Obfuscating Simple Functionalities from Knowledge Assumptions

Abstract

This paper shows how to obfuscate several simple functionalities from a new Knowledge of OrthogonALity Assumption (KOALA) in cyclic groups which is shown to hold in the Generic Group Model. Specifically, we give simpler and stronger security proofs for obfuscation schemes for point functions, general-output point functions and pattern matching with wildcards. We also revisit the work of Bishop et al. (CRYPTO 2018) on obfuscating the pattern matching with wildcards functionality. We improve upon the construction and the analysis in several ways:

• attacks and stronger guarantees: We show that the construction achieves virtual black-box security for a simulator that runs in time roughly $2^{n/2}$, as well as distributional security for larger classes of distributions. We give attacks that show that our results are tight.

• weaker assumptions: We prove security under KOALA.

• better efficiency: We also provide a construction that outputs $n+1$ instead of 2n group elements.

We obtain our results by first obfuscating a simpler “big subset functionality”, for which we establish full virtual black-box security; this yields a simpler and more modular analysis for pattern matching. Finally, we extend our distinguishing attacks to a large class of simple linear-in-the-exponent schemes.