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Issue No.10 - October (2010 vol.59)
pp: 1297-1308
Omran Ahmadi , University College Dublin, Ireland
We present low complexity formulae for the computation of cubing and cube root over $\F_{3^m}$ constructed using special classes of irreducible trinomials, tetranomials and pentanomials. We show that for all those special classes of polynomials, field cubing and field cube root operation have the same computational complexity when implemented in hardware or software platforms. As one of the main applications of these two field arithmetic operations lies in pairing-based cryptography, we also give in this paper a selection of irreducible polynomials that lead to low cost field cubing and field cube root computations for supersingular elliptic curves defined over $\F_{3^m}$, where $m$ is a prime number in the pairing-based cryptographic range of interest, namely, $m$ ∈ [47, 541].
Finite field arithmetic, cubing, cube root, characteristic three, cryptography.
Omran Ahmadi, "Low Complexity Cubing and Cube Root Computation over $\F_{3^m}$ in Polynomial Basis", IEEE Transactions on Computers, vol.59, no. 10, pp. 1297-1308, October 2010, doi:10.1109/TC.2009.183
[1] O. Ahmadi, D. Hankerson, and A. Menezes, "Formulas for Cube Roots in ${\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{3^{\rm m}}$ ," Discrete Applied Math., vol. 155, no. 3, pp. 260-270, 2007.
[2] O. Ahmadi, D. Hankerson, and A. Menezes, "Software Implementation of Arithmetic in ${\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{3^{\rm m}}$ ," Proc. Int'l Workshop Arithmetic of Finite Fields (WAIFI), C. Carlet and B. Sunar, eds., vol. 4547, pp. 85-102, 2007.
[3] O. Ahmadi, D. Hankerson, and F. Rodríguez-Henríquez, "Parallel Formulations of Scalar Multiplication on Koblitz Curves," J. Universal Computer Science, special issue on cryptography in computer system security, vol. 14, pp. 481-504, 2008.
[4] O. Ahmadi and A. Menezes, "Irreducible Polynomials of Maximum Weight," Utilitas Mathematica, vol. 72, pp. 111-123, 2007.
[5] R. Avanzi, "Another Look at Square Roots (and Other Less Common Operations) in Fields of Even Characteristic," Proc. Selected Areas in Cryptography, C.M. Adams, A. Miri, and M.J. Wiener, eds., vol. 4876, pp. 138-154, 2007.
[6] R. Avanzi, "Another Look at Square Roots and Traces (and Quadratic Equations) in Fields of Even Characteristic," Cryptology ePrint Archive, Report 2007/103, http:/, 2007.
[7] P.S.L.M. Barreto, "A Note on Efficient Computation of Cube Roots in Characteristic 3," Cryptology ePrint Archive, Report 2004/305, http:/, 2004.
[8] P.S.L.M. Barreto, H.Y. Kim, B. Lynn, and M. Scott, "Efficient Algorithms for Pairing-Based Cryptosystems," Proc. 22nd Ann. Int'l Cryptology Conf. Advances in Cryptology (CRYPTO '02), M. Yung, ed., pp. 354-368, 2002.
[9] J.-L. Beuchat, N. Brisebarre, J. Detrey, E. Okamoto, and F. Rodríguez-Henríquez, "A Comparison between Hardware Accelerators for the Modified Tate Pairing over ${\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{2^m}$ and ${\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_{3^m}$ ," Pairing Based Cryptography—Pairing 2008, S.D. Galbraith and K.G. Paterson, eds., pp. 297-315, Springer, 2008.
[10] J.-L. Beuchat, N. Brisebarre, J. Detrey, E. Okamoto, M. Shirase, and T. Takagi, "Algorithms and Arithmetic Operators for Computing the $\eta_t$ Pairing in Characteristic Three," IEEE Trans. Computers, special section on special-purpose hardware for cryptography and cryptanalysis, vol. 57, no. 11, pp. 1454-1468, Nov. 2008.
[11] J.-L. Beuchat, J. Detrey, N. Estibals, E. Okamoto, and F. Rodríguez-Henríquez, "Fast Architectures for the $\eta_{T}$ Pairing over Small-Characteristic Supersingular Elliptic Curves," Cryptology ePrint Archive, Report 2009/398, 2009.
[12] J.-L. Beuchat, E. López-Trejo, L. Martínez-Ramos, S. Mitsunari, and F. Rodríguez-Henríquez, "Multi-Core Implementation of the Tate Pairing over Supersingular Elliptic Curves," Proc. Cryptology and Network Security (CANS '09), J.A. Garay, A. Miyaji, and A. Otsuka, eds., pp. 413-432, 2009.
[13] A.W. Bluher, "A Swan-Like Theorem," Finite Fields and Their Applications, vol. 12, no. 1, pp. 128-138, 2006.
[14] C. Doche, "Redundant Trinomials for Finite Fields of Characteristic 2," Proc. 10th Australasian Conf. Information Security and Privacy (ACISP '05), C. Boyd and J.-M. González Nieto, eds., vol. 3574, pp. 122-133, 2005.
[15] K. Fong, D. Hankerson, J. López, and A. Menezes, "Field Inversion and Point Halving Revisited," IEEE Trans. Computers, vol. 53, no. 8, pp. 1047-1059, Aug. 2004.
[16] D. Hankerson, A. Menezes, and M. Scott, Software Implementation of Pairings, chapter 12. IOS Press, to be published.
[17] D. Hankerson, A. Menezes, and S. Vanstone, Guide to Elliptic Cryptography. Springer-Verlag, 2004.
[18] IEEE Standards Documents, IEEE P1363: Standard Specifications for Public Key Cryptography. Draft Version D18, IEEE, http://grouper., Nov. 2004.
[19] B. Ito and S. Tsujii, "Structure of a Parallel Multipliers for a Class of Fields ${\rm GF}(2^m)$ Using Normal Bases," Information and Computers, vol. 83, pp. 21-40, 1989.
[20] A. Menezes, I. Blake, S. Gao, R. Mullin, S. Vanstone, and T. Yaghoobian, Applications of Finite Fields. Kluwer, 1993.
[21] D. Panairo and D. Thompson, "Efficient $p{\rm th}$ Root Computations in Finite Fields of Characteristic $p$ ," Designs, Codes and Cryptography, vol. 50, pp. 351-358, 2009.
[22] F. Rodríguez-Henríquez, G. Morales-Luna, and J. López, "Low-Complexity Bit-Parallel Square Root Computation over GF($2^{m}$ ) for All Trinomials," IEEE Trans. Computers, vol. 57, no. 4, pp. 472-480, Apr. 2008.
[23] M. Scott, "Optimal Irreducible Polynomials for GF($2^m$ ) Arithmetic," Cryptology ePrint Archive, Report 2007/192, http:/, 2007.
[24] J.H. Silverman, "Fast Multiplication in Finite Fields ${\rm GF}(2^{\rm n}$ )," Proc. Workshop Cryptographic Hardware and Embedded Systems (CHES), Ç.K. Koç and C. Paar, eds., vol. 1717, pp. 122-134, 1999.
[25] J. von zur Gathen, "Irreducible Trinomials over Finite Fields," Math. of Computation, vol. 72, no. 243, pp. 1443-1452, 2003.
[26] L.C. Washington, Elliptic Curves—Number Theory and Cryptography, second ed. CRC Press, 2008.
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