# Learning with Errors: A Lattice-Based Keystone of Post-Quantum Cryptography
## 1 Introduction
Some techniques and approaches that are used involve code-based cryptography, multivariate polynomial cryptography, hash-based cryptography and lattice-based cryptography [13]. Early discussions of lattices can be found in the 18 th century in the works of mathematicians such as [PERSON] and [PERSON], and in the late 19 th and early 20 th centuries, [PERSON] played a pivotal role in advancing the study of lattices and their geometrical properties [14]. The field of lattice theory, as a branch of abstract algebra and order theory, gained momentum in the mid-20 th century with the contributions of [PERSON] [15] and [PERSON] [16]. It was not until the early 21 st century that lattice-based cryptography emerged as a post-quantum cryptographic solution. The hardness of certain lattice problems, such as the shortest vector problem (SVP), the closest vector problem (CVP) and the learning with errors (LWE) problem, formed the basis for developing cryptographic schemes resistant to quantum attacks. The learning with errors (LWE) cryptosystem is critically important for post-quantum cryptography and is regarded as one of the leading candidates for creating cryptographic schemes resilient to attacks from both classical and quantum computers. It is based on the namesake LWE problem, which is a fundamental problem in lattice-based cryptography. The LWE cryptographic scheme is a cornerstone of many cryptographic protocols whose security is founded on the belief that solving LWE on a classical or quantum computer is computationally difficult, even for quantum computers, thus providing resistance to attacks from powerful quantum algorithms, like [PERSON]'s algorithm. Various cryptographic primitives crucial for post-quantum security, both encryption and digital signature schemes, exploit the complexity of solving LWE to offer security guarantees in a post-quantum setting. These schemes are characterized by their efficiency and practicality and undergo rigorous analysis and scrutiny to ensure their viability and security. Ongoing research focuses on refining LWE-based cryptographic constructions, optimizing their efficiency and exploring new variants to enhance security further. This active research aims to strengthen and improve the practicality of LWE-based schemes. In this sense, we introduce a variant of the learning with errors cryptographic protocol without modification of its original structure by adding an extra step in the procedure of key generation in the algorithm. In this manuscript, we present a variation of the LWE cryptosystem by introducing a transformation to the protocol's key generation procedure. This extra phase preserves the security and efficiency of the protocol without increasing its complexity because the operations added to it are simply matrix additions and multiplications and thus do not interact with the remaining parts of the LWE algorithm's structure. After selecting a specific transformation mapping \(f\), we provide discrete implementations in small dimensions and propose specific cases of our variation. The rest of the paper is organized as follows. In Section 2, we denote the main definitions and the essential mathematical background, while in Section 3, the basic lattice theory and the computational problems in lattices are presented. In Section 4, we introduce the most widely known and analyzed lattice-based cryptosystems, and in Section 5, we present the learning with errors problem and the original LWE cryptosystem, and we propose a new efficient variant of it. Finally, Section 6 concludes this work, and some future work directions are given.