Cryptanalysis of a Homomorphic Public-Key Cryptosystem

The aims of this research are to give a precise description of a new homomorphic public-key encryption scheme proposed by Grigoriev and Ponomarenko [7] in 2004 and to break Grigoriev and Ponomarenko homomorphic public-key cryptosystem. Firstly, we prove some properties of linear fractional transformations. We analyze the X_n-representation algorithm which is used in the decryption scheme of Grigoriev and Ponomarenko homomorphic public-key cryptosystem and by these properties of the linear fractional transformations, we correct and modify the X_n-representation algorithm. We implement the modified X_n-representation algorithm by programming it and we prove the correctness of the modified X_n-representation algorithm. Secondly, we find an explicit formula to compute the X(n,S)-representations of elements of the group \Lambda_n. The X(n,S)-representation algorithm is used in the decryption scheme of Grigoriev and Ponomarenko homomorphic public-key cryptosystem and we modify the X(n,S)-representation algorithm. We implement the modified X(n,S)-representation algorithm by programming it and we justify the modified X(n,S)-representation algorithm. By these two modified X_n-representation algorithm and X(n,S)-representation algorithm, we make its decryption scheme more efficient. Thirdly, by using those properties of the linear fractional transformations, we design new X_1-representation algorithms I and II and we mainly use these two X_1-representation algorithms to break Grigoriev and Ponomarenko homomorphic public-key cryptosystem. We implement the algorithms by programming them and we prove the correctness of these two algorithms. Fourthly, we analyze Grigoriev and Ponomarenko homomorphic public-key cryptosystem and we give a clear description of Grigoriev and Ponomarenko scheme with a practical example. We also consider implementation issues for its practical applications. Lastly, we show several attack methods with examples and experiments according as the attack methods and so we break Grigoriev and Ponomarenko homomorphic public-key cryptosystem