Obfuscating Low-Rank Matrix Branching Programs

In this work, we seek to extend the capabilities of the “core obfuscator” from the work of Garg, Gentry, Halevi, Raykova, Sahai, and Waters (FOCS 2013), and all subsequent works constructing general-purpose obfuscators. This core obfuscator builds upon approximate multilinear maps, and applies to matrix branching programs. All previous works, however, limited the applicability of such core obfuscators to matrix branching programs where each matrix was of full rank. As we illustrate by example, this limitation is quite problematic, and intuitively limits the core obfuscator to obfuscating matrix branching programs that cannot “forget.” At a technical level, this limitation arises in previous work because all previous work relies on Kilian’s statistical simulation theorem, which is false when applied to matrices not of full rank. In our work, we build the first core obfuscator that can apply to matrix branching programs where matrices can be of arbitrary rank. We prove security of our obfuscator in the generic multilinear model, demonstrating a new proof technique that bypasses Kilian’s statistical simulation theorem. Furthermore, our obfuscator achieves two other notable advances over previous work: • Our construction allows for non-square matrices of arbitrary dimensions. We also show that this flexibility yields concrete efficiency gains. • Our construction allows for a single obfuscation to yield multiple bits of output. All previous work yielded only one bit of output. Our work leads to significant efficiency gains for obfuscation. Furthermore, our work can be applied to achieve efficiency gains even in applications not directly using obfuscation.