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SPIE Conference on Quantum and Nonlinear Optics

Generating Gaussian error in lattice cryptography with quantum random number generator

Jingxiang Huang, Min Gu, Zhenhong Liu, Bingqi Yan

Published Oct 12, 2023• 0 citations
Publisher

Abstract

The Gaussian Error, represented by random numbers with a Gaussian distribution, plays a critical role in the security of modern lattice-based cryptography. Lattice-based cryptography is the most important category of post-quantum cryptography capable of resisting attacks from both classical and quantum.While current schemes of generating Gaussian Error used in lattice cryptography face severe challenges. On the one hand, the current scheme produces an approximate Gaussian distribution rather than the rigorously proven Gaussian distribution required in lattice cryptography, posing a security risk. Meanwhile, the Gaussian Error of current schemes is obtained by using algorithms to compute uniformly distributed random numbers, which lack provable randomness and also pose security risks. On the other hand, the generation rate of Gaussian Error is not high enough for many applications of cryptographic systems in current schemes. To address these challenges, this paper proposes a novel scheme that uses a quantum random number generator(QRNG) to generate the Gaussian Error, which follows a theoretically proven Gaussian distribution and satisfies rigorous security justification in lattice cryptography. Furthermore, based on the quantum mechanical process of measuring vacuum fluctuations and the principle of minimum entropy under classical noise conditions, the Gaussian Error obtained by our scheme is theoretically unpredictable, further enhancing the security of lattice cryptography compared to the current schemes. Finally, an experiment for generating Gaussian Error was constructed, and our experimental results demonstrate that the Gaussian Error generation rate is 1 G/s, which achieves higher speed among existing works.

