that, when there is an optimal normal basis, the proposed bit-serial and hybrid multiplier architectures have very low space complexity. Constant multiplication is also considered and is shown to have an advantage in using the redundant representation.

[1] G.B. Agnew, R. Beth, R.C. Mullin, and S.A. Vanstone, “Arithmetic Operations in$\big. {\rm GF}(2^m)\bigr.$,” J. Cryptology, vol. 6, pp. 3-13, 1993.
[2] G.B. Agnew, R.C. Mullin, I. Onyszchuk, and S.A. Vanstone, “An Implementation for a Fast Public Key Cryptosystem,” J. Cryptology, vol. 3, pp. 63-79, 1991.
[3] D.W. Ash, I.F. Blake, and S.A. Vanstone, “Low Complexity Normal Bases,” Discrete Applied Math., vol. 25, pp. 191-210, 1989.
[4] G. Drolet, “A New Representation of Elements of Finite Fields$\big. {\rm GF}(2^m)\bigr.$Yielding Small Complexity Arithmetic Circuits,” IEEE Trans. Computers, vol. 47, no. 9, pp. 938-946, Sept. 1998.
[5] 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.
[6] S. Gao and S. Vanstone, “On Orders of Optimal Normal Basis Generators,” Math. Computation, vol. 64, no. 2, pp. 1227-1233, 1995.
[7] S. Gao, J. von zur Gathen, and D. Panario, “Gauss Periods and Fast Exponentiation in Finite Fields,” Lecture Notes in Computer Science, vol. 911, pp. 311-322, 1995.
[8] S. Gao, J. von zur Gathen, D. Panario, and V. Shoup, “Algorithms for Exponentiation in Finite Fields,” J. Symbolic Computation, vol. 29, pp. 879-889, 2000.
[9] W. Geiselmann and D. Gollmann, “VLSI Design for Exponentiation in$\big. {\rm GF}(2^m)\bigr.$,” Proc. AUSCRYPT '90, pp. 398-405, 1990.
[10] W. Geiselmann and H. Lukhaub, “Redundant Representation of Finite Fields,” Proc. Public Key Cryptography, Fourth Int'l Workshop Practice and Theory in Public Key Cryptosystems (PKC 2001), K. Kim, ed. pp. 339-352, 2001.
[11] M.A. Hasan, M. Wang, and V.K. Bhargava, Modular Construction of Low Complexity Parallel Multipliers for a Class of Finite Fields$GF(2^m)$ IEEE Trans. Computers, vol. 41, no. 8, pp. 962-971, Aug. 1992.
[12] M.A. Hasan, M.Z. Wang, and V.K. Bhargava, “A Modified Massey-Omura Parallel Multiplier for a Class of Finite Fields,” IEEE Trans. Computers, vol. 42, no. 10, pp. 1278-1280, Oct. 1993.
[13] 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.
[14] T. Itoh and S. Tsujii, “Structure of Parallel Multipliers for a Class of Finite Fields$GF(2^m)$,” Information and Computation, vol. 83, pp. 21-40, 1989.
[15] R. Lidl and H. Niederreiter, Finite Fields. Reading, Mass.: Addison-Wesley, 1983.
[16] J.L. Massey and J.K. Omura, “Computational Method and Apparatus for Finite Field Arithmetic,” US Patent No. 4587627, 1984.
[17] R.C. Mullin,I.M. Onyszchuk,S.A. Vanstone, and R.M. Wilson,"Optimal Normal Bases inGF(pn)," Discrete Applied Maths., pp. 142-169, 1988/89.
[18] I.M. Onyszchuk, R.C. Mullin, and S.A. Vanstone, “Computational Method and Apparatus for Finite Field Multiplication,” US Patent No. 4,745,568, 1988.
[19] J.H. Silverman, “Fast Multiplication in Finite Fields$\big. {\rm GF}(2^N)\bigr.$,” Proc. Cryptographic Hardware and Embedded Systems, First Int'l Workshop (CHES '99), ÇK. Koçand C. Paar, eds., pp. 122-134, 1999.
[20] 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.
[21] A. Wassermann, “Konstruktion von Normalbasen,” Bayreuther Mathematische Schriften, pp. 155-164, 1990.
[22] J.K. Wolf, “Efficient Circuits for Multiplying in$\big. {\rm GF}(2^m)\bigr.$for Certain Values of$\big. m\bigr.$,” Discrete Math., vols. 106/107, pp. 497-502, 1992.
[23] H. Wu, M.A. Hasan, and I.F. Blake, “Highly Regular Architectures for Finite Field Computation Using Redundant Basis,” Proc. Cryptographic Hardware and Embedded Systems, First Int'l Workshop (CHES '99), ÇK. Koçand C. Paar, eds., pp. 269-279, 1999.

Index Terms:
Finite field arithmetic, cyclotomic ring, redundant set, normal basis, multiplier, squaring.
Citation:
Huapeng Wu, M. Anwar Hasan, Ian F. Blake, Shuhong Gao, "Finite Field Multiplier Using Redundant Representation," IEEE Transactions on Computers, vol. 51, no. 11, pp. 1306-1316, Nov. 2002, doi:10.1109/TC.2002.1047755
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