Linear Complexity of Sequences on Koblitz Curves of Genus 2

Vishnupriya Anupindi

doi:10.2478/udt-2022-0010

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Linear Complexity of Sequences on Koblitz Curves of Genus 2

Uniform distribution theory

Volume 17 (2022): Issue 2 (December 2022)

By: Vishnupriya Anupindi

Open Access

|Dec 2022

Abstract

In this paper, we consider the hyperelliptic analogue of the Frobenius endomorphism generator and show that it produces sequences with large linear complexity on the Jacobian of genus 2 curves.

References

[1] ANUPINDI, V.—MÉRAI, L.: Linear complexity of some sequences derived from hyper-elliptic curves of genus 2, Cryptogr. Commun. 14 (2022), 117–134.10.1007/s12095-021-00521-y
Search in Google Scholar Back to article
[2] Handbook of Elliptic and Hyperelliptic Curve Cryptography. (H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, F. Vercauteren, eds.). Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2006.
Search in Google Scholar Back to article
[3] GALBRAITH, S. D.: Mathematics of Public Key Cryptography. Cambridge University Press, Cambridge, 2012.10.1017/CBO9781139012843
Search in Google Scholar Back to article
[4] GRANT, D.: Formal groups in genus two,J.Reine Angew. Math. 411 (1990), 96–121.
Search in Google Scholar Back to article
[5] GÜNTHER, C.—LANGE, T.—STEIN, A.: Speeding up the arithmetic on Koblitz curves of genus two.In: Selected Areas in Cryptography (Waterloo, ON, 2000),In: Lecture Notes in Comput. Sci. Vol. 2012, Springer, Berlin, 2001, pp. 106–117.
Search in Google Scholar Back to article
[6] HARTSHORNE, R.: Algebraic Geometry.In: Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.10.1007/978-1-4757-3849-0
Search in Google Scholar Back to article
[7] HINDRY, M.—SILVERMAN, J. H.: Diophantine Geometry (An introduction). In: Graduate Texts in Math. Vol. 201, Springer-Verlag, New York, 2000.
Search in Google Scholar Back to article
[8] KOBLITZ, N.: CM-curves with good cryptographic properties. In:Advances in Cryptology—CRYPTO ’91 (Santa Barbara, CA, 1991),In:Lecture Notes in Comput. Sci. Vol. 576, Springer-Verlag, Berlin, 1992. pp. 279–287.10.1007/3-540-46766-1_22
Search in Google Scholar Back to article
[9] KOBLITZ, N.: Algebraic Aspects of Cryptography. (With an appendix by Alfred J. Menezes, Yi-Hong Wu and Robert J. Zuccherato). In: Algorithms and Computation in Mathematics Vol. 3, Springer-Verlag, Berlin, 1998.10.1007/978-3-662-03642-6_1
Search in Google Scholar Back to article
[10] LACHAUD, G.—ROLLAND, R.: On the number of points of algebraic sets over finite fields, J. Pure Appl. Algebra 219 (2015), 5117–5136.10.1016/j.jpaa.2015.05.008
Search in Google Scholar Back to article
[11] LANGE, T.: Koblitz curve cryptosystems, Finite Fields Appl. 11 (2005), 200–229.10.1016/j.ffa.2004.07.001
Search in Google Scholar Back to article
[12] LANGE, T.—SHPARLINSKI, I.: Collisions in fast generation of ideal classes and points on hyperelliptic and elliptic curves, Appl. Algebra Engrg. Comm. Comput. 15 (2005), 329–337.10.1007/s00200-004-0161-9
Search in Google Scholar Back to article
[13] LANGE, T.—SHPARLINSKI, I. E.: Certain exponential sums and random walks on elliptic curves, Canad. J. Math. 57 (2005), 338–350.10.4153/CJM-2005-015-8
Search in Google Scholar Back to article
[14] LANGE, T.—SHPARLINSKI, I. E.: Distribution of some sequences of points on elliptic curves,J. Math. Cryptol. 1 (2007), 1–11.10.1515/JMC.2007.001
Search in Google Scholar Back to article
[15] MEIDL, W.—WINTERHOF, A.: Linear Complexity of Sequences and Multisequences. In: Handbook of Finite Fields. CRC Press, Boca Raton, 2013.
Search in Google Scholar Back to article
[16] MÉRAI, L.: On pseudorandom properties of certain sequences of points on elliptic curve. In: Arithmetic of Finite Fields,In: Lecture Notes in Comput. Sci. Vol. 10064, Springer, Cham, 2016. pp. 54–63.
Search in Google Scholar Back to article
[17] MÜLLER, V.: Fast multiplication on elliptic curves over small fields of characteristic two, J. Cryptology 11 (1998), 219–234.10.1007/s001459900045
Search in Google Scholar Back to article
[18] MUMFORD, D.: Tata Lectures on Theta. II. Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura, Modern Birkh¨auser Classics. Birkh¨auser Boston, Inc., Boston, MA, 2007. (Reprint of the 1984 original).
Search in Google Scholar Back to article
[19] NIEDERREITER, H.: Linear complexity and related complexity measures for sequences. In: Progress in Cryptology—INDOCRYPT 2003, Lecture Notes in Comput. Sci. Vol. 2904, Springer-Verlag, Berlin, 2003, pp. 1–17.
Search in Google Scholar Back to article
[20] SMART, N. P.: Elliptic curve cryptosystems over small fields of odd characteristic, J.Cryptology 12 (1999), 141–151.10.1007/PL00003820
Search in Google Scholar Back to article
[21] SOLINAS,J.A.: Efficient arithmetic on Koblitz curves. Towards a quarter-century of public key cryptography. vol. 19, 2000, pp. 195–249.10.1007/978-1-4757-6856-5_6
Search in Google Scholar Back to article
[22] STICHTENOTH, H.: Algebraic function fields and codes (2nd edition) In: Graduate Texts in Mathematics Vol. 254, Springer-Verlag, Berlin, 2009.10.1007/978-3-540-76878-4
Search in Google Scholar Back to article
[23] WINTERHOF, A.: Linear complexity and related complexity measures.In:Selected Topics in Information and Coding Theory, Ser. Coding Theory Crypto. Vol. 7 World Sci. Publ. Hackensack, NJ, 2010, pp. 3–40.10.1142/9789812837172_0001
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/udt-2022-0010 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X

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Language: English

Page range: 1 - 20

Submitted on: Mar 4, 2022

Accepted on: Mar 25, 2022

Published on: Dec 12, 2022

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

Related subjects:

Algebra and number theory,

Discrete mathematics,

Probability and statistics,

Numerical and computational mathematics

© 2022 Vishnupriya Anupindi, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 17 (2022): Issue 2 (December 2022)