Plaintext Non-Delayed Chaotic Cipher

• PNDCC is defined by a diffusion process that employs ciphertext feedback without plaintext delay, forming a canonical permutation–diffusion–permutation architecture.
• The scheme achieves favorable statistical properties, yet its homogeneous algebraic structure enables various differential and chosen-ciphertext attacks.
• Introducing plaintext delay, as in PDCC, is proposed as a countermeasure to break exploitable positional dependency chains and enhance security.

Searching arXiv for the specified paper and closely related material to ground the article in current research. Plaintext non-delayed chaotic cipher (PNDCC) denotes any chaos-based encryption scheme whose diffusion stage has the form

$C(k) = f_D\big(P(k),\, C(k-1),\, K(k)\big),$

with no dependence on $P(k-1), P(k-2), \ldots$ and with explicit feedback from the previous ciphertext $C(k-1)$. In the literature on chaos-based image ciphers, the diffusion function $f_D$ is described as invariably taking this structure. The term therefore refers not to a single algorithm but to a family of designs characterized by plaintext non-delay and ciphertext feedback in diffusion, together with one or more permutation stages surrounding that diffusion process (Wang et al., 14 Sep 2025).

1. Formal definition and canonical architecture

A PNDCC is defined by the absence of plaintext delay terms in diffusion and the presence of a ciphertext feedback term. The paper presents a canonical three-stage permutation–diffusion–permutation construction. Let the plaintext image be scanned in raster order into the one-dimensional sequence $P(1 \ldots M \cdot N)$, let the chaotic key stream or dynamic key be $K(1 \ldots M \cdot N)$, and let the permuting maps be $S_0$ and $S_1$. Encryption proceeds as follows:

• Pre-permutation:

$P_0(k) = P\big(S_0(k)\big)$

• Diffusion:

$C_1(k) = f_D\big(P_0(k),\, C_1(k-1),\, K(k)\big)$

• Post-permutation:

$P(k-1), P(k-2), \ldots$0

In the running example used for cryptanalysis, the diffusion is a pure XOR/difference form:

$P(k-1), P(k-2), \ldots$1

Letting $P(k-1), P(k-2), \ldots$2 be an IV or zero, the full encryption is given by

$P(k-1), P(k-2), \ldots$3

This architecture is significant because it isolates a structural pattern common in chaos-based image encryption: permutation is used to obscure positional relationships, while diffusion with ciphertext feedback is intended to propagate local plaintext changes throughout the image. A plausible implication is that the security of many proposed chaotic image ciphers depends less on their specific chaotic generator than on the algebraic and positional properties induced by this architecture.

2. Security rationale and the limits of statistical validation

Typical security claims for PNDCC designs rely on statistical tests. Designers show that, after encryption, the cipher-image has a flat histogram, near-zero pixel correlation, high NPCR/UACI, and near-ideal entropy, and then conclude that the cipher is secure. The paper argues that these indicators all rely on the same underlying mechanism: the feedback of $P(k-1), P(k-2), \ldots$4 propagates every plaintext bit throughout the image. According to this argument, good statistical metrics are thereby guaranteed regardless of whether $P(k-1), P(k-2), \ldots$5 or the permutations are cryptographically strong (Wang et al., 14 Sep 2025).

The central conclusion is explicit: passing these statistical tests is a necessary but not a sufficient condition for modern security. This directly addresses a recurrent misconception in the chaotic-cipher literature, namely that histogram flattening, entropy values, or differential-statistical indicators can substitute for cryptanalytic resistance. The paper positions PNDCC as a case in which empirical image-statistics can be favorable while multiple attacks remain effective.

This distinction is important because the claimed rationale for PNDCC often rests on diffusion strength understood in a visual or statistical sense. The results summarized in the paper suggest that such reasoning is incomplete when the diffusion equation remains plaintext non-delayed. In that setting, the feedback structure can improve statistical appearance while preserving exploitable algebraic or positional regularities.

3. Cryptanalysis of a three-stage PNDCC

The paper studies a representative three-stage PNDCC and reports four distinct attacks against it, even though all of its statistical indicators show good performance. The attacks are: a differential attack based on homogeneous operations, an S-PTC attack, an impulse-step-based differential attack (ISBDA), and a chain attack (Wang et al., 14 Sep 2025).

The differential attack based on homogeneous operations applies when $P(k-1), P(k-2), \ldots$6 uses a homogeneous group operation, such as all additions/subtractions modulo $P(k-1), P(k-2), \ldots$7 or all XORs. For two chosen plaintexts whose difference $P(k-1), P(k-2), \ldots$8 has only one nonzero component, the ciphertext difference $P(k-1), P(k-2), \ldots$9 can be predicted and peeled off stage by stage. The derivation given in the paper assumes

$C(k-1)$0

so only pixel $C(k-1)$1 is flipped. If $C(k-1)$2 and $C(k-1)$3 are the two ciphertext streams, then

$C(k-1)$4

and

$C(k-1)$5

From this, one can locate $C(k-1)$6 by scanning the first position where $C(k-1)$7, recover $C(k-1)$8, and then recover $C(k-1)$9 by solving linear equations for the remaining positions.

