Cryptanalysis of two cryptosystems based on multiple intractability assumptions

Two public key cryptosystems based on the two intractable number-theoretic problems, integer factorisation and simultaneous Diophantine approximation, were proposed in 2005 and 2009, respectively. In this study, the authors break these two cryptosystems for the recommended minimum parameters by solving the corresponding modular linear equations with small unknowns. For the first scheme, the public modulus is factorised and the secret key is recovered with the Gauss algorithm. By using the LLL basis reduction algorithm for a seven-dimensional lattice, the public modulus in the second scheme is also factorised and the plaintext is recovered from a ciphertext. The author´s attacks are efficient and verified by experiments which were done within 5s.