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JOURNAL OF ELECTRONICS INFORMATION TECHNOLOGY

On the Hardness of the Asymmetric Learning With Errors Problem

Jiang Zhang, Shuqin Fan

Published Feb 19, 2020• 0 citations
Publisher

Abstract

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.

DOI
10.11999/jeit190685

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•Dec 5, 2025•0 citations
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.
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).
This chapter introduces post-quantum cryptography. It describes lattice-based cryptographic constructions of post-quantum cryptography. Lattice-based cryptographic constructions are one of candidates of post-quantum cryptography, which are believed to be secure against quantum computers. Learning with errors (LWE) is one of the most promising primitives in many usages due to its lightweight operation and rigorous security reduction against the worst-case of the lattice problems that are considered to be hard to solve even after the advance of quantum computers. After the research about the connection between the LWE problem and some lattice problems, some variants of LWE, of which the secret distributions are modified from the uniform distribution, were proposed. Due to the efficiency and compactness of ring-LWE, many lattice-based cryptosystems are constructed as ring-LWE based, rather than LWE-based. The chapter explains a ring variant of Lizard scheme, called RLizard, IND-CCA secure encryption scheme.
The Learning with Errors (LWE) problem is a fundamental hardness assumption for post-quantum lattice-based cryptography. Standard LWE schemes rely on discrete Gaussian error distributions, which incur high sampling overhead. This paper explores the feasibility of replacing the Gaussian distribution with a novel non-Gaussian alternative—the Step Distribution—within Regev’s classic LWE public-key encryption scheme. The Step Distribution partitions the error range into equal-width steps, with probabilities proportional to Gaussian weights at step midpoints. The distribution is formally defined, its asymptotic convergence to the discrete Gaussian is proved, and conditions under which the decryption error probability becomes negligible are derived. The Step Distribution offers a tunable parameter—the number of steps—that allows a trade-off between security (closeness to the Gaussian) and computational efficiency. This work provides a flexible error distribution for LWE-based cryptosystems, particularly suitable for resource-constrained environments.
The Learning with Errors (LWE) problem is a special average-case lattice problem, which has proved to be as hard as worst-case lattice problems. By this property, it has become the cornerstones for a series of cryptographic applications under the security assumption of worst-case lattice problems. However, the efficiency of these applications are not high. Therefore, how to improve the efficiency of cryptographic applications based on LWE is an urgent problem to be solved. In this paper, a new variant of LWE is proposed. By using trapdoor composed by lattice-based matrix and its dual orthogonal matrix, we construct a more efficient public key encryption (PKE) scheme based on the variant. Furthermore, by the hardness of the variant problem, we prove the new PKE scheme is chosen plaintext attack (CPA) secure. At last, we analyze the homomorphism property of the new PKE scheme.
Ever since the appearance of quantum computers, prime factoring and discrete logarithm-based cryptography have been questioned, giving birth to the so-called post-quantum cryptography. The most prominent field in post-quantum cryptography is lattice-based cryptography, protocols that are proved to be as difficult to break as certain hard lattice problems like Learning with Errors (LWE) or Ring Learning with Errors (R-LWE). Furthermore, the application of cryptographic techniques to different areas, like electronic voting, has also nourished a great interest in distributed cryptography. In this work, we will give two original threshold protocols based in the lattice problem R-LWE: one for key generation and one for decryption. We will prove them both correct and secure under the assumption of hardness of some well-known lattice problems. Finally, we will give a rough implementation of the protocols in C to give some tentative results about their viability, in particular our model generates keys in the order of 103 ms and decrypts and encrypts in the order of 102 ms.
The security of conventional public-key cryptography schemes is based on mathematical problems which are considered to be infeasible to solve by classical computers in a reasonable amount of time. However, the rise of quantum computers might break this security in the future. Post-quantum cryptography (PQC) aims to achieve secure schemes which are resistant against attacks assisted by quantum computers. Public-key cryptography refers to asymmetric cryptography in which encryption is done with a public key and decryption is done with a private key. Characteristic properties of public-key cryptography include the possibility to derive the private key from public information by solving some presumably hard problem. The hardness of the problem constitutes the basis for the system's security. In the post-quantum era, the security must be based on problems that are too hard to solve even for quantum computers. In this thesis, we will study lattice-based cryptography which is believed to serve as a good candidate for secure post-quantum public-key cryptography. Our main attention is in the problem of learning with errors, which is in the base of some lattice-based schemes and is linked to the hard problem of finding short vectors in a lattice.