[Paper Review] Quantum McEliece public-key encryption scheme

Li Yang, Min Liang|arXiv (Cornell University)|Jan 19, 2015

Cryptography and Data Security17 references3 citations

TL;DR

This paper proposes a quantum McEliece public-key encryption (QPKE) scheme for encrypting quantum messages using classical keys, based on the hardness of NP-complete problems in coding theory. It introduces a double-encryption technique to enhance security, proving the improved scheme is more secure than the original but less secure than the quantum one-time pad, while maintaining efficiency and avoiding unnecessary quantum error correction functionality.

ABSTRACT

This paper investigates a quantum version of McEliece public-key encryption (PKE) scheme, and analyzes its security. As is well known, the security of classical McEliece PKE is not stronger than the onewayness of related classical one-way function. We prove the security of quantum McEliece PKE ranks between them. Moreover, we propose the double-encryption technique to improve its security, and the security of the improved scheme is proved to be between the original scheme and the quantum one-time pad.

Motivation & Objective

• To design a quantum public-key encryption scheme capable of securely encrypting quantum messages using classical keys.
• To analyze the security of the original quantum McEliece PKE scheme proposed in Ref. [8], which is based on classical error-correcting codes.
• To improve the security of the original scheme using a double-encryption technique that introduces additional randomness during encryption.
• To demonstrate that the improved scheme maintains practicality while offering stronger resistance to known attacks than the original.
• To clarify misconceptions about the original scheme’s vulnerability to classical attacks when used for classical message encryption.

Proposed method

• The scheme uses a classical McEliece-like structure: a public key derived from a Goppa code via scrambling with invertible and permutation matrices.
• Quantum messages are encrypted by applying a unitary transformation based on the public key, using a random vector to mask the message.
• The double-encryption technique applies two independent random masks in sequence, increasing the entropy of the ciphertext and thwarting chosen-plaintext attacks.
• Security is analyzed in terms of computational indistinguishability, with bounds derived from the hardness of decoding random linear codes (an NP-complete problem).
• The scheme avoids quantum error correction functionality, simplifying encoding/decoding while preserving security for quantum message encryption.
• Theoretical analysis shows that repeated use of double-encryption reduces security, as the advantage of randomness diminishes over multiple encryptions.

Experimental results

Research questions

• RQ1 Is the original quantum McEliece PKE scheme secure against quantum adversaries, and how does its security compare to classical McEliece?
• RQ2 Can the security of the quantum McEliece PKE be enhanced without introducing quantum error correction, which is not required for encryption?
• RQ3 What is the impact of multiple encryptions on the security of the double-encryption variant?
• RQ4 Why do classical attacks like Korzhik-Turkin and message-resend attacks fail against the quantum McEliece PKE?
• RQ5 How does the security of the double-encryption scheme compare to the quantum one-time pad and the original scheme?

Key findings

• The original quantum McEliece PKE is at least as secure as its classical counterpart, with security bounded by the one-wayness of the underlying classical one-way function.
• The double-encryption scheme improves security beyond the original scheme, but its security degrades with repeated use due to reduced randomness advantage.
• The improved scheme is proven to be more secure than the original but less secure than the quantum one-time pad, establishing a security hierarchy.
• Classical attacks such as Korzhik-Turkin, message-resend, and related-message attacks fail against the quantum McEliece PKE due to the non-reusability of quantum states and the structure of the encryption process.
• The scheme is more efficient and simpler than Fujita’s quantum McEliece variant because it omits quantum error correction, which is unnecessary for public-key encryption.
• The scheme can be used to encrypt classical messages as a 'quantum envelope' with enhanced security compared to classical McEliece, due to the incompatibility of classical attacks with quantum states.