Algebraically Structured LWE, Revisited

Chris Peikert, Zachary Pepin

Yan Wang, Mingqiang Wang

Andrew Mendelsohn, Cong Ling

We construct a provably-secure structured variant of Learning with Errors (LWE) using nonassociative cyclic division algebras, assuming the hardness of worst-case structured lattice problems, for which we are able to give a full search-to-decision reduction, improving upon the construction of Grover et al. named `Cyclic Learning with Errors' (CLWE). We are thus able to create structured LWE over cyclic algebras without any restriction on the size of secret spaces, which was required for CLWE as a result of its restricted security proof. We reduce the shortest independent vectors problem in ideal lattices, obtained from ideals in orders of such algebras, to the decision variant of LWE defined for nonassociative CDAs. We believe this variant has greater security and greater freedom with parameter choices than CLWE, and greater asymptotic efficiency of multiplication than module LWE. Our reduction requires new results in the ideal theory of such nonassociative algebras, which may be of independent interest. We then adapt an LPR-like PKE scheme to hold for nonassociative spaces, and discuss the efficiency and security of our construction, showing that it is immune to certain subfield attacks. Finally, we give example parameters to construct algebras for cryptographic use.

Gérald Gavin, Stéphane Bonnevay

Chris Brzuska, Akın Ünal, Ivy K. Y. Woo

Rachid El Bansarkhani, Johannes A Buchmann

Abstract. Lattice-based encryption schemes still suffer from a low message throughput per ciphertext. This is mainly due to the fact that the underlying schemes do not tap the full potentials of LWE. Many constructions still follow the one-time-pad approach considering LWE instances as random vectors added to a message, most often encoded bit vectors. Recently, at Financial Crypto 2015 El Bansarkhani et al. proposed a novel encryption scheme based on the A-LWE assumption (Augmented LWE), where data is embedded into the error term without changing its target distributions. By this novelty it is possible to encrypt much more data as compared to the traditional one-time-pad approach and it is even possible to combine both concepts. In this paper we revisit this approach and propose amongst others several algebraic techniques in order to significantly improve the message throughput per ciphertext. Furthermore, we give a thorough security analysis as well as an efficient implementation of the CCA1-secure encryption scheme instantiated with the most efficient trapdoor construction. In particular, we attest that it even outperforms the CPA-secure encryption scheme from Lindner and Peikert presented at CT-RSA 2011.