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Arash ReyhaniMasoleh, M. Anwar Hasan, "Fast Normal Basis Multiplication Using General Purpose Processors," IEEE Transactions on Computers, vol. 52, no. 11, pp. 13791390, November, 2003.  
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@article{ 10.1109/TC.2003.1244936, author = {Arash ReyhaniMasoleh and M. Anwar Hasan}, title = {Fast Normal Basis Multiplication Using General Purpose Processors}, journal ={IEEE Transactions on Computers}, volume = {52}, number = {11}, issn = {00189340}, year = {2003}, pages = {13791390}, doi = {http://doi.ieeecomputersociety.org/10.1109/TC.2003.1244936}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
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TY  JOUR JO  IEEE Transactions on Computers TI  Fast Normal Basis Multiplication Using General Purpose Processors IS  11 SN  00189340 SP1379 EP1390 EPD  13791390 A1  Arash ReyhaniMasoleh, A1  M. Anwar Hasan, PY  2003 KW  Finite field multiplication KW  normal basis KW  software algorithms KW  ECDSA KW  composite fields. VL  52 JA  IEEE Transactions on Computers ER   
Abstract—For cryptographic applications, normal bases have received considerable attention, especially for hardware implementation. In this article, we consider fast software algorithms for normal basis multiplication over the extended binary field GF (2^m). We present a vectorlevel algorithm which essentially eliminates the bitwise inner products needed in the conventional approach to the normal basis multiplication. We then present another algorithm which significantly reduces the dynamic instruction counts. Both algorithms utilize the full width of the datapath of the general purpose processor on which the software is to be executed. We also consider composite fields and present an algorithm which can provide further speedups and an added flexibility toward hardwaresoftware codesign of processors for very large finite fields.
[1] G.B. Agnew, R.C. Mullin, I.M. Onyszchuk, and S.A. Vanstone, An Implementation for a Fast PublicKey Cryptosystem J. Cryptology, vol. 3, pp. 6379, 1991.
[2] G.B. Agnew, R.C. Mullin, and S.A. Vanstone, An Implementation of Elliptic Curve Cryptosystems over$F_{2^{155}}$ IEEE J. Selected Areas in Comm., vol. 11, no. 5, pp. 804813, June 1993.
[3] K. Aoki and K. Ohta, Fast Arithmetic Operations over$F_{2^n}$for Software Implementation Proc. Fourth Ann. Workshop Selected Areas in Cryptography (SAC' 97), 1997.
[4] D.W. Ash, I.F. Blake, and S.A. Vanstone, “Low Complexity Normal Bases,” Discrete Applied Math., vol. 25, pp. 191210, 1989.
[5] M. Ciet and J.J. Quisquater, F. Sica, A Secure Family of Composite Finite Fields Suitable for Fast Implementation of Elliptic Curve Cryptography Proc. Indocrypt 2001, pp. 108116, Dec. 2001.
[6] S.D. Galbraith and N. Smart, A Cryptographic Application of Weil Descent Proc. Seventh IMA Conf. Cryptography and Coding, pp. 191200, 1999.
[7] S. Gao and H.W. Lenstra Jr., Optimal Normal Bases Designs, Codes and Cryptography, vol. 2, pp. 315323, 1992.
[8] J. Guajardo and C. Paar, “Efficient Algorithms for Elliptic Curve Cryptosystems,” Advances in Cryptology—CRYPTO 97, B.S. Kaliski, ed., pp. 342356, 1997.
[9] M.A. Hasan, LookUp TableBased Large Finite Field Multiplication in Memory Constrained Cryptosystems IEEE Trans. Computers, vol. 49, no. 7, pp. 749758, July 2000.
[10] M.A. Hasan, M.Z. Wang, and V.K. Bhargava, “A Modified MasseyOmura Parallel Multiplier for a Class of Finite Fields,” IEEE Trans. Computers, vol. 42, no. 10, pp. 12781280, Oct. 1993.
[11] IEEE Std 13632000, IEEE Standard Specifications for PublicKey Cryptography, IEEE Computer Soc., Aug. 2000.
[12] D. Johnson, A. Menezes, and S. Vanstone, The Elliptic Curve Digital Signature Algorithm (ECDSA) Int'l J. Information Security, vol. 1, pp. 3663, 2001.
[13] E. Knudsen, “Elliptic Scalar Multiplication Using Point Halving,” Proc. Advances in Cryptology—Asiacrypt '99, pp. 135149, 1999.
