Cracking a hierarchical chaotic image encryption algorithm based on permutation

• Security performance of an encryption algorithm called HCIE is analyzed in detail.
• Hierarchical permutation-only encryption schemes are less secure than normal ones.
• Security of HCIE against ciphertext-only attack was reported being overestimated.

In year 2000, an efficient hierarchical chaotic image encryption (HCIE) algorithm was proposed, which divides a plain-image of size M×NM×N with T possible value levels into K blocks of the same size and then operates position permutation on two levels: intra-block and inter-block. As a typical position permutation-only encryption algorithm, it has received intensive attention. The present paper analyzes specific security performance of HCIE against ciphertext-only attack and known/chosen-plaintext attack. It is found that only  O(⌈logT(M·N/K)⌉)O(⌈logT(M·N/K)⌉)known/chosen plain-images are sufficient to achieve a good performance, and the computational complexity is  O(M·N·⌈logT(M·N/K)⌉)O(M·N·⌈logT(M·N/K)⌉), which effectively demonstrates that hierarchical permutation-only image encryption algorithms are less secure than normal (i.e., non-hierarchical) ones. Detailed experiment results are given to verify the feasibility of the known-plaintext attack. In addition, it is pointed out that the security of HCIE against ciphertext-only attack was much overestimated.