Highlights of the project’s achievements include:
• An optimal batch verification theorem. This new protocol allows for the verification of k statements, at a cost that grows only poly-logarithmically with k. The statements need to be in UP, meaning that there is at most a single valid witness (or classical proof) for each statement. This is important both as a foundational question, and also because batch verification is a powerful tool towards constructing power proof systems for general computations.
• Cryptographically sound arguments. A central pursuit in the study of cryptography is proving the soundness of cryptographic systems based on well-founded mathematical assumptions, such as the hardness of factoring or of lattice problem. One central question along these lines is proving the security of the Fiat-Shamir transformation, which can be used to minimize interaction in a cryptographic protocol.
The project proposed the first secure instantiation of this transformation based on Lattice Assumptions. It led to several subsequent breakthroughs in constructing and proving security for non-interactive protocols, resolving important open problems in the field.
The project also constructed the first argument systems that can out-perform unconditionally sound proof systems based only on the “minimal” cryptographic assumption that a one-way function exists. This was achieved by constructing constant-round arguments for bounded-depth computations from one-way functions.
• Verifiable data science. Traditionally, research on proof systems has focused on the computational complexity of the task at hand. That is, the critical resources were computational (e.g. running time or space). On the other hand, data science is becoming a major focus of computer science and algorithmic research, and there the critical resource is often access to an unknown data distribution. The project has developed proof systems for data science tasks such as machine learning, property testing and distribution testing. These new development allow the outcomes of complex analyses to be verified using only very limited access to the underlying distribution.
• Sound foundations for algorithmic fairness. Trust in automated systems is a growing scientific and societal concern. One focus is the issue of algorithmic fairness: are the results or resource allocations computed by algorithms fair, or do they discriminate against disadvantaged or protected populations? Our project has been tackling these concerns by formulating provable fairness guarantees, and constructing protocols and algorithms that are sound, in the sense that they meet these definitions, and can thus be verified. This builds a bridge between the social / societal fairness literature and the cryptographic literature on proof systems. It has already led to new work on fair risk prediction in medical contexts.
• The complexity of finding a Nash equilibrium. The project established a new connection between proof systems and the hardness of computational problems related to finding Nash Equilibria, a basic problem in game theory and economics (as well as, more broadly, the complexity class PPAD). Together with follow-up works, this led to a new understanding that these problems are no easier to solve than cryptographically hard lattice problems, resolving a long standing open question.
