Constructing Good Gottesman-Kitaev-Preskill Codes from the NTRU Cryptosystem

Core Concepts

The authors introduce a new class of random Gottesman-Kitaev-Preskill (GKP) codes derived from the cryptanalysis of the NTRU cryptosystem, which exhibit constant rate and average distance scaling proportional to the square root of the number of bosonic modes.

Abstract

The authors introduce a new class of random Gottesman-Kitaev-Preskill (GKP) codes derived from the cryptanalysis of the NTRU cryptosystem. These NTRU-GKP codes have the following key properties: They exhibit constant rate and average distance scaling proportional to the square root of the number of bosonic modes, which is equivalent to the distance scaling of a GKP code obtained by concatenating single-mode GKP codes into a qubit-quantum error correcting code with linear distance. Decoding for a stochastic displacement noise model is equivalent to decrypting the NTRU cryptosystem, such that every random instance of the code naturally comes with an efficient decoder. The authors discuss the computational hardness of decoding GKP codes in general and propose a simple public key quantum communication protocol with security inherited from the NTRU cryptosystem as a new application of the NTRU-GKP codes. The article is structured as follows: Section 2 introduces the basic principles of the GKP code and the relevant aspects of the general decoding problem. Section 3 discusses notable examples of GKP codes and their properties. Section 4 introduces the NTRU cryptosystem and shows that random NTRU lattices can be used to construct a family of good randomized GKP codes. Section 5 discusses the decoding problem for the NTRU-GKP codes and the proposed quantum public key cryptosystem. Section 6 concludes and provides an outlook.

Stats

We introduce a new class of random Gottesman-Kitaev-Preskill (GKP) codes derived from the cryptanalysis of the NTRU cryptosystem. The derived codes are good in that they exhibit constant rate and average distance scaling ∆∝√n with high probability, where n is the number of bosonic modes. The derived class of NTRU-GKP codes has the additional property that decoding for a stochastic displacement noise model is equivalent to decrypting the NTRU cryptosystem.

Quotes

"The derived codes are good in that they exhibit constant rate and average distance scaling ∆∝√n with high probability, where n is the number of bosonic modes, which is a distance scaling equivalent to that of a GKP code obtained by concatenating single mode GKP codes into a qubit-quantum error correcting code with linear distance." "The derived class of NTRU-GKP codes has the additional property that decoding for a stochastic displacement noise model is equivalent to decrypting the NTRU cryptosystem, such that every random instance of the code naturally comes with an efficient decoder."

Key Insights Distilled From

Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem

by Jonathan Con... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2303.02432.pdf

Deeper Inquiries

How can the NTRU-GKP codes be extended or generalized to achieve even better distance scaling or encoding rates?

The NTRU-GKP codes can be extended or generalized in several ways to achieve better distance scaling or encoding rates. One approach is to explore different variations of the NTRU cryptosystem, such as using different polynomial rings or modifying the parameters of the system. By adjusting the parameters, such as the degree of the polynomial, the modulus, or the choice of the irreducible polynomial, it may be possible to optimize the performance of the NTRU-GKP codes. Another strategy is to investigate concatenated codes, where the NTRU-GKP codes are combined with other quantum error correcting codes to enhance their performance. By concatenating the NTRU-GKP codes with other codes that have complementary properties, it may be possible to achieve better distance scaling and encoding rates. Furthermore, exploring different decoding algorithms or error correction techniques tailored specifically for NTRU-GKP codes could also improve their performance. By developing more efficient decoding methods or error correction strategies, the overall performance of the NTRU-GKP codes can be enhanced.

What are the potential limitations or drawbacks of using the NTRU cryptosystem as the basis for constructing quantum error correcting codes?

While the NTRU cryptosystem offers several advantages for constructing quantum error correcting codes, there are also potential limitations and drawbacks to consider. One limitation is the reliance on the hardness of the polynomial factorization problem for security. If advances in quantum computing or new mathematical techniques make polynomial factorization easier, the security of the NTRU-based quantum error correcting codes could be compromised. Another drawback is the sensitivity of the NTRU cryptosystem to parameter choices. The performance of the NTRU-based quantum error correcting codes can be heavily influenced by the selection of parameters such as the modulus, the degree of the polynomial, and the choice of the irreducible polynomial. Suboptimal parameter choices could lead to weaker error correction capabilities or reduced security. Additionally, the computational complexity of the NTRU cryptosystem may pose challenges for practical implementations of quantum error correcting codes based on NTRU. The computational overhead required for key generation, encryption, and decryption in the NTRU cryptosystem could impact the efficiency and scalability of the quantum error correcting codes.

What other connections or applications might exist between post-quantum cryptography and the design of robust quantum error correcting codes?

Post-quantum cryptography and the design of robust quantum error correcting codes are closely related fields with several potential connections and applications. One key connection is the shared goal of developing secure and reliable cryptographic systems in the presence of quantum threats. By leveraging techniques and principles from post-quantum cryptography, researchers can enhance the security and resilience of quantum error correcting codes against quantum attacks. Furthermore, advancements in post-quantum cryptography, such as the development of new encryption schemes or signature algorithms, can inspire innovations in the design of quantum error correcting codes. By adapting cryptographic techniques or protocols from post-quantum cryptography, researchers can improve the error correction capabilities and performance of quantum codes. Moreover, the intersection of post-quantum cryptography and quantum error correction opens up possibilities for developing hybrid cryptographic systems that combine the strengths of both fields. By integrating post-quantum cryptographic algorithms with robust quantum error correcting codes, researchers can create more secure and efficient cryptographic solutions that are resilient to both classical and quantum attacks.