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Montgomery Multiplier and Squarer for a Class of Finite Fields
May 2002 (vol. 51 no. 5)
pp. 521-529

Montgomery multiplication in {\rm GF}(2^m) is defined by a(x)b(x)r^{-1}(x)\bmod{f(x)}, where the field is generated by a root of the irreducible polynomial f(x), a(x) and b(x) are two field elements in {\rm GF}(2^m), and r(x) is a fixed field element in {\rm GF}(2^m). In this paper, first, a slightly generalized Montgomery multiplication algorithm in {\rm GF}(2^m) is presented. Then, by choosing r(x) according to f(x), we show that efficient architectures of bit-parallel Montgomery multiplier and squarer can be obtained for the fields generated with an irreducible trinomial. Complexities of the Montgomery multiplier and squarer in terms of gate counts and time delay of the circuits are investigated and found to be as good as or better than that of previous proposals for the same class of fields.

[1] http://csrc.nist.govencryption, 2001.
[2] A.V. Aho,J.E. Hopcroft, and J.D. Ullman,The Design and Analysis of Computer Algorithms.Reading, Mass.: Addison-Wesley, 1974.
[3] ÇK. Koç and T. Acar, “Fast Software Exponentiation in${\rm GF}(2^k)$,” Proc. 13th Symp. Computer Arithmetic, pp. 279-287, July 1997.
[4] Ç.K. Koç and T. Acar, “Montgomery Multplication in$\big. GF(2^k)\bigr.$,” Design, Codes, and Cryptography, vol. 14, no. 1, pp. 57-69, 1998.
[5] Ç.K. Koç and B. Sunar, Low-Complexity Bit-Parallel Canonical and Normal Basis Multipliers for a Class of Finite Fields IEEE Trans. Computers, vol. 47, no. 3, pp. 353-356, Mar. 1998.
[6] 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.
[7] B.S. Kaliski Jr., “The Montgomery Inverse and Its Applications,” IEEE Trans. Computers, vol. 44, no. 8, pp. 1,064-1,065, Aug. 1995.
[8] A. Karatsuba and Y. Ofman, “Multiplication of Multidigit Numbers on Automata,” Sov. Phys.-Dokl. (English translation), vol. 7, no. 7, pp. 595-596, 1963.
[9] J.L. Massey and J.K. Omura, “Computational Method and Apparatus for Finite Field Arithmetic,” US Patent No. 4587627, 1984.
[10] A.J. Menezes, I.F. Blake, X. Gao, R.C. Mullin, S.A. Vanstone, and T. Yaghoobian, Applications of Finite Fields. Kluwer Academic, 1993.
[11] P.L. Montgomery, “Modular Multiplication without Trial Division,” Math. Computation, vol. 44, pp. 519-521, 1985.
[12] C. Paar, “Efficient VLSI Architectures for Bit-Parallel Computation in Galois Fields,” PhD thesis, VDI-Verlag, Düsseldorf, 1994.
[13] C. Paar, P. Fleischmann, and P. Roelse, “Efficient Multiplier Architectures for Galois Fields,” IEEE Trans. Computers, vol. 47, no. 2, pp. 162-170, Feb. 1998.
[14] E. Savas and Ç.K. Koç, “The Montgomery Modular Inverse—Revisited,” IEEE Trans. Computers, vol. 49, no. 7, pp. 763-766, July 2000.
[15] E. Savas, A.F. Tenca, and Ç.K. Koç, “A Scalable and Unified Multiplier Architecture for Finite Fields$\big. GF(p)\bigr.$and$\big. GF(2^m)\bigr.$,” Proc. Workshop Cryptographic Hardware and Embedded Systems (CHES 2000), Ç.K. Koçand C. Paar, eds., pp. 277-292, 2000.
[16] B. Sunar and Ç.K. Koç, Mastrovito Multiplier for All Trinomials IEEE Trans. Computers, vol. 48, no. 5, pp. 522-527, May 1999.
[17] B. Sunar and Ç.K. Koç, An Efficient Optimal Normal Basis Type II Multiplier IEEE Trans. Computers, vol. 50, no. 1, pp. 83-87, Jan. 2001.
[18] 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.
[19] M. Wang and I.F. Blake,"Bit-Serial Multiplication in Finite Fields," SIAM J. Discrete Maths., vol. 3, pp. 140-148, Feb. 1990.
[20] H. Wu, Low Complexity Bit-Parallel Finite Field Arithmetic Using Polynomial Basis Cryptographic Hardware and Embedded Systems, Ç.K. Koçand C. Paar, eds., pp. 280-291, Berlin: Springer-Verlag, 1999.
[21] H. Wu, “Montgomery Multiplier and Squarer in$\big. {\rm GF}(2^m)\bigr.$,” Proc. Cryptographic Hardware and Embedded Systems (CHES 2000), pp. 264-276, Aug. 2000.
[22] H. Wu, M.A. Hasan, and I.F. Blake, New Low-Complexity Bit-Parallel Finite Field Multipliers Using Weakly Dual Bases IEEE Trans. Computers, vol. 47, no. 11, pp. 1223-1233, Nov. 1998.

Index Terms:
Finite fields arithmetic, hardware architecture, Montgomery multiplication, elliptic curve cryptography
Citation:
H. Wu, "Montgomery Multiplier and Squarer for a Class of Finite Fields," IEEE Transactions on Computers, vol. 51, no. 5, pp. 521-529, May 2002, doi:10.1109/TC.2002.1004591
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