Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Delegating computation reliably : paradigms and constructions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 285-297).In an emerging computing paradigm, computational capabilities, from processing power to storage capacities, are offered to users over communication networks as a service. This new paradigm holds enormous promise for increasing the utility of computationally weak devices. A natural approach is for weak devices to delegate expensive tasks, such as storing a large file or running a complex computation, to more powerful entities (say servers) connected to the same network. While the delegation approach seems promising, it raises an immediate concern: when and how can a weak device verify that a computational task was completed correctly? This practically motivated question touches on foundational questions in cryptography and complexity theory. The focus of this thesis is verifying the correctness of delegated computations. We construct efficient protocols (interactive proofs) for delegating computational tasks. In particular, we present: e A protocol for delegating any computation, where the work needed to verify the correctness of the output is linear in the input length, polynomial in the computation's depth, and only poly-logarithmic in the computation's size. The space needed for verification is only logarithmic in the computation size. Thus, for any computation of polynomial size and poly-logarithmic depth (the rich complexity class N/C), the work required to verify the correctness of the output is only quasi-linear in the input length. The work required to prove the output's correctness is only polynomial in the original computation's size. This protocol also has applications to constructing one-round arguments for delegating computation, and efficient zero-knowledge proofs. * A general transformation, reducing the parallel running time (or computation depth) of the verifier in protocols for delegating computation (interactive proofs) to be constant. Next, we explore the power of the delegation paradigm in settings where mutually distrustful parties interact. In particular, we consider the settings of checking the correctness of computer programs and of designing error-correcting codes. We show: * A new methodology for checking the correctness of programs (program checking), in which work is delegated from the program checker to the untrusted program being checked. Using this methodology we obtain program checkers for an entire complexity class (the class of N/C¹-computations that are WNC-hard), and for a slew of specific functions such as matrix multiplication, inversion, determinant and rank, as well as graph functions such as connectivity, perfect matching and bounded-degree graph isomorphism. * A methodology for designing error-correcting codes with efficient decoding procedures, in which work is delegated from the decoder to the encoder. We use this methodology to obtain constant-depth (AC⁰) locally decodable and locally-list decodable codes. We also show that the parameters of these codes are optimal (up to polynomial factors) for constant-depth decoding.by Guy N. Rothblum.Ph.D

Quantum information in security protocols

Information security deals with the protection of our digital infrastructure. Achieving meaningful real-world security requires powerful cryptographic models that can give strong security guarantees and it requires accuracy of the model. Substantial engineering effort is required to ensure that a deployment meets the requirements imposed by the model. Quantum information impacts the field of security in two major ways. First, it allows more efficient cryptanalysis of currently widely deployed systems. New "post-quantum" cryptographic algorithms are designed to be secure against quantum attacks, but do not require quantum technology to be implemented. Since post-quantum algorithms have different properties, substantial effort is required to integrate these in the existing infrastructure. Second, quantum cryptography leverages quantum-mechanical properties to build new cryptographic systems with potential advantages, however these require a more substantial overhaul of the infrastructure. In this thesis I highlight the necessity of both the mathematical rigour and the engineering efforts that go into security protocols in the context of quantum information. This is done in three different contexts. First, I analyze the impact of key exhaustion attacks against quantum key distribution, showing that they can lead to substantial loss of security. I also provide two mitigations that thwart such key exhaustion attacks by computationally bounded adversaries, without compromising the information theoretically secure properties of the protocol output. I give various security considerations for secure implementation of the mitigations. Second, I consider how quantum adversaries can successfully attack quantum distance bounding protocols that had previously been claimed to be secure by informal reasoning. This highlights the need for mathematical rigour in the analysis of quantum adversaries. Third, I propose a post-quantum replacement for the socialist millionaire protocol in secure messaging. The protocol prevents some of the usability problems that have been observed in other key authentication ceremonies. The post-quantum replacement utilizes techniques from private set intersection to build a protocol from primitives that have seen much scrutiny from the cryptographic community

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum