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Cryptanalysis of an Image Encryption Scheme Based on a Compound Chaotic Sequence
Chengqing Li, Shujun Li, Guanrong Chen, Wolfgang A. Halang
TL;DR
The paper examines whether a recently proposed compound-chaotic image cipher is secure. It analyzes the scheme and reports that three chosen plain-images suffice to break it, alongside other security weaknesses. The authors conclude that it is not secure enough for applications requiring a high level of security.
Problem
The security of a recently proposed compound-chaotic image encryption scheme is under study in the context of protecting bulky, redundant multimedia data.
Source: §1 · pp. 1–2Method
Results
The scheme can be broken with three chosen plain-images and also has weak and equivalent keys, low plaintext sensitivity, and an insufficiently random generated sequence.
Source: §5 · p. 10Takeaways & Limitations
The authors conclude that the scheme is not secure enough and do not recommend it for applications requiring a high level of security.
Source: §5 · p. 10The plaintext-sensitivity concern is qualified by key reuse: with a non-repeated key, two slightly different plaintexts are rarely encrypted by the same keystream.
Source: §3.4 · pp. 7–8
Abstract
from arXiv · showhide
Recently, an image encryption scheme based on a compound chaotic sequence was proposed. In this paper, the security of the scheme is studied and the following problems are found: (1) a differential chosen-plaintext attack can break the scheme with only three chosen plain-images; (2) there is a number of weak keys and some equivalent keys for encryption; (3) the scheme is not sensitive to the changes of plain-images; and (4) the compound chaotic sequence does not work as a good random number resource.
1 Introduction
The paper examines the security of a compound-chaotic image encryption scheme against the challenges of protecting bulky, redundant multimedia data. It reports attacks and other weaknesses in the scheme.
1 Introduction
- The paper studies a recently proposed image encryption scheme based on a compound chaotic sequence.Image encryption is motivated by multimedia data properties, including the bulky size and strong redundancy of uncompressed data.Source: §1 · pp. 1–2
- The scheme combines XOR pixel substitutions with circular-shift permutations of rows and columns.A compound pseudo-random sequence controls the XOR substitutions, while two chaotic maps determine the row and column shifts, respectively.Source: §1 · p. 2
- The reported weaknesses include a break with three chosen plain-images, weak and equivalent keys, low plaintext sensitivity, and insufficient randomness.The paper also outlines its organization: scheme description, security analysis, a differential chosen-plaintext attack, and conclusions.Source: §1 · p. 2
2 The image encryption scheme under study
The section specifies the image representation and describes the cipher's chaotic-map initialization and encryption components.
2 The image encryption scheme under study
- The analysis assumes an M × N 8-bit grayscale image, representing a 24-bit RGB image as a 3M × N grayscale array for encryption.The authors state that the algorithm itself is independent of plain-image structure and can encrypt any 2-D byte array.Source: §2 · p. 2
- The secret key consists of two 10^-14-precision floating-point initial states for maps f0(x) = 8x^4 − 8x^2 + 1 and f1(y) = 4y^3 − 3y.The states are x0 and y0, each in [−1, 1].Source: §2 · p. 3
- Initialization iterates a compound map from k0 = k1 = 0, updating one chaotic-map index according to whether xk0 + yk1 is negative.The iterations construct the compound chaotic sequence; the index for the selected map is incremented at each step.Source: §2 · p. 3
- Encryption combines an XOR substitution part with permutation parts that use horizontal circular shifts.The supplied passages identify the XOR and horizontal-shift components but do not provide their full operational steps.Source: §2 · pp. 3–4
(3) Permutation part – vertical circular shift operations
The vertical-shift stage takes the intermediate image as input and produces the cipher-image; the encryption procedure also has a compact combined form.
