Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Cryptography and Communications
Erschienen in: Cryptography and Communications 3/2019

18.03.2019

On classical Kloosterman sums

verfasst von: V. A. Zinoviev

Erschienen in: Cryptography and Communications | Ausgabe 3/2019

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

It is a survey on the Kloosterman sums. We consider the known results on the classical Kloosterman sums, i.e. the Kloosterman sums over the finite fields \(\mathbb {F}_{q}\) of order q = pm, p ≥ 2 is prime.
Vorheriger Artikel Factorization using binary decision diagrams
Nächster Artikel Computing the number of finite field elements with prescribed absolute trace and co-trace

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Literatur
1.
2.
Bassalygo, L.A., Zinoviev, V.A.: On divisibility of exponential sums of polynomials of special type over finite fields of characteristic 2. In: Augot, D., Canteaut, A. (eds.) Seventh International Workshop on Coding and Cryptography, WCC-2011, April 11-15, 2011, Paris, France, Proceedings, INRIA, pp 389–396 (2011)
3.
Bassalygo, L.A., Zinoviev, V.A.: On divisibility of exponential sums of polynomials of special type over finite fields of characteristic 2. Des. Codes Crypt. 66(1-3), 129–143 (2013)CrossRefMATH
4.
Bassalygo, L.A., Zinoviev, V.A.: On Kloosterman sums over finite fields of characteristic 3. In: Proceedings of Thirteenth International Workshop. Algebraic and Combinatorial Coding Theory, ACCT - 13 (June 15 - 21, Pomorie, Bulgaria), pp. 83–87 (2012)
5.
Bassalygo, L.A., Zinoviev, V.A.: On Kloosterman sums over finite fields of characteristic 3. Discret. Appl. Math. 216(part 3), 518–523 (2017)MathSciNetCrossRefMATH
6.
7.
Borisov, L., Borisov, Y.: On some properties of the Kloosterman sums over \(GF(p)\), The International Workshop on Algebraic and Combinatorial Coding Theory, ACCT - 2018, Svetlogorsk, Russia (2018)
8.
Cao, X., Hollmann, H.D.L., Xiang, Q.: New Kloosterman sum identities and equalities over finite fields. Finite Fields Appl. 14, 823–833 (2008)MathSciNetCrossRefMATH
9.
10.
11.
12.
13.
14.
15.
Charpin, P., Helleseth, T., Zinoviev, V.A.: On cosets of weight 4 of binary BCH codes of length \(2^{m}\) (m odd) with minimal distance 8 and exponential sums. Probl. Inf. Transm. 41(4), 301–320 (2005)MathSciNetCrossRefMATH
16.
Charpin, P., Helleseth, T., Zinoviev, V.A.: Propagation characteristics of \(x^{-1}\). Finite Fields and Their Applications 13, 366–381 (2007)MathSciNetCrossRefMATH
17.
Charpin, P., Helleseth, T., Zinoviev, V.A.: The divisibility modulo 24 of Kloosterman sums on \(\mathbb {F}_{2^{m}}\), m odd. Journal of Combinatorial Theory Ser. A 114(2), 322–338 (2007)MathSciNetCrossRefMATH
18.
Charpin, P., Helleseth, T., Zinoviev, V.A.: On cosets of weight 4 of binary primitive BCH codes of length \(2^{m}\) (m even) with minimum distance 8 and exponential sums. SIAM J. Discret. Math. 23(1), 59–78 (2008)CrossRefMATH
19.
Charpin, P., Helleseth, T., Zinoviev, V.A.: On divisibility properties of classical binary Kloosterman sums. Discret. Math. 309(12), 3975–3984 (2009)MathSciNetCrossRefMATH
20.
