# Learning with Errors: A Lattice-Based Keystone of Post-Quantum Cryptography
## 4 Lattice-Based Cryptography  
Although lattices have been studied for centuries, lattice-based cryptographic protocols are considered to be one of the most modern and futuristic tools in the science of cryptology. One of the main arguments for this fact is that lattice-based protocols are considered to be resistant algorithms under quantum attacks and have immediately become a choice for post-quantum cryptography, as we discussed above. However, there are several reasons why lattice-based cryptography is widely studied and preferred over other forms of cryptography. An additional factor supporting the preference for lattice-based cryptography is the main structure of a lattice-based protocol. The majority of these algorithms are efficient and parallelizable. Moreover, they are characterized by simplicity, and their operations are uncomplicated as they are based on linear and simple operations on matrices and vectors modulo nearly small integers. Another important feature of lattice-based cryptographic algorithms is the variety of applications in the field of encryption. Lattice-based protocols can successfully perform fully homomorphic encryption, i.e., the computation of encrypted data or ciphertext without revealing any information about the data. Consequently, lattice-based cryptographic schemes are flexible, versatile and powerful for encryption, digital signatures, key transport and authenticated key exchange. In recent years, significant steps have been taken in the construction of this type of cryptosystem based on lattice theory, though it remains an open field of research [34]. The cryptographic schemes that are based on lattices' structure offer strong security guarantees. They are based on the hypothesis that some mathematical problems in lattice structures are difficult. The security of lattice-based cryptography depends on the worst-case hardness assumptions of the lattice problems. Lattice-based cryptographic schemes will continue to provide a solid basis for safe communication and data protection in a post-quantum world if these certain computational problems are still difficult to resolve, even for quantum computing devices. Lattice cryptography is gaining the interest of scientists and cryptographers because of the interesting and practical properties of lattices. The development of lattice cryptography has been significant and fundamental to the art of encryption. As a result of this evolution, various cryptosystems have been introduced that are now considered cornerstones for post-quantum cryptography. Subsequently, we present some primitive, well-known and performed constructions of lattice-based cryptography. In 1996, [PERSON] proposed the first cryptographic primitive based on lattices and gave the worst-case reduction [17], and in 1997, [PERSON] and [PERSON] introduced a lattice-based asymmetric key cryptosystem that is performed in \(\mathbb{R}^{n}\)[35]. This work attracted a lot of interest because of its unexpected security proof that was predicated on the worst-case assumptions [35]. [PERSON]'s brilliant idea inspired numerous researchers to turn their academic interest to lattice-based cryptosystems. Although the presented work is indeed amazing, it is regarded as impractical and inefficient due to its long-lasting key generation process, sluggish encryption and large key sizes in larger dimensions. The cryptosystem is not at all functional in dimensions where the lattice problems given in the security proof are hard to solve. Around the same time, in 1996, [PERSON], [PERSON] and [PERSON] proposed another public key cryptographic scheme that can be explained via specially constructed lattices, the NTRU cryptosystem [36]. The NTRU is one of the fastest public key cryptography systems and is based on the shortest vector problem in lattices. It uses polynomial rings to encrypt and decrypt data, as the operations of the cryptographic protocol take place in the ring of polynomials.