Abstract
An efficient algorithm is presented which for any finite field Fq of small characteristic finds an extension F q s of polynomially bounded degree and an element α∈ F q s of exponentially large multiplicative order. The construction makes use of certain analogues of Gauss periods of a special type. This can be considered as another step towards solving the celebrated problem of finding primitive roots in finite fields efficiently.
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I. F. Blake, X.H. Gao, A. J. Menezes, R. C. Mullin, S. A. Vanstone and T. Yaghoobian, Applications of finite fields, Kluwer Acad. Publ., 1993.
S. Feisel, J. von zur Gathen and A. Shokrollahi, ‘Normal bases via general Gauss periods’, Math. Comp., 68 (1999), 271–290.
S. Gao, J. von zur Gathen and D. Panario, ‘Gauss periods and fast exponentiation in finite fields’, Proceedings LATIN ’95, Springer Lecture Notes in Comp. Sci., 911 (1995), 311–322.
S. Gao, J. von zur Gathen and D. Panario, ‘Gauss periods: Orders and cryptographical applications’, Math. Comp., 67 (1998), 343–352.
S. Gao and H. W. Lenstra, ‘Optimal normal bases’, Designs, Codes and Cryptography, 2 (1992), 315–323.
S. Gao and G. L. Mullen, ‘Dickson polynomials and irreducible polynomials over finite fields’, J. Number Theory, 49 (1994), 118–132.
S. Gao and S. Vanstone, ‘On orders of optimal normal basis generators’, Math. Comp., 64 (1995), 1227–1233.
J. von zur Gathen and M. J. Nöcker, ‘Exponentiation in finite fields: Theory and practice’, Proceedings AAECC’97, Springer Lecture Notes in Comp. Sci., 1255 (1997), 88–113.
J. von zur Gathen and I. E. Shparlinski, ‘Orders of Gauss periods in finite fields’, Appl. Algebra in Engin., Commun. and Comp., 9 (1998), 15–24.
N. M. Korobov, ‘Exponential sums with exponential functions and the distribution of digits in periodic fractions’, Matem. Zametki, 8 (1970), 641–652. Matem. Notes, 8 (1970), 831–837.
N. M. Korobov, ‘On the distribution of digits in periodic fractions’, Matem. Sbornik, 89 (1972), 654–670. Mat. USSR-Sb., 18 (1972), 659–676.
I. E. Shparlinski, ‘On finding primitive roots in finite fields’, Theor. Comp. Sci. 157 (1996), 273–275.
I. E. Shparlinski Finite fields: Theory and computation, Kluwer Acad. Publ., Dordrecht, 1999.
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von zur Gathen, J., Shparlinski, I. (1999). Constructing Elements of Large Order in Finite Fields. In: Fossorier, M., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1999. Lecture Notes in Computer Science, vol 1719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46796-3_38
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DOI: https://doi.org/10.1007/3-540-46796-3_38
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