Several works have characterized weak instances of the Ring-LWE problem by exploring vulnerabilities arising from the use of algebraic structures. Although these weak instances are not addressed by worst-case hardness theorems, enabling other ring instantiations enlarges the scope of possible applications and favors the diversification of security assumptions. In this work, we extend the Ring-LWE problem in lattice-based cryptography to include algebraic lattices, realized through twisted embeddings. We define the class of problems Twisted Ring-LWE, which replaces the canonical embedding by an extended form. By doing so, we allow the Ring-LWE problem to be used over maximal real subfields of cyclotomic number fields. We prove that Twisted Ring-LWE is secure by providing a security reduction from Ring-LWE to Twisted Ring-LWE in both search and decision forms. It is also shown that the twist factor does not affect the asymptotic approximation factors in the worst-case to average-case reductions. Thus, Twisted Ring-LWE maintains the consolidated hardness guarantee of Ring-LWE and increases the existing scope of algebraic lattices that can be considered for cryptographic applications. Additionally, we expand on the results of Ducas and Durmus (Public-Key Cryptography, 2012) on spherical Gaussian distributions to the proposed class of lattices under certain restrictions. As a result, sampling from a spherical Gaussian distribution can be done directly in the respective number field while maintaining its format and standard deviation when seen in Zn via twisted embeddings.

EntropyAug 26, 202114 citations

Cryptography with Ring Algorithm – LWE

Paramita Lumban Gaol, Stephen Buulolo, Aritmen Andreas L. Manurung +1 more

Ring Learning With Errors (Ring-LWE) is one of the basic schemes used to develop cryptographic algorithms that are resistant to quantum attacks, offering better computational efficiency compared to standard Learning With Errors (LWE). Ring-LWE uses the algebraic characteristics of polynomial rings to create a robust and fast encryption system. This study implements and evaluates the Ring-LWE algorithm in the context of public key exchange and encryption. Simulations are performed with various security parameters, including modulus size and error rate, to test the resistance to brute force attacks and lattice-based attacks. The experimental results show that the Ring-LWE algorithm can achieve a balance between security and computational efficiency, with shorter processing time than conventional LWE while maintaining a high level of security even in the face of quantum computers. From this study, it can be concluded that Ring-LWE is an excellent choice for application in post-quantum security systems, especially in encrypted communication and digital authentication applications. The next step can be directed at optimizing resource utilization and testing against various types of attacks.