Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Berchtold, J.: The Bernstein basis in set-theoretic geometric modelling. PhD thesis, University of Bath (2000)
Bernstein, S.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Comm. Kharkov Math. Soc. 13(1-2), 49–194 (1912)
Brent, R.P.: A FORTRAN multiple–precision arithmetic package. ACM Transactions on Mathematical Software 4(1), 57–70 (1978)
Claessen, K., Hughes, J.: QuickCheck: a lightweight tool for random testing of Haskell programs. ACM SIGPLAN Notices (2000)
Collberg, C., Thomborson, C., Low, D.: A taxonomy of obfuscating transformations. Technical Report 148, Department of Computer Science, University of Auckland (July 1997)
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)
Drape, S.: Obfuscation of Abstract Data-Types. DPhil thesis, Oxford University Computing Laboratory (2004)
Drape, S.: Generalising the array split obfuscation. Information Sciences 177(1), 202–219 (2007)
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)
Farouki, R.T., Rajan, V.T.: Algorithms for polynomials in Bernstein form. Computer Aided Geometric Design 5, 1–26 (1988)
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)
Geisow, A.: Surface Interrogations. PhD thesis, University of East Anglia (1983)
Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms. Springer, Heidelberg (2009)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
LiDIA Group, Darmstadt University of Technology, www.cdc.informatik.tu-darmstadt.de/TI/LiDIA
Lorentz, G.G.: Bernstein Polynomials. Chelsea Publishing Company, New York (1986)
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)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Drape, S., Voiculescu, I. (2009). Creating Transformations for Matrix Obfuscation. In: Palsberg, J., Su, Z. (eds) Static Analysis. SAS 2009. Lecture Notes in Computer Science, vol 5673. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03237-0_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-03237-0_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03236-3
Online ISBN: 978-3-642-03237-0
eBook Packages: Computer ScienceComputer Science (R0)