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Lecture notes in computer science

Pseudorandomness with Proof of Destruction and Applications

Amit Behera, Zvika Brakerski, Or Sattath, Omri Shmueli

Published Jan 1, 2023• 10 citations
Publisher
DOI
10.1007/978-3-031-48624-1_5

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Pseudorandom functions ( PRFs ) are one of the foundational concepts in theoretical computer science, with numerous applications in complexity theory and cryptography. In this work, we study the security of PRFs when evaluated on quantum superpositions of inputs. The classical techniques for arguing the security of PRFs do not carry over to this setting, even if the underlying building blocks are quantum resistant. We therefore develop a new proof technique to show that many of the classical PRF constructions remain secure when evaluated on superpositions.
Quantum pseudorandomness is an emerging research area. Ji, Liu, and Song defined pseudorandom states (PRSs) and pseudorandom unitaries (PRUs) as quantum analogs of pseudorandom generators and pseudorandom functions. A unitary oracle separation result between one-way functions and PRSs/PRUs, established by Kretschmer, suggests that certain quantum primitives may remain secure even if classical cryptography is compromised. This insight has spurred extensive work on quantum pseudorandomness and its applications in quantum cryptography. Many constructions of PRSs have been established under standard assumptions, yet building a secure PRU was a long-standing open problem. This dissertation aims to narrow the gap between PRSs and PRUs and presents results that go beyond PRSs. We introduce Pseudorandom State Scramblers (PRSSs), a new primitive that lies between PRSs and PRUs. A PRSS maps any pure state to a pseudorandom state, a property shared with PRUs but not with PRSs. We present a construction of PRSSs inspired by the well-known Kac’s walk, and in particular, we develop a parallel variant that significantly accelerates the mixing time, enabling an efficient construction. PRSSs support cryptographic tasks not known to be achievable from PRSs alone, including a quantum encryption scheme and a succinct quantum state commitment. Additionally, when suitable classical randomness is provided, our construction exhibits a special dispersing property not known to be satisfied by any existing construction of quantum pseudorandom primitives. Our subsequent work shows that, without asymptotically increasing the number of steps, our construction based on the parallel Kac’s walk yields PRUs with standard or even strong security. The proof builds on a recently developed technique for establishing adaptive security, known as the path-recording method. This result provides an alternative construction of PRUs and further showcases the power of this proof technique. In addition, this dissertation includes two side projects. The first revisits the Hidden Subgroup Problem over ℤn, providing a simplified analysis of a known quantum algorithm using elementary lattice tools. The second establishes a quantum analogue of a classical impossibility result for statistical non-interactive zero-knowledge arguments, showing limitations of black-box reductions under classical-query quantum adversaries.
Verifiable Random Functions (VRFs) and Oblivious Pseudorandom Functions (OPRFs) are essential cryptographic primitives in privacy-preserving applications such as anonymous authentication, private set intersection (PSI), and decentralized identity. Existing constructions, however, rely on number-theoretic assumptions that are vulnerable to quantum attacks. This PhD research project focuses on constructing efficient and practical VRFs and OPRFs from lattice-based assumptions to ensure post-quantum security. A key obstacle in these constructions is the overhead of zero-knowledge proofs (ZKPs), particularly range proofs, which are costly in terms of size and prover complexity. To address this, we investigate probabilistic techniques that relax exact correctness. In particular, we explore approximate range proofs and algebraic transformations, such as using automorphisms in polynomial rings to simulate inner product arguments via polynomial multiplication. These methods enable more efficient and scalable lattice-based constructions of VRFs, including group and context-bound variants, as well as OPRFs. The goal is to make these primitives practical for deployment in post-quantum privacy-preserving systems.