Charpin, P., Gong, G.: Hyperbent functions, Kloosterman sums, and Dickson polynomials. IEEE Trans. Inf. Theory 54(9), 4230–4238 (2008)MathSciNetCrossRefMATH
21.
Charpin, P., Tietavainen, A., Zinoviev, V.A.: On binary cyclic codes with distance 3. Probl. Inf. Transm. 33(4), 287–296 (1997)MathSciNetMATH
22.
Chinen, K., Toyokazu, H.: Hyper-kloosterman sums and their applications to the coding theory. Appl. Algebra Eng. Commun. Comput. 12, 381–390 (2001)MathSciNetCrossRefMATH
23.
24.
25.
26.
Deligne, P.: Applications de la formule des traces aux sommes trigonometriques. Lecture Notes in Math., Springer 569, 168–232 (1977)MathSciNetCrossRefMATH
27.
Dillon, J.F.: Elementary Hadamard Difference Sets. Ph. D. thesis, Univ. Maryland, College Park (1974)MATH
28.
Dillon, J.F.: Elementary Hadamard difference sets. In: Proc. Sixth Southeastern Conference on Combinatorics, Graph theory, and Computing. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man. pp. 237–249 (1975)
29.
Ding, C., Helleseth, T., Niederreiter, H., Xing, C.: The minimum distance of the duals of binary irreducible cyclic codes. IEEE Trans. Inform. Theory 48, 2679–2689 (2002)MathSciNetCrossRefMATH
30.
Dobbertin, H., Felke, P., Helleseth, T., Rosendalh, P.: Niho type crosscorrelation functions via Dickson polynomials and Kloosterman sums. IEEE Trans. Inf. Theory 52(2), 613–627 (2006)CrossRefMATH
31.
Enge, A.: Elliptic Curves and Their Applications to Cryptography: an Introduction. Klumer Academic Publishers, Boston (1999)CrossRefMATH
32.
33.
34.
35.
Flori, J.-P., Mesnager, S., Cohen, G.: Binary Kloosterman sums with value 4. In: IMA Int. Conf. 2011, pp. 61–78
36.
37.
38.
Gauss, C.F.: Disquisitiones Arithmeticae. Fleischer, 1801; English Translation, Yale University Press, Leipzig (1966)MATH
39.
40.
Göloğlu, F., McGuire, G., Moloney, R.: Ternary Kloosterman sums modulo 18 using Stickelberger’s theorem. Acta Arith. 148(3), 269–279 (2011)MathSciNetCrossRefMATH
41.
Göloğlu, F., McGuire, G., Moloney, R.: Binary Kloosterman sums using Stickelberger’s theorem and the Gross-Koblitz formula. Acta Arith. 148(3), 269–279 (2011)MathSciNetCrossRefMATH
42.
Göloğlu, F., McGuire, G., Moloney, R.: Some results on Kloosterman sums and their minimal polynomials. In: Augot, D., Canteaut, A. (eds.) Seventh International Workshop on Coding and Cryptography, WCC 2011”, April 11-15, 2011, Paris, France, Proceedings, INRIA, pp 403–412 (2011)
43.
Göloğlu, F., McGuire, G., Moloney, R.: Some congruences on Kloosterman sums and their characteristic polynomials. J. Number Theory 143, 1596–1607 (2013). arXiv:1006.​1802v1 [math.NT] 9 Jun 2010MathSciNetCrossRefMATH
44.
Göloğlu, F., Lisonĕk, P., McGuire, G., Moloney, R.: Binary Kloosterman sums modulo 256 and coefficients of the characteristic polynomial. IEEE Trans. Inf. Theory 58(4), 2516–2523 (2012)MathSciNetCrossRefMATH
45.
Van der Geer, G., Van Der Vlugt, M.: Kloosterman sums and the p-torsion of certain Jacobians. Math. Ann. 290(3), 549–563 (1991)MathSciNetCrossRefMATH
46.
47.
Helleseth, T.: Some results about the crosscorrelation function between two maximal linear sequences. Discrete Math. 16, 209–232 (1976)MathSciNetCrossRefMATH
48.
Helleseth, T., Zinoviev, V.A.: On \(Z_{4}\)-Linear goethals codes and Kloosterman sums. Des. Codes Crypt. 17(1-3), 246–262 (1999)MATH
