On the (Im)possibility of Obfuscating Programs

Informally, an obfuscator O is an (efficient, probabilistic) “compiler” that takes as input a program (or circuit) P and produces a new program O(P) that has the same functionality as P yet is “unintelligible” in some sense. Obfuscators, if they exist, would have a wide variety of cryptographic and complexity-theoretic applications, ranging from software protection to homomorphic encryption to complexity-theoretic analogues of Rice's theorem. Most of these applications are based on an interpretation of the “unintelligibility” condition in obfuscation as meaning that O(P) is a “virtual black box,” in the sense that anything one can efficiently compute given O(P), one could also efficiently compute given oracle access to P. In this work, we initiate a theoretical investigation of obfuscation. Our main result is that, even under very weak formalizations of the above intuition, obfuscation is impossible. We prove this by constructing a family of efficient programs P that are unobfuscatable in the sense that (a) given any efficient program P' that computes the same function as a program P ∈ p, the “source code” P can be efficiently reconstructed, yet (b) given oracle access to a (randomly selected) program P ∈ p, no efficient algorithm can reconstruct P (or even distinguish a certain bit in the code from random) except with negligible probability. We extend our impossibility result in a number of ways, including even obfuscators that (a) are not necessarily computable in polynomial time, (b) only approximately preserve the functionality, and (c) only need to work for very restricted models of computation (TC0). We also rule out several potential applications of obfuscators, by constructing “unobfuscatable” signature schemes, encryption schemes, and pseudorandom function families.Engineering and Applied Science

Obfuscation for Cryptographic Purposes

An obfuscation of a function F should satisfy two requirements: firstly, using it should be possible to evaluate F; secondly, should not reveal anything about F that cannot be learnt from oracle access to F. Several definitions for obfuscation exist. However, most of them are either too weak for or incompatible with cryptographic applications, or have been shown impossible to achieve, or both. We give a new definition of obfuscation and argue for its reasonability and usefulness. In particular, we show that it is strong enough for cryptographic applications, yet we show that it has the potential for interesting positive results. We illustrat

Impossibility of Quantum Virtual Black-Box Obfuscation of Classical Circuits

Virtual black-box obfuscation is a strong cryptographic primitive: it encrypts a circuit while maintaining its full input/output functionality. A remarkable result by Barak et al. (Crypto 2001) shows that a general obfuscator that obfuscates classical circuits into classical circuits cannot exist. A promising direction that circumvents this impossibility result is to obfuscate classical circuits into quantum states, which would potentially be better capable of hiding information about the obfuscated circuit. We show that, under the assumption that learning-with-errors (LWE) is hard for quantum computers, this quantum variant of virtual black-box obfuscation of classical circuits is generally impossible. On the way, we show that under the presence of dependent classical auxiliary input, even the small class of classical point functions cannot be quantum virtual black-box obfuscated.Comment: v2: Add the notion of decomposable public keys, which allows our impossibility to hold without assuming circular security for QFHE. We also fix an auxiliary lemma (2.9 in v2) where a square root was missing (this does not influence the main result

On Statistically Secure Obfuscation with Approximate Correctness

Goldwasser and Rothblum (TCC \u2707) prove that statistical indistinguishability obfuscation (iO) cannot exist if the obfuscator must maintain perfect correctness (under a widely believed complexity theoretic assumption: $\mathcal{NP} \not\subseteq \mathcal{SZK}\subseteq\mathcal{AM}\cap\mathbf{co}\mathcal{AM}$). However, for many applications of iO, such as constructing public-key encryption from one-way functions (one of the main open problems in theoretical cryptography), approximate correctness is sufficient. It had been unknown thus far whether statistical approximate iO (saiO) can exist. We show that saiO does not exist, even for a minimal correctness requirement, if $\mathcal{NP} \not\subseteq \mathcal{AM}\cap\mathbf{co}\mathcal{AM}$, and if one-way functions exist. A simple complementary observation shows that if one-way functions do not exist, then average-case saiO exists. Technically, previous approaches utilized the behavior of the obfuscator on evasive functions, for which saiO always exists. We overcome this barrier by using a PRF as a baseline for the obfuscated program. We broaden our study and consider relaxed notions of security for iO. We introduce the notion of correlation obfuscation, where the obfuscations of equivalent circuits only need to be mildly correlated (rather than statistically indistinguishable). Perhaps surprisingly, we show that correlation obfuscators exist via a trivial construction for some parameter regimes, whereas our impossibility result extends to other regimes. Interestingly, within the gap between the parameters regimes that we show possible and impossible, there is a small fraction of parameters that still allow to build public-key encryption from one-way functions and thus deserve further investigation

