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Cryptanalyzing image encryption using chaotic logistic map

Chengqing Li, Tao Xie, Qi Liu, Ge Cheng

arXiv:1301.4089v2cs.CR

TL;DR

Security scrutiny of chaos-based encryption remains limited, and the paper examines a logistic-map image cipher’s vulnerability. It uses the stable distribution of logistic-map states to narrow the cipher’s sub-key scopes under the known-plaintext condition, while noting that two propositions are verified only over finite image-size ranges.

  • Problem

    Security scrutiny of chaos-based encryption is limited, although reported schemes have security problems and can require cryptanalysis tailored to their structure and chaos systems.

    Source: §1 · p. 1

  • Method

    The paper re-analyzes a logistic- and standard-map image cipher by exploiting stable distribution of logistic-map chaotic states and the equivalent-key setting.

    Source: §1 · p. 2, §3 · p. 3

  • Results

    The cipher’s sub-key scopes can be narrowed further with relatively low computation under the same known-plaintext condition.

    Source: §1 · p. 2, §4 · p. 5

  • Takeaways & Limitations

    The analysis is presented as an example of insecurity in a logistic-map image encryption scheme and as useful for studying security of other such schemes.

    Source: §4 · p. 5

    Two propositions are verified computationally only for stated image-size ranges; beyond those ranges, they remain conjectures.

    Source: §3 · p. 4

Abstract

from arXiv · show

Chaotic behavior arises from very simple non-linear dynamical equation of logistic map which makes it was used often in designing chaotic image encryption schemes. However, some properties of chaotic maps can also facilitate cryptanalysis especially when they are implemented in digital domain. Utilizing stable distribution of the chaotic states generated by iterating the logistic map, this paper presents a typical example to show insecurity of an image encryption scheme using chaotic logistic map. This work will push encryption and chaos be combined in a more effective way.

1 Introduction

Multimedia data pose distinct encryption challenges, while chaos-based schemes have received limited security scrutiny despite reported vulnerabilities. The paper re-evaluates a logistic-map image cipher and narrows its sub-key scope under the previously reported known/chosen-plaintext condition.

1 Introduction

  • Multimedia encryption is motivated by data redundancy, large size, and special processing-system requirements that traditional text encryption does not efficiently address.These differences have made specialized multimedia encryption a research topic.Source: §1 · p. 1
  • Only about one tenth of proposed chaos-based encryption schemes had been reported as cryptanalyzed, leaving security scrutiny a bottleneck for the field.Conventional cryptanalysis skills cannot be directly applied because these schemes have different structures and may have security defects tied to their chaos systems.Source: §1 · p. 1
  • The logistic map is a simple discrete quadratic recurrence with complex dynamics and has been widely used in encryption schemes and pseudo-random number generators.Prior work has also reported security deficiencies caused by the logistic map.Source: §1 · pp. 1–2
  • The paper re-evaluates a logistic- and standard-map image cipher and finds that an efficient brute-force search can narrow all sub-key scopes much further.The prior equivalent key was reported recoverable from one known/chosen plaintext and ciphertext pair, but usable only for ciphertexts no larger than that image.Source: §1 · p. 2

2 The image encryption scheme under study

The scheme encrypts an RGB image through four XOR-based confusion and diffusion operations, using keys and chaotic states derived during initialization.

2 The image encryption scheme under study

  • The input is an M × N RGB true-color image represented as a matrix of three-value pixel tuples, and the chaotic key stream is arranged as an image.The stream’s channel values are formed from chaotic states; the encryption procedure applies its pixels in pixel-by-pixel confusion.Source: §2 · pp. 2–3
  • The secret key comprises floating-point x0, y0, and K plus integer L; initialization derives four XOR keys and iterates the standard map from (x0, y0).The stated ranges are x0,y0 ∈ (0, 2π), K > 18, and 100 < L < 1100.Source: §2 · p. 2
  • Encryption concatenates four operations: key-based confusion, forward horizontal diffusion, reverse vertical diffusion, and confusion using the chaotic key-stream image.The two diffusion passes mask pixels using predecessor values in their respective scan orders.Source: §2 · pp. 2–3
  • Decryption reverses the encryption procedure.The scheme’s operations are described as a simple concatenation of four encryption steps.Source: §2 · pp. 2–3

3 Cryptanalysis

The section extends a known/chosen-plaintext equivalent-key attack by narrowing secret-key subkey ranges. It combines deterministic structure in Xkey with stable logistic-map-derived self-correlation to search and verify candidates.

