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Article title CRYPTANALYSIS OF SYMMETRIC FULLY HOMOMORPHIC LINEAR CRYPTOSYSTEMS BASED ON NUMBERS FACTORIZATION PROBLEM
Authors A.V. Trepacheva
Section SECTION III. CRYPTOGRAPHIC PROTECTION OF INFORMATION
Month, Year 05, 2015 @en
Index UDC 519.7: 004: 681.5
DOI
Abstract This paper considers the fully homomorphic cryptosystems based by their authors on a factorization problem. In particular, the paper analyses the security of cryptosystems whose ciphertexts are matrices with elements modulo composite number that is hard to factorize, while encryption and decryption procedures are similarity transformations. We carefully analyze the properties of ciphertexts produced by their encryption algorithms. And on the base of this analysis main vulnerabilities of the cryptosystems are highlighted. The main focus of this paper is placed on analysis of resistance against known plaintext attack of two recently proposed cryptosystems belonging to this type. In particular, we discuss one strategy of known plaintext attack on them proposed in literature. It is based on solving of linear system modulo composite number. Our main result in comparison with predecessors is providing a strict theoretical estimation of probability to recover secret key for different number of intercepted pairs (plaintext, ciphertext) using this strategy. Also practical estimations of probability to find a key based on computer experiments are given. They correlate well with theoretical predictions. Our analysis shows that the first considered cryptosystem is vulnerable to known plaintext attack based on linear system solving. However to recover key with probability ≈1 one needs to have the number of pairs (plaintext, ciphertext) depending polylogarithmically on the number of factors in exploiting composite modulus. And for the less number of pairs the cryptosystem is secure. This to some extent corresponds to security estimations given by the authors of cryptosystem. As for the second cryptosystem, its breaking requires much less number of pairs and thus its security level doesn’t match with level stated by the authors.

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Keywords Fully homomorphic cryptosystem; known plaintext attack; secure cloud computing; factorization problem.
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