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An Efficient Basis Conversion Algorithm for Composite Fields with Given Representations
August 2005 (vol. 54 no. 8)
pp. 992-997
Berk Sunar, IEEE Computer Society
We describe an efficient method for constructing the basis conversion matrix between two given finite field representations where one is composite. We are motivated by the fact that using certain representations, e.g., low-Hamming weight polynomial or composite field representations, permits arithmetic operations such as multiplication and inversion to be computed more efficiently. An earlier work by Paar defines the conversion problem and outlines an exponential time algorithm that requires an exhaustive search in the field. Another algorithm by Sunar et al. provides a polynomial time algorithm for the limited case where the second representation is constructed (rather than initially given). The algorithm we present facilitates existing factorization algorithms and provides a randomized polynomial time algorithm to solve the basis conversion problem where the two representations are initially given. We also adapt a fast trace-based factorization algorithm to work in the composite field setting which yields a subcubic complexity algorithm for the construction of the basis conversion matrix.

[1] C. Paar, “Efficient VLSI Architectures for Bit-Parallel Computation in Galois Fields,” PhD thesis, English translation, Inst. for Experimental Math., Univ. of Essen, Essen, Germany, June 1994.
[2] R. Schroeppel, H. Orman, S. O'Malley, and O. Spatscheck, “Fast Key Exchange with Elliptic Curve Systems,” Advances in Cryptology (CRYPTO '95), D. Coppersmith, ed., pp. 43-56, 1995.
[3] J. Fan and C. Paar, “On Efficient Inversion in Tower Fields of Characteristic Two,” Proc. 1997 IEEE Int'l Symp. Information Theory, p. 20, 1997.
[4] IEEE P1363 Standard Specifications for Public Key Cryptography, 2000.
[5] B.S. Kaliski Jr. and Y.L. Yin, “Storage-Efficient Finite Field Basis Conversion,” Proc. Selected Areas in Cryptography '98, S. Tavares and H. Meijer, eds., pp. 81-93, 1999.
[6] B. Kaliski Jr. and M. Liskov, “Efficient Finite Field Basis Conversion Involving Dual Bases,” Proc. Workshop Cryptographic Hardware and Embedded Systems (CHES 1999), Ç. Koç and C. Paar, eds., pp. 135-143, Aug. 1999.
[7] M. Ben-Or, “Probabilistic Algorithms in Finite Fields,” Proc. 22nd Ann. IEEE Symp. Foundations of Computer Science, pp. 394-398, 1981.
[8] J. von zur Gathen and V. Shoup, “Computing Frobenius Maps and Factoring Polynomials,” Computational Complexity, vol. 2, pp. 187-224, 1992.
[9] E. Kaltofen and V. Shoup, “Fast Polynomial Factorization over High Algebraic Extensions of Finite Fields,” Proc. Int'l Symp. Symbolic and Algebraic Computation, 1997.
[10] A.J. Menezes, I.F. Blake, X. Gao, R.C. Mullen, S.A. Vanstone, and T. Yaghoobian, Applications of Finite Fields. Kluwer Academic, 1993.
[11] B. Sunar, E. Savaş, and Ç.K. Koç, “Constructing Composite Field Representations for Efficient Conversion,” IEEE Trans. Computers, vol. 52, no. 11, pp. 1391-1398, Nov. 2003.
[12] D.G. Cantor and H. Zassenhaus, “A New Algorithm for Factoring Polynomials over Finite Fields,” Math. Computation, vol. 36, pp. 587-592, 1981.
[13] E. Kaltofen, “Polynomial Factorization 1982-1986,” Computers in Math., Lecture Notes in Pure and Applied Math., D. Chudnovsky and R. Jenks, eds., vol. 125, pp. 285-309, New York: Marcel Dekker, 1990.
[14] E. Kaltofen, “Polynomial Factorization 1987-1991,” Proc. LATIN '92, I. Simon, ed., pp. 294-313, 1992.
[15] A. Karatsuba and Y. Ofman, “Multiplication of Multidigit Numbers on Automata,” Soviet Physiks Doklady (English translation), vol. 7, no. 7, pp. 595-596, 1963.
[16] R.P. Brent and H.T. Kung, “Fast Algorithms for Manipulating Formal Power Series,” J. ACM, vol. 4, no. 25, pp. 581-595, 1978.
[17] X. Huang and V.Y. Pan, “Fast Rectangular Matrix Multiplication and Applications,” J. Complexity, vol. 14, pp. 257-299, June 1998.

Index Terms:
Index Terms- Finite fields, change of basis, composite fields, polynomial factorization.
Berk Sunar, "An Efficient Basis Conversion Algorithm for Composite Fields with Given Representations," IEEE Transactions on Computers, vol. 54, no. 8, pp. 992-997, Aug. 2005, doi:10.1109/TC.2005.124
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