The power of quantum computers is increasing from year to year. If a sufficiently powerful quantum computer is developed, it will be possible to break the cryptosystems that are massively used today. If we still want to communicate securely, it is necessary to introduce new cryptosystems that will be resistant to quantum computer attacks. These cryptosystems must be based on problems that are considered difficult on both classical and quantum computers. Learning with errors is one of those problems. We present a cryptosystem that is based on generalization of this problem to polynomial rings over finite fields. This cryptosystem supports fully homomorphic encryption. This means that we can compute any function on cryptograms and get the same encrypted result as if we performed this function on unencrypted data. Homomorphic encryption opens up wide possibilities of usage. As an example, we present an algorithm for private set intersection, which we also implement with the SEAL library.
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