(3) Permutation part – vertical circular shift operations
- The vertical circular shifts transform I** into cipher-image I′, with source coordinates determined by the row and column shift sequences.The indices are j* = (j − S3(i)) mod N and i* = (i − S2(j*)) mod M.Source: (3) Permutation part – vertical… · p. 4
- The encryption procedure is also represented in a compact form, and decryption reverses the encryption after the same initialization.The inverse indices are i* = (i + S2(j)) mod M and j* = (j + S3(i*)) mod N.Source: (3) Permutation part – vertical… · p. 4
3 Some security problems
The section identifies weaknesses in the compound chaotic sequence, key space, and plaintext sensitivity, and argues that theory cited for a different map does not apply to the implemented scheme.
3.1 Insufficient randomness of the compound chaotic sequence
- The cited four randomness tests came from an intermediate FIPS PUB 140-2 edition and were later removed; evidence shown for one key was limited, and repeat results disagreed with Table 2.Source: §3.1 · pp. 4–5
- 100 randomly generated sequences failed to establish the compound chaotic function as a good random number generator under the test suite at significance level 0.01.The tests were performed on sequences generated for encrypting 256 × 256 images.Source: §3.1 · p. 5, Table 2 · p. 6
3.2 Weak keys
- Some keys make encryption parts fail because chaotic-map fixed points produce constant or otherwise degenerate substitution and shift sequences.Examples include y0 = 1, which gives S1(k) ≡ 255 and S3(i) ≡ N − 1, and y0 = −1, which gives S3(i) ≡ 0.Source: §3.2 · p. 6
3.3 Equivalent keys
- The keys (x0, y0) and (−x0, y0) are equivalent when |y0| ≥ |x0|, within the rational-number field considered.Source: §3.3 · p. 7
3.4 Low sensitivity to plaintext changes
- Changing one bit of a plaintext pixel affects only the corresponding bit of one cipher-image pixel; without key reuse, slightly different plaintexts rarely share a keystream.Source: §3.4 · pp. 7–8
3.5 A remark on the compound chaotic map
- The scheme iterates f0 and f1 separately but not independently, so theoretical results for the different compound map F(x) do not apply to this design.Source: §3.5 · p. 8
4 Differential chosen-plaintext attack
The attack cancels the scheme’s XOR substitutions by differencing chosen plaintext–ciphertext pairs, leaving the row and column permutations to recover. The authors report that three chosen plain-images suffice to break the scheme.
4 Differential chosen-plaintext attack
- Only three chosen plain-images break the scheme, despite its authors’ claim that it withstands chosen-plaintext attack efficiently.Fig. 2 demonstrates the proposed attack.Source: §4 · pp. 8–9, Figure 2 · p. 10
- XORing two ciphertexts cancels the XOR substitutions, leaving the correspondingly shifted plaintext difference and exposing the permutation operations.The paper describes this as a divide-and-conquer attack that breaks the encryption components separately.Source: §4 · p. 8
- Two differential images recover vertical shifts when each row of I1 ⊕ I2 is constant and each column has an identifiable pattern.The shift is read from the new position of the sole black pixel in each column.Source: §4 · p. 9
- After vertical shifts are recovered, I1 and I3 reveal horizontal shifts when each column of I1 ⊕ I3 is constant and each row has an identifiable pattern.The shift is read from the new position of the sole black pixel in each row.Source: §4 · p. 9
- Once shifts are known, the cipher reduces to an XOR-based stream cipher, and experiments show recovered sequences decrypting a new cipher-image.The demonstration uses 256 × 256 chosen images, including Peppers, and reports the decrypted result in Fig. 2h.Source: §4 · p. 9
5 Conclusion
The authors conclude that the scheme has multiple security weaknesses and is not secure enough. They therefore do not recommend it for applications requiring a high level of security.
5 Conclusion
- The scheme can be broken with only three chosen plain-images.Source: §5 · p. 10
- The scheme has weak and equivalent keys and is not sufficiently sensitive to changes in plain-images.Source: §5 · p. 10
- Its compound chaotic sequence is not sufficiently random for secure encryption, so the authors advise against using the scheme where high security is required.Source: §5 · p. 10