# Learning with Errors: A Lattice-Based Keystone of Post-Quantum Cryptography
## 1 Introduction  
The evolution of quantum computing involves a convergence of advancements in quantum theory, hardware engineering, algorithm development and interdisciplinary research. The field continues to progress rapidly, and while large-scale, fault-tolerant quantum devices suitable for broad applications may be some years away, the steady evolution suggests promising future potential for quantum technologies in various domains, like cryptography, economy, drug development, geology and others. The institutionalization by law of the United States' government in the 2018 National Quantum Initiative Act, \"to accelerate quantum research and development for the economic and national security of the United States\" [1], is indicative of this potential. Quantum computing uses quantum physics properties to process information in ways that classical computers cannot [2]. The foundations of quantum devices lie in the principles of quantum mechanics, such as qubits--the equivalent of the bit in a classical computer--superposition and entanglement. With the development of quantum algorithms, quantum machines process vast amounts of information simultaneously, providing potential speedups for highly interconnected and parallel computations [3]. Quantum computers have immense potential to revolutionize various fields, like cryptography, and optimize complex systems more efficiently than classical algorithms, offering speedups for certain types of tasks. Since antiquity, cryptography has played a crucial role in everyday life, ensuring the confidentiality, integrity and authenticity of information exchanged over communication channels or devices [4]. Cryptographic schemes encompass various methodologies, algorithms and protocols to protect information and secure data. A principal issue in cryptography is the secure process of sharing cryptographic keys between communicating parties to enable encrypted data. Key exchange ensures that both parties establish a shared secret key without exposing it to potential eavesdroppers or adversaries, and great progress has been made in this field with the evolution of quantum cryptography. Since itsinitial presentation in 1982 [5], the term \"quantum cryptography\" has drawn the attention, investigation and financial support of academics, governments and businesses. In 1984, [PERSON] and [PERSON] introduced the first quantum key distribution protocol [6], a groundbreaking concept that paved the way for the appearance and development of other protocols [7; 8]. Quantum cryptography, nevertheless, continues to be a pressing matter under investigation, with promising solutions in the fields of quantum key distribution, quantum encryption [9; 10] and quantum digital signatures [11]. The above recent advances in the quantum key distribution process and quantum encryption procedure have improved the secret key rate and the signature rate and offer security advantages over typical schemes. However, with the introduction of a sufficiently large quantum computer, there is no longer any security in present encryption systems. [PERSON]'s algorithm discovery demonstrated that a quantum computer, if realized at scale, could solve certain problems significantly faster than classical computers, like the integer factorization problem and the discrete logarithm problem [12]. The potential impact of [PERSON]'s algorithm on cryptography is major, as it poses a threat to the security of widely used cryptographic schemes, such as RSA and elliptic curve cryptography (ECC). So, this impact has spurred interest in post-quantum cryptography, which aims to develop encryption algorithms and cryptographic protocols that are resistant to quantum attacks. As the development of quantum computing progresses, post-quantum cryptography aims to develop encryption algorithms and cryptographic protocols that remain secure even in the presence of quantum computers.