The S-PTC attack, expanded as Set–Plaintext–Ciphertext attack, is a chosen-ciphertext codebook method. A ciphertext codebook is prepared by encrypting a basis of $f_D$0 chosen-ciphertext images: for each position $f_D$1, let $f_D$2 be all-zero except at index $f_D$3, where every possible byte value $f_D$4 is inserted, and observe the corresponding plaintext $f_D$5. Because the PNDCC is non-delayed in plaintext, $f_D$6 depends only on $f_D$7 but no future plaintext. The workflow is described as follows: first, query $f_D$8 decryptions to build a full lookup table $f_D$9; second, from $P(1 \ldots M \cdot N)$0, trace backward through $P(1 \ldots M \cdot N)$1 to recover the pre-permutation; third, obtain $P(1 \ldots M \cdot N)$2 from $P(1 \ldots M \cdot N)$3 and $P(1 \ldots M \cdot N)$4.

These two attacks already show that the vulnerability of the example is not confined to a single analytic technique. One attack leverages algebraic homogeneity; the other exploits invertibility and non-delayed dependence under chosen-ciphertext access. This suggests that PNDCC weaknesses can emerge through multiple, structurally independent avenues.

4. Impulse-step and chain-based attacks

The impulse-step-based differential attack (ISBDA) is introduced through an impulse-step model. An impulse is injected into the plaintext at position $P(1 \ldots M \cdot N)$5 by setting $P(1 \ldots M \cdot N)$6 while all other plaintext positions satisfy $P(1 \ldots M \cdot N)$7. This is compared with the reference plaintext $P(1 \ldots M \cdot N)$8 everywhere. If

$P(1 \ldots M \cdot N)$9

at position $K(1 \ldots M \cdot N)$0 and zero elsewhere, then under the diffusion rule,

$K(1 \ldots M \cdot N)$1

The paper states that one obtains

$K(1 \ldots M \cdot N)$2

and

$K(1 \ldots M \cdot N)$3

The ISBDA workflow is given in four steps. First, for $K(1 \ldots M \cdot N)$4, encrypt two images: an all-$K(1 \ldots M \cdot N)$5 image and an impulse image at $K(1 \ldots M \cdot N)$6. Second, compute $K(1 \ldots M \cdot N)$7 and count the number of nonzero outputs; this gives the position of the impulse after pre-permutation and hence recovers $K(1 \ldots M \cdot N)$8. Third, once $K(1 \ldots M \cdot N)$9 is known, choose $S_0$0 to neutralize diffusion; the decrypted intermediate $S_0$1 yields post-permutation $S_0$2. Fourth, recover $S_0$3 by plain algebra. The complexity is stated as $S_0$4 chosen-plaintext–ciphertext pairs (Wang et al., 14 Sep 2025).

The chain attack is presented as the paper’s second novel contribution. Its key idea is to view the entire multi-stage decryption as a positional decryption machine. Each ciphertext position $S_0$5 maps to a set of possible plaintext positions $S_0$6. By injecting single-point perturbations under chosen-ciphertext attack, one recovers for each $S_0$7 the set $S_0$8. These sets form a chain of length $S_0$9 in which each consecutive set overlaps in exactly two positions, as formalized by Definition 1 and Theorem 1 together with linking rules 1–6.

The recovery procedure is described step by step. One first picks $S_1$0, injects $S_1$1 at $S_1$2, and records the positions where plaintext changed, yielding a set of the form

$S_1$3

Collecting these sets for $S_1$4 produces a broken chain of sets. These sets are then reordered via the six linking rules so that adjacent sets overlap in exactly two elements; the ordering index corresponds to $S_1$5. From the global ordering, one reads off $S_1$6 stage by stage. Once all permutations are known, the diffusion reduces to a system of linear or XOR equations in $S_1$7. By reusing the same $S_1$8 queries, specifically all-$S_1$9 and all-$P_0(k) = P\big(S_0(k)\big)$0, one solves for $P_0(k) = P\big(S_0(k)\big)$1 at each position independently.

The significance of ISBDA and the chain attack lies in the way they exploit positional propagation rather than only local algebra. In both attacks, the absence of plaintext delay yields a directional dependence pattern that can be observed, reordered, and ultimately inverted.

5. Generalization to multi-stage PNDCC

The paper extends the analysis from the three-stage example to any PNDCC with an arbitrary number $P_0(k) = P\big(S_0(k)\big)$2 of permute–diffuse stages. Such a system has a generalized decryption iteration of the form

$P_0(k) = P\big(S_0(k)\big)$3

where the composite map $P_0(k) = P\big(S_0(k)\big)$4 encodes the chaining of all $P_0(k) = P\big(S_0(k)\big)$5 permutations (Wang et al., 14 Sep 2025).

