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

TL;DR

This paper establishes an equivalence between zero-knowledge codes and quantum CSS codes, revealing new connections in error correction and cryptography, and demonstrates how this can lead to explicit zero-knowledge locally-testable codes.

Contribution

It proves the equivalence between zero-knowledge codes and quantum CSS codes, enabling new constructions of zero-knowledge locally-testable codes.

Findings

Zero-knowledge codes and quantum CSS codes are equivalent.

This equivalence enables explicit construction of asymptotically-good zero-knowledge locally-testable codes.

The result bridges classical cryptographic codes and quantum error correction theories.

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.