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2022 | OriginalPaper | Buchkapitel

Interpolation of the Zech’s Logarithm: Explicit Forms

verfasst von : Gerasimos C. Meletiou, Michael Th. Rassias

Erschienen in: Approximation and Computation in Science and Engineering

Verlag: Springer International Publishing

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Abstract

Zech’s logarithm is a function closely related to the Discrete Logarithm. It has applications in communications, cryptography, and computing. In this paper, we provide polynomial and exponential formulas for Zech’s logarithm over prime fields.

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Literatur
1.
Zurück zum Zitat Z. Chang et al., Binary de Bruijn sequences via Zech’s logarithms (2019). arXiv:1705.03150 Z. Chang et al., Binary de Bruijn sequences via Zech’s logarithms (2019). arXiv:1705.03150
2.
Zurück zum Zitat J.H. Conway, A tabulation of some information concerning finite fields, in Computers in Mathematical Research, ed. by R.F. Churchouse, J.C. Herz (North-Holland, Amsterdam, 1968), pp. 37–50 J.H. Conway, A tabulation of some information concerning finite fields, in Computers in Mathematical Research, ed. by R.F. Churchouse, J.C. Herz (North-Holland, Amsterdam, 1968), pp. 37–50
4.
Zurück zum Zitat T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inform. Theory 31, 469–472 (1985)MathSciNetCrossRefMATH T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inform. Theory 31, 469–472 (1985)MathSciNetCrossRefMATH
5.
Zurück zum Zitat D. Hachenberger, D. Jungnickel, Topics in Galois Fields. Algorithms and Computation in Mathematics (Springer, Berlin, 2020) D. Hachenberger, D. Jungnickel, Topics in Galois Fields. Algorithms and Computation in Mathematics (Springer, Berlin, 2020)
7.
Zurück zum Zitat K. Huber, Solving equations in finite fields and some results concerning the structure of GF(pm). IEEE Trans. Inform. Theory 38, 1154–1162 (1992)MathSciNetCrossRefMATH K. Huber, Solving equations in finite fields and some results concerning the structure of GF(pm). IEEE Trans. Inform. Theory 38, 1154–1162 (1992)MathSciNetCrossRefMATH
8.
Zurück zum Zitat K. Huber et al., Method and device for generating a pseudo random sequence using a discrete logarithm. Patent, EP1342153 (2003), WO/2002/046912 (2002), US20040054703 (2004) K. Huber et al., Method and device for generating a pseudo random sequence using a discrete logarithm. Patent, EP1342153 (2003), WO/2002/046912 (2002), US20040054703 (2004)
9.
Zurück zum Zitat K. Imamura, A method for computing addition tables in GF (pn). IEEE Trans. Inform. Theory IT-26, 367–369 (1980)CrossRefMATH K. Imamura, A method for computing addition tables in GF (pn). IEEE Trans. Inform. Theory IT-26, 367–369 (1980)CrossRefMATH
10.
Zurück zum Zitat A. Joux, A. Odlyzko, C. Pierrot, The Past, evolving Present and Future of Discrete Logarithm. Open Problems in Mathematics and Computational Science (2014), pp. 5–36 A. Joux, A. Odlyzko, C. Pierrot, The Past, evolving Present and Future of Discrete Logarithm. Open Problems in Mathematics and Computational Science (2014), pp. 5–36
11.
Zurück zum Zitat D. Knuth, The Art of Computer Programming, vol III (1969) D. Knuth, The Art of Computer Programming, vol III (1969)
12.
Zurück zum Zitat C.D. Lee, Zech logarithmic decoding of triple-error-correcting binary cyclic codes. IEEE Commun. Lett. 12(10), 776– 778 (2008)CrossRef C.D. Lee, Zech logarithmic decoding of triple-error-correcting binary cyclic codes. IEEE Commun. Lett. 12(10), 776– 778 (2008)CrossRef
13.
Zurück zum Zitat F.J. MacWiIIiams, N.J.A. Sloane, The Theory of Error-Correcting Codes (North Holland, Amsterdam, 1977) F.J. MacWiIIiams, N.J.A. Sloane, The Theory of Error-Correcting Codes (North Holland, Amsterdam, 1977)
14.
Zurück zum Zitat G.C. Meletiou, A polynomial representation for exponents in Zp. Bull. Greek Math. Soc. 34, 59–63 (1992)MathSciNetMATH G.C. Meletiou, A polynomial representation for exponents in Zp. Bull. Greek Math. Soc. 34, 59–63 (1992)MathSciNetMATH
15.
Zurück zum Zitat J.F. Nash, Jr., M.Th. Rassias (eds.), Open Problems in Mathematics (Springer, Berlin, 2016)MATH J.F. Nash, Jr., M.Th. Rassias (eds.), Open Problems in Mathematics (Springer, Berlin, 2016)MATH
17.
Zurück zum Zitat A.M. Odlyzko, Discrete logarithms in finite fields and their cryptographic significance, in Proc. EUROCRYPT 84. Lecture Notes in Computer Science, vol. 209 (Springer, Berlin, Heidelberg, New York, 1985), pp. 224–314 A.M. Odlyzko, Discrete logarithms in finite fields and their cryptographic significance, in Proc. EUROCRYPT 84. Lecture Notes in Computer Science, vol. 209 (Springer, Berlin, Heidelberg, New York, 1985), pp. 224–314
18.
Zurück zum Zitat S.C. Pohlig, M.E. Hellman, An improved algorithm for computing logarithms over GF (p) and its cryptographic significance. Inform. Theory 24, 106–110 (1978)MathSciNetCrossRefMATH S.C. Pohlig, M.E. Hellman, An improved algorithm for computing logarithms over GF (p) and its cryptographic significance. Inform. Theory 24, 106–110 (1978)MathSciNetCrossRefMATH
19.
Zurück zum Zitat J. Szmidt, Nonlinear feedback shift registers and Zech’s logarithms, in Proceedings of International Conference on Military Communications and Information Systems (ICMCIS) (2019) J. Szmidt, Nonlinear feedback shift registers and Zech’s logarithms, in Proceedings of International Conference on Military Communications and Information Systems (ICMCIS) (2019)
Metadaten
Titel
Interpolation of the Zech’s Logarithm: Explicit Forms
verfasst von
Gerasimos C. Meletiou
Michael Th. Rassias
Copyright-Jahr
2022
DOI
https://doi.org/10.1007/978-3-030-84122-5_33