Integral cryptanalysis of reduced FOX block cipher

FOX is a family of block ciphers presented recently, which is based upon some results of provable security and has high performances on various platforms. In this paper, we construct some distinguishers between 3-round FOX and a random permutation of the blocks space. By using integral attack and collision-searching techniques, the distinguishers are used to attack 4, 5, 6 and 7-round FOX64, 4 and 5-round FOX128. The attack is more efficient than previous integral attacks on FOX. The complexity of improved integral attack is 2(77.6) on 4-round FOX128, 2(205.6) against 5-round FOX128 respectively. For FOX64, the complexity of improved integral attack is 2(45.4) on 4-round FOX64, 2(109.4) against 5-round FOX64, 2(173.4) against 6-round FOX64, 2(237.4) against 7-round FOX64 respectively. Therefore, 4-round FOX64/64, 5-round FOX64/128, 6-round FOX64/192, 7-round FOX64/256 and 5-round FOX128/256 are not immune to the attack in this paper.