3 Cryptanalysis

  • The equivalent key Ikey decrypts same-secret-key ciphertexts of size at most M × N, and the paper further narrows the ranges of x0, y0, K, and L under that condition.The equivalent key does not depend on the plaintext or ciphertext.Source: §3 · p. 3
  • The enhanced attack brute-forces {Xkey(i)} and verifies candidates using stable self-correlation in the logistic-map-derived blue channel.The blue-channel sequence is obtained from logistic-map states.Source: §3 · p. 3
  • Each I* value comes from one of the 15 nonempty XOR combinations of four Xkey values, providing a deterministic relationship for the candidate search.The listed combinations comprise the four individual values and their pairwise, three-way, and four-way XORs.Source: §3 · p. 3
  • Under M = N, Proposition 1 shows a relationship between T and N; Proposition 2 indicates that only a very minor portion of M,N combinations make not every element of {Xkey(i)} appear.Source: §3 · pp. 3–4
  • The attack relies on stable distributions of logistic-map trajectories and Δk across initial values, while Δk is sensitive to changes in successive transformed states.The passages report similar Δk distributions for randomly chosen initial values and sensitivity to changes in either transformed state.Source: §3 · pp. 3–4
  • The propositions are computer-verified for N = 3 through 1204 and 3 ≤ M, N ≤ 1204, respectively; their larger scopes are left as conjectures.The first proposition's wider range in N and the second proposition's wider range in M and N remain conjectural.Source: §3 · p. 4

1. Obtain Ikey via

The attack tests candidate key-byte values against the expected distribution of Δ_k, then uses the selected values to narrow the secret-key parameters.

1. Obtain Ikey via

  • Candidate key-byte values are tested by comparing the resulting Δ_k distribution with the expected distribution in Table 1.The procedure generates an estimated CKSB sequence before calculating the distribution.Source: §1 · p. 4, Table 1 · p. 5
  • Once Xkey values are determined, K and L follow from Xkey(2) and Xkey(3), while x0 and y0 are narrowed to corresponding intervals.The stated ranges are K = Xkey(2) mod 256, L = Xkey(3) + n·256 for n = 1–4, and one 2π/256-wide interval for each initial value.Source: §1 · pp. 4–5
  • In the reported experiment, the attack reduced its complexity to O(2^36) and completed within eleven minutes on the specified PC.The PC had a 2.83 GHz CPU and 2.98 GB RAM.Source: §1 · p. 5
  • The experiment selected {Xkey(i)} for i = 0,…,3 as {162, 54, 109, 110}, narrowing K, L, x0, and y0 to the reported ranges.The reported bounds include K = 109 mod 256, L = 110 + n·256 for n = 1–3, and intervals for x0 and y0.Source: §1 · p. 5

4 Conclusion

The paper re-analyzes a logistic-map image-encryption scheme and finds that all sub-keys can be narrowed further with relatively low computation under the same known-plaintext condition. The authors say the results can help study the security of other logistic-map image-encryption schemes.

4 Conclusion

  • Under the same known-plaintext condition, the analysis narrows the scope of all sub-keys further with relatively low computation.The paper re-analyzes a scheme for which an equivalent secret key had already been reported recoverable from one known-plaintext/ciphertext pair.Source: §4 · p. 5
  • The authors say these results will help study the security of other image-encryption schemes based on the logistic map.Source: §4 · p. 5
  • Table 2 gives the distribution of ∆k for 16 possible neighboring values of {162, 54, 108, 110}.Source: Table 2 · p. 6
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