Improved Integral Attacks on MISTY1

We present several integral attacks on MISTY1 using the FO Relation, which is derived from Sakurai-Zheng Property used in previous attacks. The FO Relation is a more precise form of the Sakurai-Zheng Property such that the functions in the FO Relation depend on 16-bit inputs instead of 32-bit inputs used in previous attacks, and that the functions do not change for different keys while previous works used different functions. Thus we can reduce the attack complexity by a factor of 2. We use the FO Relation to improve the 5-round integral attack. The data complexity of our attack, 2 chosen plaintexts, is the same as previous attack, but the running time is reduced from 2 encryptions to 2 encryptions. The attack is then extended by one more round with a data complexity of 2 chosen plaintexts and a time complexity of 2 encryptions. By exploring the key schedule weakness of the cipher, we also present a chosen ciphertext attack on 6-round MISTY1 with all the FL layers with a data complexity of 2 chosen ciphertexts and a time complexity of 2 encryptions. Compared with other attacks on 6-round MISTY1, our attack has the least data complexity.