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arXiv:2401.03703

On Lattices, Learning with Errors, Random Linear Codes, and Cryptography

Oded Regev

Published Jan 8, 2024• 4 citations
arXivHTMLPDF

Abstract

Our main result is a reduction from worst-case lattice problems such as GapSVP and SIVP to a certain learning problem. This learning problem is a natural extension of the `learning from parity with error' problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for GapSVP and SIVP. A main open question is whether this reduction can be made classical (i.e., non-quantum). We also present a (classical) public-key cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worst-case quantum hardness of GapSVP and SIVP. The new cryptosystem is much more efficient than previous lattice-based cryptosystems: the public key is of size $\tilde{O}(n^2)$ and encrypting a message increases its size by a factor of $\tilde{O}(n)$ (in previous cryptosystems these values are $\tilde{O}(n^4)$ and $\tilde{O}(n^2)$, respectively). In fact, under the assumption that all parties share a random bit string of length $\tilde{O}(n^2)$, the size of the public key can be reduced to $\tilde{O}(n)$.

DOI
10.48550/arxiv.2401.03703

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Our main result is a reduction from worst-case lattice problems such as GapSVP and SIVP to a certain learning problem. This learning problem is a natural extension of the `learning from parity with error' problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for GapSVP and SIVP. A main open question is whether this reduction can be made classical (i.e., non-quantum). We also present a (classical) public-key cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worst-case quantum hardness of GapSVP and SIVP. The new cryptosystem is much more efficient than previous lattice-based cryptosystems: the public key is of size O~(n2)\tilde{O}(n^2)O~(n2) and encrypting a message increases its size by a factor of O~(n)\tilde{O}(n)O~(n) (in previous cryptosystems these values are O~(n4)\tilde{O}(n^4)O~(n4) and O~(n2)\tilde{O}(n^2)O~(n2), respectively). In fact, under the assumption that all parties share a random bit string of length O~(n2)\tilde{O}(n^2)O~(n2), the size of the public key can be reduced to O~(n)\tilde{O}(n)O~(n).
Due to the advantages such as the worst-case hardness assumption, lattice-based cryptography is believed to the most promising research direction in post-quantum cryptography. As one of the two main hard problems commonly used in lattice-based cryptography, Learning With Errors (LWE) problem is widely used in constructing numerous cryptosystems. In order to improve the efficiency of lattice-based cryptosystems, Zhang et al. (2019) introduced the Asymmetric LWE (ALWE) problem. In this paper, the relation between the ALWE problem and the standard LWE problem is studied, and it shows that for certain error distributions the two problems are polynomially equivent, which paves the way for constructing secure lattice-based cryptosystems from the ALWE problem.
Cryptography is important for data confidentiality, integrity, and authentication. Public key cryptosystems allow for the encryption and decryption of data using two different keys, one that is public and one that is private. This is beneficial because there is no need to securely distribute a secret key. However, the development of quantum computers implies that many public-key cryptosystems for which security depends on the hardness of solving math problems will no longer be secure. It is important to develop systems that have harder math problems which cannot be solved by a quantum computer. In this project, two public-key cryptosystems which are candidates for quantum-resistance were implemented using Rust. The security of the McEliece system is based on the hardness of decoding a linear code which is an NP-hard problem, and the security of the Regev system is based off of the Learning with Errors problem which is as hard as several worst-case lattice problems [1], [2]. Tests were run to verify the correctness of the implemented systems and experiments were run to analyze the cost of replacing pre-quantum systems with post- quantum systems.
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The landscape of modern cryptography is fundamentally challenged by the advent of quantum computing. Current public-key cryptosystems, which form the bedrock of secure digital communication, rely on the computational hardness of problems such as integer factorization (RSA) and the discrete logarithm problem (ECC). However, Shor's algorithm, if implemented on a sufficiently powerful quantum computer, can efficiently break these schemes, rendering them insecure. This paper explores the critical transition from these classical number theory problems to lattice-based cryptography, presenting a comprehensive analysis of the motivations, theoretical underpinnings, and practical implications for forging post-quantum cryptographic security. We delve into the mathematical foundations of lattice problems, such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE), which are believed to resist known quantum algorithms. Through a thorough literature review, we trace the evolution of cryptographic thought, detailing the vulnerabilities of pre-quantum schemes and the emergence of lattice-based constructions as leading candidates for post-quantum standardization. The methodology outlines a comparative theoretical analysis of security, efficiency, and robustness against both classical and quantum attacks. Results demonstrate the compelling advantages of lattice-based schemes in terms of provable security reductions and resistance to quantum threats, albeit with considerations for increased key and ciphertext sizes. The discussion highlights the challenges and opportunities in the deployment and standardization of these new cryptographic primitives, emphasizing their critical role in securing future digital infrastructures. This work concludes that lattice-based cryptography offers a robust and theoretically sound pathway to establishing enduring security in a quantum-threatened world.
Our main result is a quantum public-key encryption scheme based on the Extrapolated Dihedral Coset problem (EDCP) which is equivalent, under quantum polynomial-time reductions, to the Learning With Errors (LWE) problem. For limited number of public keys (roughly linear in the security parameter), the proposed scheme is information-theoretically secure. For polynomial number of public keys, breaking the scheme is as hard as solving the LWE problem. The public keys in our scheme are quantum states of size O~(n)\tilde{O}(n)O~(n) qubits. The key generation and decryption algorithms require O~(n)\tilde{O}(n)O~(n) qubit operations while the encryption algorithm takes O(1)O(1)O(1) qubit operations.