49.
Helleseth, T., Zinoviev, V.A.: New Kloosterman Sums identities over \(\mathbb {F}_{2^{m}}\) for all m. Finite Fields Appl. 9, 187–193 (2003)MathSciNetCrossRefMATH
50.
Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inform. Theory 52(5), 2018–2032 (2006)MathSciNetCrossRefMATH
51.
Hollmann, H.D.L., Xiang, Q.: Kloosterman sum identities over \(\mathbb {F}_{2^{m}}\). Discrete Math. 279(1-3), 277–286 (2004)MathSciNetCrossRefMATH
52.
53.
Katz, N.M., Livné, R.: Sommes de Kloosterman et courbes elliptiques universelles caracteristiques 2 and 3. C. R. Acad. Sci. Paris Ser. I Math. 309(11), 723–726 (1989)MathSciNetMATH
54.
55.
Kloosterman, H.D.: On the representation of numbers in the form \(ax^{2} + by^{2} + cz^{2} + dt^{2}\). Acta Math. 49, 407–464 (1926)CrossRef
56.
Kojo, M.R.S.: Modular curves and identities of classical Kloosterman sums. unpublished manuscript (2002)
57.
Kononen, K.P., Rinta-aho, M.J., Väänänen, K.O.: On integer values of Kloosterman sums. IEEE Trans. Inform. Theory 56(8), 4011–4013 (2010)MathSciNetCrossRefMATH
58.
Kononen, K.P., Rinta-aho, M.J., Väänänen, K.O.: On the degree of a Kloosterman sum as an algebraic integer, arXiv:1107.​0188v1 [math.NT] (2011)
59.
Kumar, P.V., Scholtz, R.A., Welch, L.R.: Generalized bent functions and their properties. Journal Combinatorial Theory, Ser. A 40(1), 90–107 (1985)MathSciNetCrossRefMATH
60.
Lachaud, G., Wolfmann, J.: Sommes de Kloosterman, courbes elliptiques et codes cycliques en caractéristique 2. C. R. Acad. Sci. Paris (I) 305, 881–883 (1987)MathSciNetMATH
61.
62.
Lachaud, G., Wolfmann, J.: The weights of the orthogonals of the extended quadratic binary goppa codes. IEEE Trans. Inform. Theory 36(3), 686–692 (1990)MathSciNetCrossRefMATH
63.
Lahtonen, J., Ojala, L.: Seminar notes University of Helsinki (2001)
64.
65.
66.
D.H., Lehmer, E.: The cyclotomy of Kloosterman sums. Acta Arithmetica XII (4), 385–407 (1967)MathSciNetMATH
67.
D.H., Lehmer, E.: The cyclotomy of hyper-Kloosterman sums. Acta Arithmetica XIV(4), 89–111 (1968)MathSciNetMATH
68.
Leonard, P.A., Williams, K.S.: Quartics over \({{GF}}(2^{m})\). Proc. Amer. Math. Soc. 36, 347–350 (1972)MathSciNetMATH
69.
Lidl, R., Niederreiter, H.: Finite Fields, Encyclopedia of Mathematics and Its Applications, vol. 20. Addison Wesley, Reading (1983)MATH
70.
Lisonĕk, ek, P.: On the connection between Kloosterman sums and elliptic curves. In: Golomb, S. et al. (eds.) Proceedings of the 5th International Conference on Sequences and Their Applications (SETA 2008), Lecture Notes in Computer Science, vol. 5203, pp. 182–187. Springer (2008)
71.
72.
73.
Maier, R.S.: On rationally parametrized modular equations. J. Ramanujan Math. Soc. 24(1), 1–73 (2009)MathSciNetMATH
74.
Menezes, A.: Elliptic Curve Public Key Cryptosystems. Kluwer Academic Publishers, Boston (1993)CrossRefMATH
75.
Mesnager, S.: A new family of hyper-bent boolean functions in polynomial form. In: Parker, M.G. (ed.) IMA Int. Conf., Lecture Notes in Computer Science, vol. 5921, pp 402–417. Springer, Berlin (2009)
76.
Mesnager, S.: A new class of bent boolean functions in polynomial forms. In: Proceedings of International Workshop on Coding and Cryptography, WCC-2009, pp. 5–18 (2009)
