Abstract
We present an efficient exponentiation algorithm in a finite field GF(q n) using a Gauss period of type (n,1). Though the Gauss period α of type (n,1) in GF(q n) is never primitive, a computational evidence says that there always exists a sparse polynomial (especially, a trinomial) of α which is a primitive element in GF(q n). Our idea is easily generalized to the field determined by a root of unity over GF(q) with redundant basis technique. Consequently, we find primitive elements which yield a fast exponentiation algorithm for many finite fields GF(q n), where a Gauss period of type (n,k) exists only for larger values of k or the existing Gauss period is not primitive and has large index in the multiplicative group GF(q n)×.
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References
Brickell, E.F., Gordon, D.M., McCurley, K.S., Wilson, D.B.: Fast exponentiation with precomputation. In: Rueppel, R.A. (ed.) EUROCRYPT 1992. LNCS, vol. 658, pp. 200–207. Springer, Heidelberg (1992)
Tenenbaum, G.: Introduction to analytic and probabilistic number theory. Cambridge Univ. Press, Cambridge (1995)
Gao, S., von zur Gathen, J., Panario, D.: Gauss periods and fast exponentiation in finite fields. In: Baeza-Yates, R., Poblete, P.V., Goles, E. (eds.) LATIN 1995. LNCS, vol. 911, pp. 311–322. Springer, Heidelberg (1995)
Gao, S., von zur Gathen, J., Panario, D.: Orders and cryptographical applications. Math. Comp. 67, 343–352 (1998)
Gao, S., Vanstone, S.: On orders of optimal normal basis generators. Math. Comp. 64, 1227–1233 (1995)
Lim, C.H., Lee, P.J.: More flexible exponentiation with precomputation. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 95–107. Springer, Heidelberg (1994)
de Rooij, P.: Efficient exponentiation using precomputation and vector addition chains. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 389–399. Springer, Heidelberg (1994)
Kwon, S., Kim, C.H., Hong, C.P.: Efficient exponentiation for a class of finite fields GF(2n) determined by Gauss periods. In: Walter, C.D., Koç, Ç.K., Paar, C. (eds.) CHES 2003. LNCS, vol. 2779, pp. 228–242. Springer, Heidelberg (2003) (to appear)
Menezes, A.J., Blake, I.F., Gao, S., Mullin, R.C., Vanstone, S.A., Yaghoobian, T.: Applications of finite fields. Kluwer Academic Publisher, Dordrecht (1993)
Feisel, S., von zur Gathen, J., Shokrollahi, M.: Normal bases via general Gauss periods. Math. Comp. 68, 271–290 (1999)
von zur Gathen, J., Shparlinski, I.: Constructing elements of large order in finite fields. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC 1999. LNCS, vol. 1719, pp. 404–409. Springer, Heidelberg (1997)
von zur Gathen, J., Nöcker, M.J.: Exponentiation in finite fields: Theory and Practice. In: Mattson, H.F., Mora, T. (eds.) AAECC 1997. LNCS, vol. 1255, pp. 88–133. Springer, Heidelberg (1997)
von zur Gathen, J., Shparlinski, I.: Orders of Gauss periods in finite fields. In: Staples, J., Katoh, N., Eades, P., Moffat, A. (eds.) ISAAC 1995. LNCS, vol. 1004, pp. 208–215. Springer, Heidelberg (1995)
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Kwon, S., Kim, C.H., Hong, C.P. (2003). Gauss Period, Sparse Polynomial, Redundant Basis, and Efficient Exponentiation for a Class of Finite Fields with Small Characteristic. In: Ibaraki, T., Katoh, N., Ono, H. (eds) Algorithms and Computation. ISAAC 2003. Lecture Notes in Computer Science, vol 2906. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24587-2_75
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DOI: https://doi.org/10.1007/978-3-540-24587-2_75
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