DOI
10.1117/12.2687335

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The Gaussian Error, represented by random numbers with a Gaussian distribution, plays a critical role in the security of modern lattice-based cryptography. Lattice-based cryptography is the most important category of post-quantum cryptography capable of resisting attacks from both classical and quantum.While current schemes of generating Gaussian Error used in lattice cryptography face severe challenges. On the one hand, the current scheme produces an approximate Gaussian distribution rather than the rigorously proven Gaussian distribution required in lattice cryptography, posing a security risk. Meanwhile, the Gaussian Error of current schemes is obtained by using algorithms to compute uniformly distributed random numbers, which lack provable randomness and also pose security risks. On the other hand, the generation rate of Gaussian Error is not high enough for many applications of cryptographic systems in current schemes. To address these challenges, this paper proposes a novel scheme that uses a quantum random number generator(QRNG) to generate the Gaussian Error, which follows a theoretically proven Gaussian distribution and satisfies rigorous security justification in lattice cryptography. Furthermore, based on the quantum mechanical process of measuring vacuum fluctuations and the principle of minimum entropy under classical noise conditions, the Gaussian Error obtained by our scheme is theoretically unpredictable, further enhancing the security of lattice cryptography compared to the current schemes. Finally, an experiment for generating Gaussian Error was constructed, and our experimental results demonstrate that the Gaussian Error generation rate is 1 G/s, which achieves higher speed among existing works.
Quantum Key Distribution(QKD) thrives to achieve perfect secrecy of One time Pad (OTP) through quantum processes. One of the crucial components of QKD are Quantum Random Number Generators(QRNG) for generation of keys. Unfortunately, these QRNG does not immediately produce usable bits rather it produces raw bits with high entropy but low uniformity which can be hardly used by any cryptographic system. A lot of pre-processing is required before the random numbers generated by QRNG to be usable. This causes a bottle neck in random number generation rate as well as QKD system relying on it. To avoid this lacuna of post-processing methods employed as a central part of Quantum Random Number Generators alternative approaches that satisfy the entropy(non determinism) and quantum security is explored. Pseudorandom generators based on quantum secure primitives could be an alternative to the post-processing problem as PRNGs are way more faster than any random number generator employing physical randomness (quantum mechanical process in QRNG) as well as it can provide uniform bits required for cryptography application. In this work we propose a pseudorandom generator based on post quantum primitives. The central theme of this random number generator is designing PRNG with non deterministic entropy generated through hard lattice problem - Learning with errors. We leverage the non determinism by Gaussian errors of LWE to construct non-deterministic PRNG satisfying the entropy requirement of QKD. Further, the paper concludes by evaluating the PRNG through Die-Harder Test.
Quantum random number generators represent one of the most promising techniques for producing truly and unique random numbers, essential in many applications such as quantum communication and cryptography protocols. In this work, we report an experimental implementation of vacuum-based quantum random number generator. By measuring the quantum fluctuations of the radiation ground state, a mixture of a perfectly Gaussian distributed random signal—originating from the intrinsic randomness in the quantum measurement— and an undesired classical electronic noise is obtained. Using an adequate post-processing, the undesired classical contribution of the electronic noise is removed to produce final, quasi-uniformly distributed random bit sequences. The high-quality character of the generated random bits is confirmed by applying the statistical tests of the NIST SP 800-22 test battery.
Random number generators (RNG) are an important resource in many areas: cryptography (both quantum and classical), probabilistic computation (Monte Carlo methods), numerical simulations, industrial testing and labeling, hazard games, scientific research etc. Because today's computers are deterministic, they can not create random numbers unless complemented with a physical RNG. Randomness of a RNG can be defined and scientifically characterized and measured. Especially valuable is the information-theoretic provable RNG which, at state of the art, seem to be possible only by harvest of randomness inherent to certain (simple) quantum systems and such a generator we call Quantum RNG (QRNG). On the other hand, current industry standards dictate use of RNGs based on free running oscillators (FRO) whose randomness is derived from electronics noise present in logic circuits and which, although quantum in nature, cannot be strictly proven. This approach is currently used in FPGA and ASIC chips. We compare weak and strong aspects of the two approaches for use in cryptography and in general. We also give an alternative definition of randomness, discuss usage of single photon detectors in realization of QRNGs and give several examples where QRNG can significantly improve security of a cryptographic system.
This paper explores the design of cryptographic algorithms leveraging the inherent randomness of quantum random number generators (QRNGs). Traditional cryptographic algorithms often rely on pseudo-random number generators (PRNGs), which are deterministic and therefore vulnerable to attacks if their internal state is compromised. QRNGs, on the other hand, derive their randomness from quantum mechanical phenomena, offering a fundamentally unpredictable source of numbers. This research investigates the application of QRNGs to enhance cryptographic security, focusing on constructing algorithms that exploit this unique property. The core claim is that utilizing QRNG characteristics allows for the design of more secure cryptographic algorithms. The proposed mechanism involves capitalizing on the non-deterministic nature of QRNGs to build algorithms with heightened security levels. This approach presents a novel security guarantee method by incorporating true randomness into cryptographic processes. We examine specific algorithm designs and analyze their security implications compared to traditional PRNG-based approaches. The paper concludes with a discussion on the challenges and future directions of utilizing QRNGs in cryptography.
The development of quantum-resistant solutions is imperative as the emergence of quantum computing presents a substantial risk to existing cryptography systems. Lattice-based cryptography, especially schemes based on the Learning with Errors (LWE) problem, is one of the most promising methods. To guarantee long-term security, even LWE-based methods could need to be strengthened further as quantum algorithms advance. By mixing Gaussian and discrete uniform distributions to create a mixed error distribution, this work enhances the classic LWE problem. The experimental findings show that, with a slight rise in computing overhead, the mixed error distribution improves the security of the LWE problem by strengthening its resistance to quantum techniques. By presenting a novel approach for enhancing the resilience of cryptographic methods in the quantum era, this research contributes to the continuing work in post-quantum cryptography. Moreover, it introduces the direction of future model improvements and provides multidisciplinary methods for increasing the complexity of cryptographic algorithms.
In recent years, the advancement of quantum computing technology has posed potential security threats to RSA cryptography and elliptic curve cryptography. In response, the National Institute of Standards and Technology (NIST) published several Federal Information Processing Standards (FIPS) of post-quantum cryptography (PQC) in August 2024, including the Module-Lattice-Based Key-Encapsulation Mechanism (ML-KEM), Module-Lattice-Based Digital Signature Algorithm (ML-DSA), and Stateless Hash-Based Digital Signature Algorithm (SLH-DSA). Although these PQC algorithms are designed to resist quantum computing attacks, they may not provide adequate security in certain specialized application scenarios. To address this issue, this study proposes quantum random number generator (QRNG)-based PQC algorithms. These algorithms leverage quantum computing to generate random numbers, which serve as the foundation for key pair generation, key encapsulation, and digital signature generation. A generalized architecture of QRNG is proposed, along with the design of six QRNGs. Each generator is evaluated according to the statistical validation procedures outlined in NIST SP 800-90B, including tests for verification of entropy sources and independent and identically distributed (IID) outputs. Experimental results assess the computation time of the six QRNGs, as well as the performance of QRNG-based ML-KEM, QRNG-based ML-DSA, and QRNG-based SLH-DSA. These findings provide valuable reference data for future deployment of PQC-based Transport Layer Security in web systems.