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Known Plaintexts Attack on Polynomial based Homomorphic Encryption

This paper presents the security analysis of symmetric homomorphic cryptosystems based on polynomial ring homomorphisms.
 First we describe an approach to carry out a known plaintexts attack (KPA) on such cryptosystems in the case when the plaintexts space is a finite field. The approach uses Euclidean algorithm and Berlekamp algorithm for polynomial factorization. We give theoretical estimations of probability to recover a secret key and provide experimental results. We analyse the complexity of our attack and give recommendations how it can be implemented in parallel.
 Second we discuss how introduced approach can be adapted for the case when the plaintexts space is a ring of integers modulo composite number. Here a particular attention is given to modules that are products of two primes. Results of computer experiment are given.
 The presented known plaintexts attack on cryptosystems allows to recover secret key with overwhelming probability if the number of pairs (ciphertext, plaintext) is more than five.