Is IEACD Truly Secure, or Just a Patched Illusion?

Table of Links

Abstract and 1. Introduction

2. Description of two analyzed image encryption algorithms

3. Cryptanalysis of IEATD

4. Cryptanalysis of IEACD

5. Conclusion, Acknowledgements, and References

4. Cryptanalysis of IEACD

To cope with the insecurity problems of IEATD reported in [13], multiple confusion and diffusion operations are appended, making the algorithm become another algorithm IEACD. In fact, the original keystream generation mechanism indeed exists a serious pitfall, which leads to that the patched algorithm IEACD still cannot withstand chosen-plaintext attack. In this section, three weaknesses of IEACD are first analyzed to facilitate description of the following chosen-plaintext attack.

4.1. Three weaknesses of IEACD

• The real size of key space is much smaller than the expected one

Due to the limitation of finite-precision presentation, dynamics of any chaotic system is definitely degraded

when it is implemented in a digital device. As investigated in [14, 15, 16], the structure of the statemapping network (SMN) of a digitized chaotic system implemented with fixed-point precision e+1 is largely dominated by that with precision e. The short period problems of PRNG based on Logistic map (5) implemented in a digital device (with fixed-point arithmetic or floating-point arithmetic) were comprehensively discussed in [15]. As shown in Fig. 3, discretized Ikeda system obeys this rule also. No matter what the precision is, the SMN of discretized Ikeda system follows the following rules: 1) an SMN is composed of some weakly connected components; 2) there are some self-loops (an edge connecting a node to itself); 3) As for each connected component, there is one and only one cycle (including special cycle, selfloop), and every node evolves to it via a transient process; 4) Many nodes have two and only two parent nodes. Generating a pseudo-random number sequence by the orbits determined by a chaotic map is actually walking along a path of an SMN. Now, one can see that the period of a sequence by solving the discretized Ikeda system may be very short (even only one). So, there are a number of equivalent secret keys and invalid secret keys as for the function of IEATD and IEACD. Note that such pitfall always exists no matter how large the precision e gets.

• Insensibility of keystream generation mechanics

• Improper configuration of keystream

The two-round crossover diffusion is performed to resist chosen-plaintext attack and differential attack, but the keystream Z used in permutation is wrongly reused in the diffusion part, which makes the algorithm more insecure. From Eq. (6), (8) and (10), one has

• 4.2. Chosen-plaintext attack on IEACD*

4.2.1. Determining Z

Referring to Eq. (15) and XORing the two cipher-images, one has

In the process of determining Z 0 , as for a known or given element z 0 (i), one attempts to find its neighbor by verifying whether the corresponding equation holds. Therefore, Z 0 is reconstructed by seeking the relative positions of elements. As mentioned before, after constructing Z 0 , the permutation vector Z can be restored via z(u(i)) = z 0 (i) for i = 0 ∼ (HW − 1)

Now, the attack in case of b ≤ 1 is discussed. In fact, the special cases of b can be identified through Eq. (18a) and (18b). If b = 1, since z 0 (b − 1) is the first element in Z 0 , no element can be found via Eq. (18b). Hence, one should determine z 0 (0) via Eq. (18a) and then find the remainder of Z 0 through Eq. (18c). If b = 0, the attack is failed. Since Eq. (18a) and (18b) both no longer hold, one would directly attempt to determine z 0 (1) through

5. Conclusion

This paper analyzed security performance of an image encryption algorithm based on a first-order time-delay system IEATD and the enhanced version IEACD. Although another research group proposed a chosen-plaintext attack on IEATD, we presented an enhanced attack using the correlation between adjacent vectors of one plain-image and the corresponding cipher-image. Although IEACD is designed by the attacking group with intention to fix the security defects of IEATD, there still exist some security pitfalls, such as invalid secret keys, insensibility of keystream generation mechanics, and improper configuration of keystream. Based on these, we designed an efficient chosen-plaintext attack and verified it with extensive experiments. The serious insecurity of the two algorithms cannot be improved by simple modifications. They can work as typical counterexamples to remind us to recast scenario-oriented image encryption algorithms following the guidelines and lessons summarized in [9, 7, 3].

Acknowledgement

This work was supported by the National Natural Science Foundation of China (no. 61772447), Scientific Research Fund of Hunan Provincial Education Department (no. 20C1759), and Science and Technology Program of Changsha (no. kq2004021).

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