PPP-Completeness with Connections to Cryptography

Polynomial Pigeonhole Principle (PPP) is an important subclass of TFNP with profound connections to the complexity of the fundamental cryptographic primitives: collision-resistant hash functions and one-way permutations. In contrast to most of the other subclasses of TFNP, no complete problem is known for PPP. Our work identifies the first PPP-complete problem without any circuit or Turing Machine given explicitly in the input, and thus we answer a longstanding open question from [Papadimitriou1994]. Specifically, we show that constrained-SIS (cSIS), a generalized version of the well-known Short Integer Solution problem (SIS) from lattice-based cryptography, is PPP-complete. In order to give intuition behind our reduction for constrained-SIS, we identify another PPP-complete problem with a circuit in the input but closely related to lattice problems. We call this problem BLICHFELDT and it is the computational problem associated with Blichfeldt's fundamental theorem in the theory of lattices. Building on the inherent connection of PPP with collision-resistant hash functions, we use our completeness result to construct the first natural hash function family that captures the hardness of all collision-resistant hash functions in a worst-case sense, i.e. it is natural and universal in the worst-case. The close resemblance of our hash function family with SIS, leads us to the first candidate collision-resistant hash function that is both natural and universal in an average-case sense. Finally, our results enrich our understanding of the connections between PPP, lattice problems and other concrete cryptographic assumptions, such as the discrete logarithm problem over general groups

Obfuscating Compute-and-Compare Programs under LWE

We show how to obfuscate a large and expressive class of programs, which we call compute-and-compare programs, under the learning-with-errors (LWE) assumption. Each such program $CC[f,y]$ is parametrized by an arbitrary polynomial-time computable function $f$ along with a target value $y$ and we define $CC[f,y](x)$ to output $1$ if $f(x)=y$ and $0$ otherwise. In other words, the program performs an arbitrary computation $f$ and then compares its output against a target $y$. Our obfuscator satisfies distributional virtual-black-box security, which guarantees that the obfuscated program does not reveal any partial information about the function $f$ or the target value $y$, as long as they are chosen from some distribution where $y$ has sufficient pseudo-entropy given $f$. We also extend our result to multi-bit compute-and-compare programs $MBCC[f,y,z](x)$ which output a message $z$ if $f(x)=y$. Compute-and-compare programs are powerful enough to capture many interesting obfuscation tasks as special cases. This includes obfuscating conjunctions, and therefore we improve on the prior work of Brakerski et al. (ITCS \u2716) which constructed a conjunction obfuscator under a non-standard entropic ring-LWE assumption, while here we obfuscate a significantly broader class of programs under standard LWE. We show that our obfuscator has several interesting applications. For example, we can take any encryption scheme and publish an obfuscated plaintext equality tester that allows users to check whether an arbitrary ciphertext encrypts some target value $y$; as long as $y$ has sufficient pseudo-entropy this will not harm semantic security. We can also use our obfuscator to generically upgrade attribute-based encryption to predicate encryption with one-sided attribute-hiding security, as well as witness encryption to indistinguishability obfuscation which is secure for all null circuits. Furthermore, we show that our obfuscator gives new circular-security counter-examples for public-key bit encryption and for unbounded length key cycles. Our result uses the graph-induced multi-linear maps of Gentry, Gorbunov and Halevi (TCC \u2715), but only in a carefully restricted manner which is provably secure under LWE. Our technique is inspired by ideas introduced in a recent work of Goyal, Koppula and Waters (EUROCRYPT \u2717) in a seemingly unrelated context