Binary permutation polynomial inversion and application to obfuscation techniques

Whether it is for constant obfusation, opaque predicate or equation obfuscation, Mixed Boolean-Arithmetic (MBA) expressions are a powerful tool providing concrete ways to achieve obfuscation. Recent papers [22, 1] presented ways to mix such a tool with permutation polynomials modulo 2n in order to make the obfuscation technique more resilient to SMT solvers. However, because of limitations regarding the inversion of such permutations, the set of permutation polynomials presented suffers some restrictions. Those restrictions allow several methods of arithmetic simplification, decreasing the effectiveness of the technique at hiding information. In this work, we present general methods for permutation polynomials inversion. These methods allow us to remove some of the restrictions presented in the literature, making simplification attacks less effective. We discuss complexity and limits of these methods, and conclude that not only current simplification attacks may not be as effective as we thought, but they are still many uses of polynomial permutations in obfuscation that are yet to be explored.