Probabilistically Checkable Proofs and Applications
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zheng-kzzheng-phd-math-thesis2026.pdf
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Author(s)
Zheng, Kai Zhe
Advisor(s)
Minzer, Dor
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
Abstract
This thesis studies Probabilistically Checkable Proofs (PCPs) and related proof systems through a collection of results in local testing, hardness of approximation, coding theory, and cryptography. Its contents can roughly be divided into three parts. The first part studies local testing of algebraic codes, especially Reed-Muller codes, which have long played a foundational role in PCP theory. We study this problem, broadly, in two regimes. In the high-soundness regime, where a function passes a local test with probability close to one, we give new local tests for Reed–Muller codes defined over extension fields. Our tests improve upon the query complexity and rejection probability of all prior tests over extension fields. We also consider the Reed–Muller testing question in a new model, called the online-erasure model of [131]. Here, we construct the first testers that work for general degrees over general fields. In the high soundness regime, we also consider the question of testing direct sums, which are a different but related class of functions and prove soundness for a local test that was proposed in [75]. Finally, we shift to the low soundness regime. Here one deals with encodings that pass some local test with small, but still non-trivial probability. We give a near-optimal analysis of the cube-versus-cube agreement test in this regime. The second part of the thesis presents new PCP constructions which lead to resolutions of two long-standing open problems in hardness of approximation and coding theory. The quest for more efficient PCPs has been a major research direction, due to their many downstream applications in areas like hardness of approximation, coding theory, and cryptography. Typically one aims for the best possible tradeoffs between some subset of 1) size (the number of symbols in the proof), 2) alphabet size (the size of the alphabet from which the symbols come), 3) query complexity (the number of queries needed to check the proof), 4) soundness (the probability that a false statement is accepted), and 5) completeness (the probability that a true statement is accepted). We present several new PCP constructions with state-of-the-art parameters in several settings. First, we construct two-query PCPs with a nearly optimal alphabet–soundness tradeoff, and then extend this construction to obtain k-query PCPs for every k ⩾ 3 with similarly near-optimal parameters. These results close a long line of work on optimizing alphabet soundness tradeoff and yield improved hardness of approximation for problems including quadratic programming, boundeddegree 2-CSP, maximum independent set in claw-free graphs, various vertex-connectivity problems, and k-dimensional matching and related packing problems. Our next construction gives the first quasilinear-length 3-query PCPs of Proximity (PCPPs) with small soundness, which in turn give 3-query Relaxed Locally Decodable Codes (RLDCs) of near-quadratic length. Combined with lower bounds for standard Locally Decodable Codes (LDCs) from [7], this gives a separation between RLDCs and LDCs, answering a question of [38] from 2005. The final part turns to interactive oracle proofs motivated by cryptographic applications. These are variants of PCPs which leverage interaction to surpass efficiency barriers. There we first construct improved IOPs with strong round-by-round soundness and small standard soundness. Our results are obtained by incorporating PCP style tests, similar to those studied in the previous two parts, into the IOP framework. We then turn to a more applied setting, where we discuss Zinc+, a proof system for generating succinct proofs of general computations. Zinc+ is based on IOPs over the integers and rationals, and by operating over the rationals, it mitigates what we term the “arithmetization overhead”. This setting leads to improved performance in practical applications. We discuss the key challenges that arise when working over the rationals and provide preliminary benchmarks that highlight the performance of Zinc+.
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Massachusetts Institute of Technology. Department of Mathematics
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