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The cryptosystem based on the Learning-with-Errors (LWE) problem is considered as a post-quantum cryptosystem, because it is not based on the factoring problem with large primes which is believed to be easily solved by a quantum computer. Moreover, the LWE-based cryptosystem allows fully homomorphic arithmetics so that two encrypted variables can be added and multiplied without decrypting them. This chapter provides a comprehensive introduction to the LWE-based cryptosystem with examples. A key to the security of the LWE-based cryptosystem is the injection of random errors in the ciphertexts, which however hinders unlimited recursive operation of homomorphic arithmetics on ciphertexts due to the growth of the error. We show how this limitation can be overcome for dynamic feedback controllers that guarantee stability of the closed-loop system when the system matrix of the controller consists of integers. Finally, we illustrate through MATLAB codes how the LWE-based cryptosystem can be customized to build a secure feedback control system. This chapter is written for the control engineers who do not have background on cryptosystems.

Comprehensive Introduction to Fully Homomorphic Encryption for Dynamic Feedback Controller via LWE-Based Cryptosystem