Previous |  Up |  Next

Article

Title: Approximation of the discrete logarithm in finite fields of even characteristic by real polynomials (English)
Author: Brandstätter, Nina
Author: Winterhof, Arne
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 42
Issue: 1
Year: 2006
Pages: 43-50
Summary lang: English
.
Category: math
.
Summary: We obtain lower bounds on degree and additive complexity of real polynomials approximating the discrete logarithm in finite fields of even characteristic. These bounds complement earlier results for finite fields of odd characteristic. (English)
Keyword: Discrete logarithm
Keyword: polynomial approximation
Keyword: character sums
MSC: 11T24
MSC: 11T71
MSC: 94A60
idZBL: Zbl 1164.11073
idMR: MR2227111
.
Date available: 2008-06-06T22:47:09Z
Last updated: 2012-05-10
Stable URL: http://hdl.handle.net/10338.dmlcz/107980
.
Reference: [1] Cochrane T.: On a trigonometric inequality of Vinogradov.J. Number Theory 27 (1987), 9–16. Zbl 0629.10030, MR 0904002
Reference: [2] Coppersmith D., Shparlinski I.: On polynomial approximation of the discrete logarithm and the Diffie-Hellman mapping.J. Cryptology 13 (2000), 339–360. Zbl 1038.94007, MR 1768482
Reference: [3] Ding C., Helleseth T.: On cyclotomic generator of order $r$.Inform. Process. Lett. 66 (1998), 21–25. Zbl 1078.94511, MR 1626061
Reference: [4] Kiltz E., Winterhof A.: Polynomial interpolation of cryptographic functions related to Diffie-Hellman and discrete logarithm problem.Discrete Appl. Math. 154 (2006), 326–336. Zbl 1092.94024, MR 2194405
Reference: [5] Konyagin S., Lange T., Shparlinski I.: Linear complexity of the discrete logarithm.Des. Codes Cryptogr. 28 (2003), 135–146. Zbl 1024.11078, MR 1962801
Reference: [6] Lange T., Winterhof A.: Polynomial interpolation of the elliptic curve and XTR discrete logarithm.Lecture Notes in Comput. Sci. 2387 (2002), 137–143. Zbl 1077.94518, MR 2064510
Reference: [7] Lange T., Winterhof A.: Incomplete character sums over finite fields and their application to the interpolation of the discrete logarithm by Boolean functions.Acta Arith. 101 (2002), 223–229. Zbl 0998.11070, MR 1875841
Reference: [8] Lange T., Winterhof A.: Interpolation of the discrete logarithm in $F_q$ by Boolean functions and by polynomials in several variables modulo a divisor of $q-1$.Discrete Appl. Math. 128 (2003), 193–206. MR 1991426
Reference: [9] Meidl W., Winterhof A.: Lower bounds on the linear complexity of the discrete logarithm in finite fields.IEEE Trans. Inform. Theory 47 (2001), 2807–2811. Zbl 1032.94004, MR 1872841
Reference: [10] Meletiou G. C.: Explicit form for the discrete logarithm over the field ${\text GF}(p,k)$.Arch. Math. (Brno) 29 (1993), 25–28. Zbl 0818.11049, MR 1242625
Reference: [11] Meletiou G. C.: Explicit form for the discrete logarithm over the field ${\text GF}(p,k)$.Bul. Inst. Politeh. Iaşi. Secţ. I. Mat. Mec. Teor. Fiz. 41(45) (1995), 1–4. MR 1491193
Reference: [12] Meletiou G. C., Mullen G. L.: A note on discrete logarithms in finite fields.Appl. Algebra Engrg. Comm. Comput. 3 (1992), 75–78. Zbl 0749.11055, MR 1325748
Reference: [13] Menezes A. J., van Oorschot P. C., Vanstone S. A. : Handbook of applied cryptography.CRC Press, Boca Raton, FL 1997. Zbl 0868.94001, MR 1412797
Reference: [14] Mullen G. L., White D.: A polynomial representation for logarithms in ${\text GF}(q)$.Acta Arith. 47 (1986), 255–261. MR 0870668
Reference: [15] Niederreiter H.: A short proof for explicit formulas for discrete logarithms in finite fields.Appl. Algebra Engrg. Comm. Comput. 1 (1990), 55–57. Zbl 0726.11079, MR 1325511
Reference: [16] Niederreiter H., Winterhof A.: Incomplete character sums and polynomial interpolation of the discrete logarithm.Finite Fields Appl. 8 (2002), 184–192. MR 1894512
Reference: [17] Risler J.-J.: Khovansky’s theorem and complexity theory.Rocky Mountain J. Math. 14 (1984), 851–853. MR 0773123
Reference: [18] Risler J.-J.: Additive complexity of real polynomials.SIAM J. Comp. 14 (1985), 178–183. MR 0774937
Reference: [19] Rojas J. M.: Additive complexity and p-adic roots of polynomials.Lecture Notes in Comput. Sci. 2369 (2002), 506–516. MR 2041107
Reference: [20] Rojas J. M.: Arithmetic multivariate Descartes’ rule.Amer. J. Math. 126 (2004), 1–30. Zbl 1046.12001, MR 2033562
Reference: [21] Shparlinski I.: Number theoretic methods in cryptography. Complexity lower bounds.Birkhäuser, Basel 1999. Zbl 0912.11057, MR 1707287
Reference: [22] Shparlinski I.: Cryptographic applications of analytic number theory. Complexity lower bounds and pseudorandomness.Birkhäuser, Basel 2003. MR 1954519
Reference: [23] Winterhof A.: Some estimates for character sums and applications.Des. Codes Cryptogr. 22 (2001), 123–131. Zbl 0995.11067, MR 1813781
Reference: [24] Winterhof A.: Incomplete additive character sums and applications.In: Jungnickel, D. and Niederreiter, H. (eds.): Finite fields and applications, 462–474, Springer, Heidelberg 2001. Zbl 1019.11034, MR 1849268
Reference: [25] Winterhof A.: Polynomial interpolation of the discrete logarithm.Des. Codes Cryptogr. 25 (2002), 63–72. Zbl 1017.11065, MR 1881340
Reference: [26] Winterhof A.: A note on the linear complexity profile of the discrete logarithm in finite fields.Progress Comp. Sci. Appl. Logic 23 (2004), 359–367. Zbl 1063.11052, MR 2090662
.

Files

Files Size Format View
ArchMathRetro_042-2006-1_5.pdf 200.5Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo