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Cryptography in the Common Haar State Model: Feasibility Results and Separations

Prabhanjan Ananth, Aditya Gulati, Yao-Ting Lin

TL;DR

A construction of pseudorandom function-like states with security against computationally unbounded adversaries, as long as the adversaries only receive (a priori) bounded number of copies in the common Haar state (CHS) model is presented.

Abstract

Common random string model is a popular model in classical cryptography. We study a quantum analogue of this model called the common Haar state (CHS) model. In this model, every party participating in the cryptographic system receives many copies of one or more i.i.d Haar random states. We study feasibility and limitations of cryptographic primitives in this model and its variants: - We present a construction of pseudorandom function-like states with security against computationally unbounded adversaries, as long as the adversaries only receive (a priori) bounded number of copies. By suitably instantiating the CHS model, we obtain a new approach to construct pseudorandom function-like states in the plain model. - We present separations between pseudorandom function-like states (with super-logarithmic length) and quantum cryptographic primitives, such as interactive key agreement and bit commitment, with classical communication. To show these separations, we prove new results on the indistinguishability of identical versus independent Haar states against LOCC (local operations, classical communication) adversaries.

Cryptography in the Common Haar State Model: Feasibility Results and Separations

TL;DR

A construction of pseudorandom function-like states with security against computationally unbounded adversaries, as long as the adversaries only receive (a priori) bounded number of copies in the common Haar state (CHS) model is presented.

Abstract

Common random string model is a popular model in classical cryptography. We study a quantum analogue of this model called the common Haar state (CHS) model. In this model, every party participating in the cryptographic system receives many copies of one or more i.i.d Haar random states. We study feasibility and limitations of cryptographic primitives in this model and its variants: - We present a construction of pseudorandom function-like states with security against computationally unbounded adversaries, as long as the adversaries only receive (a priori) bounded number of copies. By suitably instantiating the CHS model, we obtain a new approach to construct pseudorandom function-like states in the plain model. - We present separations between pseudorandom function-like states (with super-logarithmic length) and quantum cryptographic primitives, such as interactive key agreement and bit commitment, with classical communication. To show these separations, we prove new results on the indistinguishability of identical versus independent Haar states against LOCC (local operations, classical communication) adversaries.
Paper Structure (88 sections, 44 theorems, 156 equations, 2 figures)

This paper contains 88 sections, 44 theorems, 156 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Common Haar State Model.
  3. Our Results
  4. Feasibility Results
  5. Pseudorandom Function-Like States with Statistical Security.
  6. Commitments.
  7. Black-Box Separations
  8. LOCC Indistinguishability.
  9. Separations.
  10. Technical Overview
  11. Pseudorandomness in the CHS Model
  12. Warmup: Pseudorandom State Generators (PRSGs).
  13. Limitations.
  14. Pseudorandom Function-like State Generators.
  15. Quantum Bit Commitments
  16. ...and 73 more sections

Key Result

Theorem 1.1

There is a statistically secure $(\lambda,m,n,\ell)$-PRFSG in the CHS model, for $m=\lambda^c$, $n \geq \lambda$ and $\ell=O\left( \frac{\lambda^{1-c}}{\log(\lambda)^{1+\varepsilon}} \right)$, for any constant $\varepsilon > 0$ and for all $c \in [0,1)$.

Figures (2)

  • Figure 1: PRFS in the CHS model
  • Figure 2: Quantum commitment scheme in the CHS model

Theorems & Definitions (97)

  • Theorem 1.1: Informal
  • Corollary 1.2
  • Theorem 1.3: Informal
  • Theorem 1.4: Informal
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 3.1: Statistically secure $(\lambda,n,\ell)$-pseudorandom state generators in the CHS model
  • Definition 3.2: Multi-key statistically secure $(\lambda,n,\ell)$-pseudorandom state generators in the CHS model
  • Remark 3.3
  • ...and 87 more