Deterministic, Efficient Variation of Circuit Components to Improve Resistance to Reverse Engineering

This research proposes two alternative methods for generating semantically equivalent circuit variants which leave the circuit\u27s internal structure pseudo-randomly determined. Component fusion deterministically selects subcircuits using a component identification algorithm and replaces them using a deterministic algorithm that generates canonical logic forms. Component encryption seeks to alter the semantics of individual circuit components using an encoding function, but preserves the overall circuit semantics by decoding signal values later in the circuit. Experiments were conducted to examine the performance of component fusion and component encryption against representative trials of subcircuit selection-and-replacement and Boundary Blurring, two previously defined methods for circuit obfuscation. Overall, results support the conclusion that both component fusion and component encryption generate more secure variants than previous methods and that these variants are more efficient in terms of required circuit delay and the power and area required for their implementation

Obfuscation Framework Based on Functionally Equivalent Combinatorial Logic Families

This thesis aims to be a few building blocks in the bridge between theoretical and practical software obfuscation that researchers will one day construct. We provide a method for random uniform selection of circuits based on a functional signature and specific construction specifiers. Additionally, this thesis includes the first formal definition of an algorithm that performs only static analysis on a program; that is analysis that does not rely on the input and output behavior of the analyzed program. This is analogous to some techniques used in real-world software reverse engineering. Finally, this thesis uses the equivalent circuit library to empirically produce some statistical data about enumerated circuit families and explains how this data may be useful to future researchers

On Strong Simulation and Composable Point Obfuscation

The Virtual Black Box (VBB) property for program obfuscators provides a strong guarantee: Anything computable by an efficient adversary given the obfuscated program can also be computed by an efficient simulator with only oracle access to the program. However, we know how to achieve this notion only for very restricted classes of programs. This work studies a simple relaxation of VBB: Allow the simulator unbounded computation time, while still allowing only polynomially many queries to the oracle. We then demonstrate the viability of this relaxed notion, which we call Virtual Grey Box (VGB), in the context of fully composable obfuscators for point programs: It is known that, w.r.t. VBB, if such obfuscators exist then there exist multi-bit point obfuscators (aka ``digital lockers\u27\u27) and subsequently also very strong variants of encryption that are resilient to various attacks, such as key leakage and key-dependent-messages. However, no composable VBB-obfuscators for point programs have been shown. We show fully composable {\em VGB}-obfuscators for point programs under a strong variant of the Decision Diffie Hellman assumption. We show they suffice for the above applications and even for extensions to the public key setting as well as for encryption schemes with resistance to certain related key attacks (RKA)

Optimizing Cryptographic Obfuscation

Cryptographic obfuscation is a powerful tool that makes programs “unintelligible” yet still runnable. It essentially gives programs the ability to keep secrets. The practical applications of obfuscation range from keeping secrets in banking applications to preventing software theft to providing secure messaging applications. The cryptographic applications of obfuscation are also vast – a tool that hides secrets in programs essentially enables all other cryptographic constructions. Despite (or perhaps due to) its power, obfuscation is currently wildly inefficient and on shaky theoretical ground. Its shaky theoretical ground in particular has resulted in a lack of engineering effort at making it more efficient. In this work, we focus largely on efficiency. We explore the concrete efficiency of multilinear maps, which are the basis of many cryptographic obfuscation constructions. Multilinear maps are mathematical objects that allow oblivious addition and multiplication of encrypted values. Using multilinear maps, we give the first ever implementations of obfuscation and multi-input functional encryption (MIFE: a variant of obfuscation) for branching programs. Along the way, we create the 5Gen framework for implementations of multilinear map-based applications. We apply the 5Gen framework to experiment with obfuscating point functions and MIFE of order-revealing encryption. We also explore efficiency in the context of obfuscators and MIFE for circuits. Circuits are more efficient than branching programs for many functions. We give the first MIFE construction for circuits and prove its security in an ideal model. Our scheme is efficient. To compare, we implement all known circuit obfuscation schemes using the 5Gen framework, and experiment with obfuscating a PRF. This results in the most complex PRF obfuscated to date – with 12 bits of security. Finally, recently Bishop et al. showed an obfuscation scheme for the specific functionality of wildcard pattern-matching [BKM+18]. This is a simple type of string matching where strings must match a pattern exactly except where there are wildcards. This obfuscation scheme simply relies on the generic group model, with no multilinear maps. Inspired by their work, and the deep connection of functional encryption to obfuscation, we give a function-private, public-key functional encryption scheme for the same wildcard pattern-matching functionality. Our scheme is the first such scheme and we prove its security in a generic model