Creating Transformations for Matrix Obfuscation

  • Stephen Drape
  • Irina Voiculescu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5673)


There are many programming situations where it would be convenient to conceal the meaning of code, or the meaning of certain variables. This can be achieved through program transformations which are grouped under the term obfuscation. Obfuscation is one of a number of techniques that can be employed to protect sensitive areas of code. This paper presents obfuscation methods for the purpose of concealing the meaning of matrices by changing the pattern of the elements.

We give two separate methods: one which, through splitting a matrix, changes its size and shape, and one which, through a change of basis in a ring of polynomials, changes the values of the matrix and any patterns formed by these. Furthermore, the paper illustrates how matrices can be used in order to obfuscate a scalar value. This is an improvement on previous methods for matrix obfuscation because we will provide a range of techniques which can be used in concert.

This paper considers obfuscations as data refinements. Thus we consider obfuscations at a more abstract level without worrying about implementation issues. For our obfuscations, we can construct proofs of correctness easily. We show how the refinement approach enables us to generalise and combine existing obfuscations. We then evaluate our methods by considering how our obfuscations perform under certain relevant program analysis-based attacks.


Obfuscation Matrix Operations Information Hiding Program Transformations 


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  1. 1.
    Barak, B., Goldreich, O., Impagliazzo, R., Rudich, S., Sahai, A., Vadhan, S.P., Yang, K.: On the (im)possibility of obfuscating programs. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 1–18. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Berchtold, J.: The Bernstein basis in set-theoretic geometric modelling. PhD thesis, University of Bath (2000)Google Scholar
  3. 3.
    Bernstein, S.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Comm. Kharkov Math. Soc. 13(1-2), 49–194 (1912)zbMATHGoogle Scholar
  4. 4.
    Brent, R.P.: A FORTRAN multiple–precision arithmetic package. ACM Transactions on Mathematical Software 4(1), 57–70 (1978)CrossRefGoogle Scholar
  5. 5.
    Claessen, K., Hughes, J.: QuickCheck: a lightweight tool for random testing of Haskell programs. ACM SIGPLAN Notices (2000)Google Scholar
  6. 6.
    Collberg, C., Thomborson, C., Low, D.: A taxonomy of obfuscating transformations. Technical Report 148, Department of Computer Science, University of Auckland (July 1997)Google Scholar
  7. 7.
    de Roever, W.-P., Engelhardt, K.: Data Refinement: Model-Oriented Proof Methods and their Comparison. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  8. 8.
    Drape, S.: Obfuscation of Abstract Data-Types. DPhil thesis, Oxford University Computing Laboratory (2004)Google Scholar
  9. 9.
    Drape, S.: Generalising the array split obfuscation. Information Sciences 177(1), 202–219 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Drape, S., Thomborson, C., Majumdar, A.: Specifying imperative data obfuscations. In: Garay, J.A., et al. (eds.) ISC 2007. LNCS, vol. 4779, pp. 299–314. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Farouki, R.T., Rajan, V.T.: Algorithms for polynomials in Bernstein form. Computer Aided Geometric Design 5, 1–26 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Garloff, J.: Convergent bounds for the range of multivariate polynomials. In: Nickel, K. (ed.) Interval Mathematics 1985. LNCS, vol. 212, pp. 37–56. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  13. 13.
    Geisow, A.: Surface Interrogations. PhD thesis, University of East Anglia (1983)Google Scholar
  14. 14.
    Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms. Springer, Heidelberg (2009)CrossRefzbMATHGoogle Scholar
  15. 15.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefzbMATHGoogle Scholar
  16. 16.
    LiDIA Group, Darmstadt University of Technology,
  17. 17.
    Lorentz, G.G.: Bernstein Polynomials. Chelsea Publishing Company, New York (1986)zbMATHGoogle Scholar
  18. 18.
    Majumdar, A., Drape, S.J., Thomborson, C.D.: Slicing obfuscations: design, correctness, and evaluation. In: DRM 2007: Proceedings of the 2007 ACM workshop on Digital Rights Management, pp. 70–81. ACM, New York (2007)CrossRefGoogle Scholar
  19. 19.
    Zettler, M., Garloff, J.: Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion. IEEE Transactions on Automatic Control 43(3), 425–431 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Stephen Drape
    • 1
  • Irina Voiculescu
    • 1
  1. 1.Computing LaboratoryOxford UniversityOxfordUK

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