Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Note on the Equivalence Between Zero-knowledge and Quantum CSS Codes

Noga Ron-Zewi, Mor Weiss

TL;DR

It is shown that (linear, perfect) zero-knowledge codes and quantum CSS codes are equivalent, and the potential of this equivalence is demonstrated by using it to obtain explicit asymptotically-good zero-knowledge locally-testable codes.

Abstract

Zero-knowledge codes, introduced by Decatur, Goldreich, and Ron (ePrint 1997), are error-correcting codes in which few codeword symbols reveal no information about the encoded message, and have been extensively used in cryptographic constructions. Quantum CSS codes, introduced by Calderbank and Shor (Phys. Rev. A 1996) and Steane (Royal Society A 1996), are error-correcting codes that allow for quantum error correction, and are also useful for applications in quantum complexity theory. In this short note, we show that (linear, perfect) zero-knowledge codes and quantum CSS codes are equivalent. We demonstrate the potential of this equivalence by using it to obtain explicit asymptotically-good zero-knowledge locally-testable codes.

A Note on the Equivalence Between Zero-knowledge and Quantum CSS Codes

TL;DR

It is shown that (linear, perfect) zero-knowledge codes and quantum CSS codes are equivalent, and the potential of this equivalence is demonstrated by using it to obtain explicit asymptotically-good zero-knowledge locally-testable codes.

Abstract

Zero-knowledge codes, introduced by Decatur, Goldreich, and Ron (ePrint 1997), are error-correcting codes in which few codeword symbols reveal no information about the encoded message, and have been extensively used in cryptographic constructions. Quantum CSS codes, introduced by Calderbank and Shor (Phys. Rev. A 1996) and Steane (Royal Society A 1996), are error-correcting codes that allow for quantum error correction, and are also useful for applications in quantum complexity theory. In this short note, we show that (linear, perfect) zero-knowledge codes and quantum CSS codes are equivalent. We demonstrate the potential of this equivalence by using it to obtain explicit asymptotically-good zero-knowledge locally-testable codes.
Paper Structure (13 sections, 5 theorems, 3 equations)

This paper contains 13 sections, 5 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Zero-knowledge codes.
  3. Quantum CSS codes.
  4. Equivalence between zero-knowledge and quantum CSS codes.
  5. Preliminaries
  6. Zero-Knowledge (ZK) Codes
  7. Error Correction in ZK Codes.
  8. Quantum CSS Codes
  9. Zero-knowledge and Quantum CSS Codes Are Equivalent
  10. Proof of Lemma \ref{['lem:equiv']}:
  11. Proof of Thm \ref{['thm:equiv']}:
  12. Application: Explicit Asymptotically-Good Zero-knowledge Locally-Testable Codes
  13. Acknowledgement.

Key Result

Theorem 3.1

Let $C \subseteq \mathbb{F}^n$ be a linear code of dimension $k$, let $G \in \mathbb{F}^{n \times k}$ be a generator matrix for $C$, let $k'<k$ be a parameter, and let ${\mathsf{Enc}}: \mathbb{F}^{k'} \to C$ be the $k'$-randomized encoding map for $G$. Let $C_X = C$, and let $C_Z \subseteq \mathbb{F

Theorems & Definitions (12)

  • Definition 2.1: Randomized Encoding Map
  • Definition 2.2: Zero-Knowledge (ZK) Code
  • Definition 2.3: CSS Code
  • Theorem 3.1: ZK Codes are CSS Codes
  • Corollary 3.2: CSS Codes are ZK Codes
  • Lemma 3.3: ZK Codes, Equivalent Formulation
  • Claim 3.4
  • Claim 3.5
  • Definition 4.1: Locally-Testable Code (LTC)
  • Theorem 4.2: Asymptotically-good quantum LTCs DLV24WLH25
  • ...and 2 more