Post-Quantum Cryptosystems: Open Problems and Solutions. Lattice-Based Cryptosystems

The paper provides an overview of the main approaches to the construction of post-quantum cryptographic systems that are currently used. The area of lattice-based cryptography is analyzed in detail. We give the description and characterization of some known lattice-based cryptosystems whose resilience is based on the complexity of the shortest vector problem, learning with errors problem, and their variations. The main approaches to solving the problems from lattice theory, on which attacks on the corresponding cryptosystems are based, are analyzed. In particular, some known theoretical estimates of time and memory complexity of lattice basis reduction and lattice sieving algorithms are presented.