[14] N. Koblitz, Elliptic Curve Cryptosystems Math. Computation, vol. 48, pp. 203209, 1987.
[15] Ç.K. Koç and T. Acar, “Montgomery Multplication in$\big. GF(2^k)\bigr.$,” Design, Codes, and Cryptography, vol. 14, no. 1, pp. 5769, 1998.
[16] C. Lee and J. Lim, A New Aspect of Dual Basis for Efficient Field Arithmetic Proc. Int'l Workshop Practice and Theory in Public Key Cryptography (PKC '99), pp. 1228, 1999.
[17] J. Lopez and R. Dahab, High Speed Software Multiplication in$F_{2^m}$ Proc. Indocrypt 2000, pp. 203212, 2000.
[18] C.C. Lu, A Search of Minimal Key Functions for Normal Basis Multipliers IEEE Trans. Computers, vol. 46, no. 5, pp. 588592, May 1997.
[19] J.L. Massey and J.K. Omura, Computational Method and Apparatus for Finite Field Arithmetic US Patent No. 4,587,627, 1986.
[20] M. Maurer, A. Menezes, and E. Teske, Analysis of the GHS Weil Descent Attack on the ECDLP over Characteristic Two Finite Fields of Composite Degree Proc. Indocrypt 2001, pp. 195213, Dec. 2001.
[21] A.J. Menezes, I.F. Blake, X. Gao, R.C. Mullin, S.A. Vanstone, and T. Yaghoobian, Applications of Finite Fields. Kluwer Academic, 1993.
[22] V.S. Miller, "Use of Elliptic Curves in Cryptography," Advances in Cryptology—Crypto 85, Lecture Notes in Computer Science, H.C. Williams, ed., Vol. 218, SpringerVerlag, New York, 1986, pp. 417426.
[23] R.C. Mullin,I.M. Onyszchuk,S.A. Vanstone, and R.M. Wilson,"Optimal Normal Bases inGF(pn)," Discrete Applied Maths., pp. 142169, 1988/89.
[24] P. Ning and Y.L. Yin, Efficient Software Implementation for Finite Field Multiplication in Normal Basis Proc. Information and Commu. Security (ICICS 2001), pp. 177181, Nov. 2001.
[25] Nat'l Inst. of Standards and Tech nology, Digital Signature Standard, FIPS Publication 1862, 2000.
[26] S. Oh, C.H. Kim, J. Lim, and D.H. Cheon, Efficient Normal Basis Multipliers in Composite Fields IEEE Trans. Computers, vol. 49, no. 10, pp. 11331138, Oct. 2000.
[27] C. Paar, P. Fleishmann, and P. SoriaRodriguez, Fast Arithmetic for PublicKey Algorithms in Galois Fields with Composite Exponents IEEE Trans. Computers, vol. 48, no. 10, pp. 10251034, Oct. 1999.
[28] A. ReyhaniMasoleh and M.A. Hasan, On Efficient Normal Basis Multiplication Proc. Indocrypt 2000, pp. 213224, Dec. 2000.
[29] A. ReyhaniMasoleh and M.A. Hasan, Fast Normal Basis Multiplication Using General Purpose Processors Technical Report CORR 200125, Dept. of C&O, Univ. of Waterloo, Canada, Apr. 2001.
[30] A. ReyhaniMasoleh and M.A. Hasan, Fast Normal Basis Multiplication Using General Purpose Processors Proc. Selected Areas in Cryptography (SAC 2001), pp. 230244, Aug. 2001.
[31] A. ReyhaniMasoleh and M.A. Hasan, A New Construction of MasseyOmura Parallel Multiplier over$GF(2^m)$ IEEE Trans. Computers, vol. 51, no. 5, pp. 511520, May 2002.
[32] M. Rosing, Implementing Elliptic Curve Cryptography. Manning Publications, 1999.
[33] R. Schroeppel, S. O'Malley, H. Orman, and O. Spatscheck, “A Fast Software Implementation for Arithmetic Operations in GF($2^n$),” Proc. Advances in Cryptology–CRYPTO '95, pp. 4356, 1995.
[34] N.P. Smart, How Secure Are Elliptic Curves over Composite Extension Fields? Proc. Eurocrypt 2001, pp. 3039, 2001.
[35] B. Sunar and Ç.K. Koç, An Efficient Optimal Normal Basis Type II Multiplier IEEE Trans. Computers, vol. 50, no. 1, pp. 8387, Jan. 2001.