Matrix analysis
Convergent bounds for the range of multivariate polynomials
Proceedings of the International Symposium on interval mathematics on Interval mathematics 1985
Algorithms for polynomials in Bernstein form
Computer Aided Geometric Design
QuickCheck: a lightweight tool for random testing of Haskell programs
ICFP '00 Proceedings of the fifth ACM SIGPLAN international conference on Functional programming
A Fortran Multiple-Precision Arithmetic Package
ACM Transactions on Mathematical Software (TOMS)
On the (Im)possibility of Obfuscating Programs
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
Slicing obfuscations: design, correctness, and evaluation
Proceedings of the 2007 ACM workshop on Digital Rights Management
Data Refinement: Model-Oriented Proof Methods and their Comparison
Data Refinement: Model-Oriented Proof Methods and their Comparison
Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms
Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms
Specifying imperative data obfuscations
ISC'07 Proceedings of the 10th international conference on Information Security
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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.