To isolate positional dependencies, the paper introduces the associated position expression

$P_0(k) = P\big(S_0(k)\big)$6

By Theorem 1, such a chain always exists if all diffusion feedback depths $P_0(k) = P\big(S_0(k)\big)$7, with each node’s set overlapping the next in two elements. The general chain-attack flow is then stated to be identical to the three-stage case: chosen-ciphertext perturbations are used to recover the positional map, the resulting sets are reordered via the linking rules, and the residual diffusion keys are solved afterward. The complexity remains $P_0(k) = P\big(S_0(k)\big)$8 queries.

This generalization is central because it shows that the vulnerabilities identified are not confined to a particular parameterization or to a single permutation–diffusion–permutation example. The argument is structural: once the decryption dependencies can be expressed purely through ciphertext positions and key terms, without delayed plaintext terms, a chain representation becomes available. A plausible implication is that increasing the number of stages does not by itself remove the core weakness if the non-delayed plaintext structure is preserved.

6. Plaintext-delayed chaotic cipher as a countermeasure

As a secure improvement, the paper proposes plaintext-delayed chaotic cipher (PDCC). The stated design principle is that, to thwart the attacks described above, one must introduce plaintext delay, meaning that $P_0(k) = P\big(S_0(k)\big)$9 must depend also on $C_1(k) = f_D\big(P_0(k),\, C_1(k-1),\, K(k)\big)$0. Concretely, each diffusion inverse in decryption is redesigned as

$C_1(k) = f_D\big(P_0(k),\, C_1(k-1),\, K(k)\big)$1

The paper’s claim is that this breaks any closed-form chain like the earlier equations because the positional dependencies now refer to unknown past plaintext (Wang et al., 14 Sep 2025).

The consequences for the attacks are stated explicitly. ISBDA fails because the impulse no longer produces a pure step until after unknown plaintext terms. The chain attack fails because chosen-ciphertext perturbation yields sets that cannot be reordered into a single chain of length $C_1(k) = f_D\big(P_0(k),\, C_1(k-1),\, K(k)\big)$2; there is no longer a deterministic, acyclic positional path through only ciphertext indices. S-PTC and codebook attacks increase sharply in complexity because the state depends on multiple past plaintext terms.

As an example, the paper states that one can set $C_1(k) = f_D\big(P_0(k),\, C_1(k-1),\, K(k)\big)$3 and choose $C_1(k) = f_D\big(P_0(k),\, C_1(k-1),\, K(k)\big)$4 as a mixed XOR+modular function, thereby ensuring heterogeneous operations. The preliminary security arguments and supporting simulations are summarized in three points: there is no closed form for $C_1(k) = f_D\big(P_0(k),\, C_1(k-1),\, K(k)\big)$5 in terms of past $C_1(k) = f_D\big(P_0(k),\, C_1(k-1),\, K(k)\big)$6 values only; statistical indicators remain good; and all linear attacks require exhaustive search in at least $C_1(k) = f_D\big(P_0(k),\, C_1(k-1),\, K(k)\big)$7 or higher.

Within the framework of the paper, PDCC is not presented merely as an implementation variant but as a change in the dependency model of diffusion. The key distinction is that PNDCC permits decryption dependencies to be expressed through ciphertext history alone, whereas PDCC introduces unknown plaintext state into that dependency graph. This suggests that the decisive design variable is the presence or absence of plaintext delay rather than the mere use of chaos, permutation, or feedback.

7. Position within chaotic image cryptography

The paper characterizes PNDCC as a widely used diffusion pattern in chaos-based image ciphers and argues that its empirical evaluation has often emphasized statistical indicators without corresponding cryptographic proofs. In that context, the reported attacks serve two roles: they break a typical three-stage permutation–diffusion–permutation instance, and they also motivate a more general critique of security evaluation based primarily on histogram, correlation, NPCR/UACI, and entropy criteria (Wang et al., 14 Sep 2025).

The broader significance of PNDCC therefore lies in the contrast between statistical adequacy and cryptanalytic security. In the formulation presented, ciphertext feedback ensures strong propagation effects, which explains the recurrently favorable statistical results. At the same time, the absence of plaintext delay leaves exploitable structure in differential behavior, chosen-ciphertext response, and positional dependency chains. The paper’s analysis treats these as family-level issues rather than isolated defects of one cipher.

For researchers, PNDCC is thus best understood as a design class defined by a specific diffusion dependency pattern. Its study connects image-oriented chaos-based encryption with standard cryptanalytic concerns: algebraic homogeneity, query complexity, positional inference, and the distinction between necessary and sufficient security conditions. The proposed transition from PNDCC to PDCC reframes secure design around delayed plaintext dependence and heterogeneous operations, rather than around statistical quality alone.