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Orthogonal matrix and its application in Bloom’s threshold scheme

  • Ahmed Mameri
  • Amar Aissani
Original Paper

Abstract

Applying the Gram–Schmidt process (also called Gram–Schmidt orthogonalization) to a matrix \(M\in GL(n, {\mathbb {R}})\), set of \(n\times n\) invertible matrices over the field of real numbers, with the usual inner product gives easily an orthogonal matrix. However, the orthogonality in the vector space \({\mathbb {F}}_{q}^k\), where \({\mathbb {F}}_{q}\) is a binary finite field, is quite tricky as there are non-zero vectors which are orthogonal to themselves. For this reason the computational variants of Gram–Schmidt orthogonalization can fail. This paper presents an algorithm for constructing random orthogonal matrices over binary finite fields. The approach is inspired from the Gram–Schmidt procedure. Since the inverse of orthogonal matrix is easy to compute, the orthogonal matrices are used to construct a proactive variant of Bloom’s threshold secret sharing scheme.

Keywords

Matrix Orthogonalization Secret sharing Cryptography 

Mathematics Subject Classification

MSC 65F25 MSC 15A33 MSC 15A09 MSC 94A62 MSC 94A60 

References

  1. 1.
    Arfken, G.: Gram Schmidt orthogonalization. In: Mathematical Methods for Physicists, 3rd edn, pp. 516–520. Academic Press, Orlando (1985)Google Scholar
  2. 2.
    Bjorck, A.: Numeric’s of Gram–Schmidt orthogonalization. J. Linear Algebra Appl. 187–198, 297–316 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bjorck, A., Pereyra, V.: Solution of Vandermonde systems of linear equations. Math. Comput. 24, 893–903 (1970)CrossRefzbMATHGoogle Scholar
  4. 4.
    Blakley, G.: Safeguarding cryptographic keys. In: Proceedings of the National Computer Conference, vol. 48, pp. 242–268 (1979)Google Scholar
  5. 5.
    Dickson, L.F.: Linear Groups with an Exposition of the Galois Field Theory. B. G. Teubner, Leipzig (1901)zbMATHGoogle Scholar
  6. 6.
    Eisinberg, A., Fedel, G.: On the inversion of the Vanermonde matrix. Appl. Math. Comput. 174, 1384–1397 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Golub, G., Vanloan, C.: Matrix Computations, 3rd edn. John Hopkins Univ. Press, Baltimore (1996)Google Scholar
  8. 8.
    Haupt, J., Bajwa, W.U., Raz, G., Nowak, R.: Toeplitz compressed sensing matrices with applications to sparse channel estimation. IEEE Trans. Inf. Theory 56(11), 5862–5875 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Herzberg, A., Jarecki, S., Krawczyk, H., Krawczyk, M.: Proactive secret sharing or: how to cope with perpetual leakage. In: Coppersmith D (Eds.) Advances in Cryptology—Crypto ’95, August, Santa Barbara, pp. 339–352 (1995)Google Scholar
  10. 10.
    Iris, A., Michael, A., Dorian, G.: A linear time matrix key agreement protocol over Small Finite Fields. Appl. Algebra Eng. Commun. Comput. 17(3), 195–203 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Iuon-Chang, L., Chin-Chen, C.: A (t, n) threshpld secret sharing system with efficient identification of cheaters. Comput. Inf. 24, 529–541 (2005)zbMATHGoogle Scholar
  12. 12.
    Kaufman, I.: The inversion of the Vandermonde matrix and the transformation to the Jordan canonical form. IEEE Trans. Autom. control 14, 774–777 (1969)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kothari, S.C.: Generalized linear threshold scheme. In: Blakley, G.R., Chaum, D. (eds.) Advances in Cryptology, CRYPTO 1984. Lecture Notes in Computer Science, vol. 196, pp. 231–241. Springer, Heidelberg, Berlin (1985)Google Scholar
  14. 14.
    Mac William, J.: Orthogonal matrices over finite fields. Am. Math. Mon. 76(2), 152–164 (1969)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ramakrishna, A.V., Prasanna, T.V.N.: Symmetric circulant matrices and publickey cryptography. Int. J. Contemp. Math. Sci. 8(12), 589–593 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shamir, A.: How to share a secret. Commun. ACM 24(11), 612–613 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Toorani, M., Falahati, A.: A secure variant of the Hill cipher. In: IEEE Symposium on Computers and Communications 2009, pp. 313–316 (2009)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Algebra and Numbers TheoryUSTHBBab Ezzouar, AlgiersAlgeria
  2. 2.Departement of Computer ScienceUSTHBBab Ezzouar, AlgiersAlgeria

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