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A Note on the Common Haar State Model

Prabhanjan Ananth, Aditya Gulati, Yao-Ting Lin

TL;DR

The work advances quantum cryptography in the CHS model by showing that statistically secure stretch PRSGs can be built even when the adversary has access to a sublinear number of Haar-state copies, with an explicit construction achieving output length exceeding the key length and secure against up to $O\left(\frac{λ}{(\log λ)^{1.01}}\right)$ copies. It also introduces multi-key PRS and a CHS-based non-interactive commitment scheme using SWAP tests, and provides a lower bound showing optimality for certain parameter choices, including impossibility results for single-copy CHS PRSGs with sufficiently many copies. The methodology relies on type-state decompositions and hybrid arguments that connect Haar-state averaging to combinatorial type constructions, yielding elementary, accessible proofs. Overall, the paper delineates feasible cryptographic primitives in CHS and highlights the nuanced trade-offs between state copy numbers, state lengths, and security guarantees in quantum settings, with potential impact on quantum public-key constructions and related primitives.

Abstract

Common random string model is a popular model in classical cryptography with many constructions proposed in this model. We study a quantum analogue of this model called the common Haar state model, which was also studied in an independent work by Chen, Coladangelo and Sattath (arXiv 2024). In this model, every party in the cryptographic system receives many copies of one or more i.i.d Haar states. Our main result is the construction of a statistically secure PRSG with: (a) the output length of the PRSG is strictly larger than the key size, (b) the security holds even if the adversary receives $O\left(\fracλ{(\log(λ))^{1.01}} \right)$ copies of the pseudorandom state. We show the optimality of our construction by showing a matching lower bound. Our construction is simple and its analysis uses elementary techniques.

A Note on the Common Haar State Model

TL;DR

The work advances quantum cryptography in the CHS model by showing that statistically secure stretch PRSGs can be built even when the adversary has access to a sublinear number of Haar-state copies, with an explicit construction achieving output length exceeding the key length and secure against up to copies. It also introduces multi-key PRS and a CHS-based non-interactive commitment scheme using SWAP tests, and provides a lower bound showing optimality for certain parameter choices, including impossibility results for single-copy CHS PRSGs with sufficiently many copies. The methodology relies on type-state decompositions and hybrid arguments that connect Haar-state averaging to combinatorial type constructions, yielding elementary, accessible proofs. Overall, the paper delineates feasible cryptographic primitives in CHS and highlights the nuanced trade-offs between state copy numbers, state lengths, and security guarantees in quantum settings, with potential impact on quantum public-key constructions and related primitives.

Abstract

Common random string model is a popular model in classical cryptography with many constructions proposed in this model. We study a quantum analogue of this model called the common Haar state model, which was also studied in an independent work by Chen, Coladangelo and Sattath (arXiv 2024). In this model, every party in the cryptographic system receives many copies of one or more i.i.d Haar states. Our main result is the construction of a statistically secure PRSG with: (a) the output length of the PRSG is strictly larger than the key size, (b) the security holds even if the adversary receives copies of the pseudorandom state. We show the optimality of our construction by showing a matching lower bound. Our construction is simple and its analysis uses elementary techniques.
Paper Structure (28 sections, 14 theorems, 43 equations, 1 figure)

This paper contains 28 sections, 14 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Work.
  3. Future Directions.
  4. Preliminaries
  5. Notation
  6. Pseudorandom State Generators
  7. Quantum Commitments
  8. Common Haar State Model
  9. Pseudorandom State Generators in the CHS model
  10. Quantum Commitments in the CHS model
  11. Symmetric Subspaces, Type States, and Haar States
  12. $\ell$-copy statistical PRS in the CHS model
  13. Construction
  14. Hybrid $1$.
  15. Hybrid $2$.
  16. ...and 13 more sections

Key Result

Theorem 1.1

There is a statistically secure PRSG in the CHS model satisfying the following: (a) the output length of PRSG is $> \lambda$ and, (b) the security holds as long as the adversary receives $O\left( \frac{\lambda}{\left( \log(\lambda) \right)^{1+\varepsilon} }\right)$ copies, for some constant $\vareps

Figures (1)

  • Figure 1: Commitment scheme in the CHS model

Theorems & Definitions (36)

  • Theorem 1.1: Informal
  • Theorem 1.2: Informal
  • Theorem 1.3: Informal
  • Definition 2.1
  • Definition 2.2: Quantum commitments in the Common Reference Quantum State (CRQS) model MNY23
  • Definition 2.3: $t$-copy statistical hiding MNY23
  • Definition 2.4: Statistical sum-binding MNY23
  • Definition 2.5: $\ell$-copy PRS in CHS model
  • Definition 2.6: Multi-key $\ell$-copy PRS in CHS model
  • Remark 2.7
  • ...and 26 more