77.
Mesnager, S.: A new class of bent and hyperbent Boolean functions in polynomial forms. Des. Codes Crypt. 59(1-3), 265–279 (2011)CrossRefMATH
78.
Mesnager, S.: Semi-bent functions from Dillon and Niho exponents, Kloosterman sums and Dickson polynomials. IEEE Trans. Inform. Theory 57(11), 7443–7458 (2011)MathSciNetCrossRefMATH
79.
Moisio, M.: On the number of rational points on some families of Fermat curves over finite fields. Finite Fields Appl. 13(3), 546–562 (2007)MathSciNetCrossRefMATH
80.
Moisio, M.: The moments of a Kloosterman sum and the weight distribution of a Zetterberg type binary cyclic code. IEEE Trans. Inform. Theory 53(2), 843–847 (2007)MathSciNetCrossRefMATH
81.
82.
Moisio, M.: The divisibility modulo 24 of Kloosterman sums on \({{GF}}(2^{m})\), m even. Finite Fields Appl. 15, 174–184 (2009)MathSciNetCrossRefMATH
83.
84.
Moisio, M.: Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm. Acta Arithmetica 132(4), 329–350 (2008)MathSciNetCrossRefMATH
85.
Moisio, M., Ranto, K.: Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros. Finite Fields Appl. 13(4), 922–935 (2007)MathSciNetCrossRefMATH
86.
Mordell, L.J.: On a special polynomial congruence and exponential sums, Calcutta Math. Soc., Golden Jubilee Commemoration Volume, Part. 1, pp. 29–32 (1963)
87.
Yasuo Morita, A: P-adic analogue of the Γ-function. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22(2), 255–266 (1975)MathSciNetMATH
88.
89.
Remijn, J.C.C.M., Tiersma, H.J.: A duality theorem for the weight distribution of some cyclic codes. IEEE Trans. Inform. Theory IT-34(4, pt. II), 1348–1351 (1988)MathSciNetCrossRefMATH
90.
Robert, A.M.: The Gross-Koblitz formula revised. Rend. Sem. Mat. Univ. Padova 105, 157–170 (2001)MathSciNet
91.
92.
93.
Shin, D.-J., Vijay Kumar, P., Helleseth, T.: 3-Designs from the \(z_{4}\)-goethals codes via a new Kloosterman sum identity. Des. Codes Crypt. 28(3), 247–263 (2003)CrossRefMATH
94.
Shin, D.-J., Sung, W.: A new Kloosterman sum identity over \(\mathbb {F}_{2^{m}}\) for odd m. Discrete Math. 268, 337–341 (2003)MathSciNetCrossRefMATH
95.
96.
97.
Stichtenoth, H.: Algebraic Function Fields and Codes. Springer, Berlin (1993)MATH
98.
99.
100.
101.
102.
103.
104.
105.
Wolfmann, J.: Codes projectivs a deux ou trois poids associes aux hyperquadriques d’une geometrie finie. Discrete Math. 13, 185–211 (1975)MathSciNetCrossRefMATH
Metadaten
Titel
On classical Kloosterman sums
verfasst von
V. A. Zinoviev
Publikationsdatum
18.03.2019
Verlag
Springer US
Erschienen in
Cryptography and Communications / Ausgabe 3/2019
Print ISSN: 1936-2447
Elektronische ISSN: 1936-2455
DOI
https://doi.org/10.1007/s12095-019-00357-7

Weitere Artikel der Ausgabe 3/2019

Cryptography and Communications 3/2019 Zur Ausgabe

OriginalPaper

Computing the number of finite field elements with prescribed absolute trace and co-trace

OriginalPaper

Affine linear cryptanalysis

OriginalPaper

Secret sharing on large girth graphs

OriginalPaper

Decomposition of permutations in a finite field

OriginalPaper

On the security of biquadratic C∗ public-key cryptosystems and its generalizations

OriginalPaper

Constructions of S-boxes with uniform sharing