Chaos-Based Adaptive S-Box Generation

• Chaos-based adaptive S-box generation is a cryptographic technique that uses chaotic systems (e.g., Logistic and Lorenz maps) to create dynamic, key-dependent substitution boxes.
• It employs various adaptation mechanisms at different granularities—such as session, round, or pixel level—to counter linear and differential cryptanalysis.
• Evaluations report enhanced nonlinearity, low adjacent-pixel correlation, and high entropy, indicating robust resistance against statistical, differential, and side-channel attacks.

Chaos-based adaptive S-box generation denotes a family of cryptographic design methods in which the substitution box is generated, selected, or modified by a chaotic dynamical system rather than fixed a priori. In the cited literature, this paradigm appears in image ciphers, AES- and CAST-128-derived designs, VPN-oriented symmetric encryption, and physical-layer security, with adaptivity defined at the level of session, round, pixel, image, device, or network slice (Ahmad et al., 2017, Fadhil et al., 9 Sep 2025, Cherkaoui et al., 15 Nov 2025). The common objective is to strengthen confusion through key dependence, nonlinearity, and unpredictability, while reported constructions differ substantially in their mathematical engines, enforcement of bijectivity, and evaluation methodology (Cassal-Quiroga et al., 2019, Dimitrov, 2020).

1. Conceptual foundations

An S-box is the nonlinear component of a block or stream cipher, and its cryptanalytic relevance is central because resistance to linear and differential attacks depends strongly on its Boolean and permutation structure (Dimitrov, 2020). Chaos-based adaptive generation replaces the conventional fixed lookup table with a dynamically produced mapping derived from a chaotic source such as the Logistic map, a piecewise linear chaotic map, the Lorenz system, a 2D exponential hyper chaotic map, a 6D hyperchaotic system, or a $\beta$-transformation (Ahmad et al., 2017, Al-Maadeed et al., 2020, Liu et al., 2021, Gholamzadeh et al., 2023, Cherkaoui et al., 15 Nov 2025).

Within this literature, “adaptive” does not denote a single mechanism. Some schemes generate an entire keyed S-box per session or per image, as in Logistic Sine Map-driven CAST-128 variants and CNN-personalized image ciphers (Fadhil et al., 9 Sep 2025, Ibrahim et al., 18 Jun 2026). Others generate many candidate S-boxes and then select one per pixel using a chaotic trajectory, as in the PWLCM- and Logistic-map-based image cipher that produces one thousand S-boxes and chooses among them dynamically (Ahmad et al., 2017). A third pattern keeps one S-box but applies a chaos-driven random operation to each substituted value, as in SRSS with CROSS (Khan et al., 2023). A fourth pattern targets hardware and communication agility, where the S-box is regenerated on demand for each session, slice, or device (Cherkaoui et al., 15 Nov 2025).

The principal rationale is that static S-boxes are structurally reusable across encryptions, whereas key- and seed-dependent constructions aim to prevent attackers from exploiting a persistent substitution layer. Several papers explicitly frame this as a defense against linear cryptanalysis, differential cryptanalysis, statistical attacks, replay of learned mappings, or large-scale precomputation (Ishaq et al., 2023, Fadhil et al., 9 Sep 2025, Cherkaoui et al., 15 Nov 2025). This suggests that chaos is used not merely as a pseudorandom source, but as a mechanism for diversification of the confusion layer itself.

2. Generative mechanisms and construction patterns

Across representative constructions, a recurring workflow combines a chaotic source, a quantization or permutation step, and a bijectivity-enforcement mechanism. The details, however, vary materially.

Approach	Chaotic mechanism	Adaptive role
PWLCM + Fisher–Yates	Piecewise linear chaotic map	Generates 1000 shuffled S-boxes
Lag-time logistic series	Two lagged Logistic-map series	Produces uniformly distributed bits for direct S-box construction
Lorenz + PGL	Lorenz system with algebraic S-box bank	Selects among 16 algebraically defined S-boxes per pixel
2D-ECM filtering	2D exponential hyper chaotic map	Generates keyed S-boxes, then rejects weak permutations
Dyadic-chaotic lifting	$\beta$-transformation with dyadic sampling	Produces on-demand 8-bit bijections

In the PWLCM-based image cipher, the map

$$y(n+1)= \begin{cases} \frac{y(n)}{p}, & 0<y(n)\le p \ \frac{1-y(n)}{1-p}, & p<y(n)<1 \end{cases}$$

drives a Fisher–Yates shuffle over the set $\{0,\ldots,255\}$. The process is repeated, with seed variation, to generate one thousand distinct $16\times16$ S-boxes (Ahmad et al., 2017). In that construction, a second chaotic mechanism, the Logistic map

$x(n+1)=\lambda x(n)(1-x(n)),$

is used for adaptive selection of the S-box index and its row and column.

A different strategy is direct symbolic construction from chaotic series. The lag-time method mixes two Logistic-map orbits, uses noncontiguous delays, combines them modulo $1$, thresholds the resulting sequence at $0.5$, groups the bits into $n$-bit blocks, and keeps the first $2^n$ unique integers to form an $\beta$0 S-box (Cassal-Quiroga et al., 2019). The stated purpose of the dual lagged series is to hide the underlying map and avoid the U-shape distribution of the raw Logistic map.

The 2D-ECM approach adds an explicit post-generation filter. After running the map and extracting

$\beta$1

duplicates are removed, and the candidate permutation is rejected if it contains fixed points, reverse fixed points, or short period rings (Liu et al., 2021). That paper also derives recursion formulas for counting S-boxes satisfying these constraints and defines such constructions as “strong” when these conditions accompany the usual six cryptographic criteria.

Other work couples chaotic control with algebraic S-box generation. One image-encryption scheme first constructs 16 S-boxes from linear fractional transformations over $\beta$2 using the group $\beta$3 and 16 primitive irreducible polynomials, then uses a Lorenz chaotic system to adaptively select which S-box is applied to a given pixel (Al-Maadeed et al., 2020). In contrast, the dyadic-chaotic lifting design starts from the $\beta$4-transformation

$\beta$5

and uses dyadic conditional sampling plus uniqueness filtering until all 256 output values are collected, yielding a full 8-bit bijection (Cherkaoui et al., 15 Nov 2025).

A further variation appears in VPN-oriented work based on the AHYB map, where an initial chaotic permutation is followed by a heuristic swap process that keeps only nonlinearity-improving exchanges (Ishaq et al., 2023). This reflects a broader movement toward hybrid designs in which chaos supplies candidate structure, while later processing enforces cryptographic criteria more directly.

3. Adaptive selection, substitution, and system integration

Adaptive generation becomes operationally significant only when coupled to an encryption architecture. The literature shows several such couplings.

In the dynamic image cipher based on chaotic S-boxes, the selected S-box element is passed through an Affine–Power–Affine transformation over $\beta$6,

$\beta$7

and then used within a substitution formula of the form

$\beta$8

where $\beta$9 introduces cipher block chaining and $$y(n+1)= \begin{cases} \frac{y(n)}{p}, & 0<y(n)\le p \ \frac{1-y(n)}{1-p}, & p<y(n)<1 \end{cases}$$0 is taken from a keyed Latin square derived from a 256-bit external key (Ahmad et al., 2017). The same design also includes reshaping, rotation, flipping, and multiple rounds.

The Lorenz-based scheme adopts a two-phase structure. First, the image is flattened, permuted by a chaotic sequence, and XOR-masked. Second, each pixel is split into 4 MSBs and 4 LSBs, which determine the row and column of an adaptively chosen S-box from the 16-element bank (Al-Maadeed et al., 2020). In this formulation, the chaotic system controls both permutation and substitution.

SRSS alters the role of chaos by shifting it from S-box generation to operation selection. It retains a single S-box and a single round, but CROSS uses a Logistic-map-derived sequence quantized modulo 3 to choose, for each pixel, which modifier constant is XORed with the S-box output:

$$y(n+1)= \begin{cases} \frac{y(n)}{p}, & 0<y(n)\le p \ \frac{1-y(n)}{1-p}, & p<y(n)<1 \end{cases}$$1

The paper’s claim is that this per-pixel random operation better breaks highly auto-correlated image regions than traditional single-S-box, multi-S-box, or multi-round substitution alone (Khan et al., 2023).

In block-cipher adaptations, the static substitution layer of legacy ciphers is replaced or supplemented by chaos-derived tables. The CAST-128 enhancement derives a bijective 256-entry S-box from the Logistic Sine Map and loads it into the block-wise encryption structure (Fadhil et al., 9 Sep 2025). AES-oriented IoT work combines Logistic-map evolution with a 3-dimensional key generation matrix, XOR, and LFSR operations to create dynamic key material and effective session-dependent S-box behavior (Rahman et al., 2021, Rahman et al., 2022). Another hybrid image scheme integrates a DPA-resistant JAYAR S-box with a 6D hyperchaotic system, using chaotic sequences for shuffling, key expansion, and S-box control (Gholamzadeh et al., 2023).

At the opposite end of the implementation spectrum, the dyadic-chaotic lifting construction is explicitly designed for table-free or small-table realization, on-the-fly diversification, and sub-millisecond regeneration in hardware-constrained environments (Cherkaoui et al., 15 Nov 2025). This widens the application space from multimedia encryption toward communication-layer security primitives.

4. Cryptographic criteria and reported performance

The recurring evaluation framework includes bijectivity, nonlinearity, strict avalanche criterion (SAC), bit independence criterion (BIC), equiprobable input/output XOR distribution or differential probability, and linear approximation probability. One paper explicitly groups these as the criteria of “good S-boxes,” together with bijectivity and maximum expected linear probability (Cassal-Quiroga et al., 2019).

Reported S-box-level values vary by design. The lag-time Logistic-map construction reports output-coordinate nonlinearities of 104, 104, 102, 104, 96, 102, 100, and 102; average SAC of about 0.5012; BIC-SAC of about 0.5066; maximum difference 5; and MELP equal to 0.0176 (Cassal-Quiroga et al., 2019). The Lorenz-plus-PGL design reports nonlinearity in the range of about 104–107, BIC and SAC values close to 0.5, and differential and linear approximation probabilities at or below 0.25, with differential probability often at 0.125 (Al-Maadeed et al., 2020). The 2D-ECM work reports average nonlinearity of approximately 103.5 and states that, among 3000 generated S-boxes, nonlinearities ranged from 103.2 to 104.1 (Liu et al., 2021). The AHYB-based VPN S-box reports minimum nonlinearity 108, maximum 110, average 109.5, SAC 0.5007, BIC-NL 103.6, LP 0.1328, and DP 0.0391 (Ishaq et al., 2023). The dyadic-chaotic lifting design reports optimal algebraic degree 7 on every output bit, average nonlinearity 102.5, maximum DDT entry 10 corresponding to probability 0.039, and maximum linear probability 0.648 (Cherkaoui et al., 15 Nov 2025).

Image-level statistics are likewise emphasized. The PWLCM/Latin-square scheme reports, for Lena, encrypted adjacent-pixel correlation 0.00673 versus 0.95679 in the original image, entropy 7.9965, and NPCR above 99%, with one example of 99.60 for a “Black” image (Ahmad et al., 2017). The Lorenz-based image cipher reports adjacent-pixel correlation approximately zero, entropy approximately 7.95, and NPCR above 99% with UACI above 33% (Al-Maadeed et al., 2020). SRSS reports entropy 7.989 and correlation 0.0007 in a single-round, single-S-box setting (Khan et al., 2023). The Logistic Sine Map CAST-128 variant reports entropy values around 7.987–7.995, NPCR around 99.58%–99.63%, UACI around 33.28%–33.57%, and encryption times below 30 ms for the listed grayscale images (Fadhil et al., 9 Sep 2025). The GH401 hybrid reports near-ideal NPCR around 99.6%, UACI around 33.37%, and entropy approximately 7.9978 (Gholamzadeh et al., 2023).

These results are typically interpreted as evidence of strong confusion, diffusion, and resistance to statistical and differential attacks. However, the significance of any single reported metric depends on how comprehensively the S-box has been evaluated.

5. Critiques, controversies, and common misconceptions

The strongest internal critique of the field is methodological. Dimitrov argues that many chaos-based S-box papers overestimate resistance to linear cryptanalysis because they report only the average coordinate nonlinearity value, $$y(n+1)= \begin{cases} \frac{y(n)}{p}, & 0<y(n)\le p \ \frac{1-y(n)}{1-p}, & p<y(n)<1 \end{cases}$$2, while actual resistance is governed by the minimum nonlinearity over all nontrivial components of the S-box (Dimitrov, 2020). That paper reports that literature values with ACNV near 110 may still have minimum nonlinearity as low as 88 to 96, and concludes that the benefit of chaos structures for nonlinearity optimization is negligible when compared with heuristic search starting from pseudorandom bijections. It also reports that hill climbing reached ACNV 114 and bandit-assisted post-processing reached ACNV 114.5, outperforming previously published chaos-function-based S-boxes in the comparison terms used by that literature (Dimitrov, 2020).

A second concern is the absence of standardized guidelines and extensive security analysis. The VPN-focused study explicitly states that existing work on dynamic S-boxes using chaotic maps lacks standardized guidelines and extensive security analysis, and that key management plus sensitivity to initial conditions require careful treatment (Ishaq et al., 2023). This suggests that favorable histogram, entropy, and correlation results do not by themselves settle cryptanalytic soundness.

A third misconception concerns architectural complexity. Several image-encryption papers implicitly assume that multiple S-boxes or multiple rounds are necessary to defeat image autocorrelation. SRSS challenges this assumption, arguing that the decisive factor is not multiplicity of S-boxes alone but per-pixel randomness in the operation applied to substituted values (Khan et al., 2023). The claim is specific to highly auto-correlated images and should be read as an architectural counterexample rather than a universal theorem.

Side-channel resistance is another underemphasized issue. Most chaos-based S-box work focuses on statistical and classical cryptanalytic metrics, whereas GH401 explicitly adds a DPA-resistant S-box optimized for transparency order and integrates it with hyperchaotic control (Gholamzadeh et al., 2023). A plausible implication is that adaptive generation alone does not automatically address leakage; the substitution layer may still require side-channel-specific design criteria.

6. Applications, hybrids, and emerging directions

Chaos-based adaptive S-box generation has expanded beyond its initial concentration in image encryption. Multimedia-oriented schemes remain prominent, including dynamic substitution with Latin squares, Lorenz-based permutation-plus-substitution, SRSS/CROSS, CAST-128 image protection, and GH401-style hyperchaotic hybrids (Ahmad et al., 2017, Al-Maadeed et al., 2020, Khan et al., 2023, Fadhil et al., 9 Sep 2025, Gholamzadeh et al., 2023). In parallel, AES-oriented IoT work uses Logistic-map-based dynamic key propagation and session-dependent substitution behavior for smart-home and edge-device scenarios (Rahman et al., 2021, Rahman et al., 2022). VPN data security and 6G physical-layer security extend the topic into communication infrastructure, where on-demand diversification and hardware amenability become central (Ishaq et al., 2023, Cherkaoui et al., 15 Nov 2025).

Hybridization is a clear trend. Algebraic S-box banks defined by primitive irreducible polynomials are combined with chaotic selectors (Al-Maadeed et al., 2020). Genetic algorithms are used to optimize transparency order and nonlinearity in chaos-controlled S-box construction (Gholamzadeh et al., 2023). Heuristic hill climbing and bandit-based optimization are proposed as more effective nonlinearity improvers than chaos alone (Dimitrov, 2020). Even work positioned beyond the chaos-based tradition compares against it directly: CNN-based dynamic S-box generation ties substitution to image content, while still using chaos maps such as Logistic or Lorenz for permutation in the broader encryption framework (Ibrahim et al., 18 Jun 2026).

The 6G-oriented dyadic-chaotic lifting proposal indicates another direction: mathematically structured chaos with explicit ergodicity arguments, uniformity guarantees, full 8-bit bijections, and hardware-conscious regeneration latency under 0.2 ms at 200 MHz for practical parameters (Cherkaoui et al., 15 Nov 2025). This differs from earlier practice in which chaotic maps often served primarily as empirical randomness sources. A plausible implication is that future work will continue moving toward formally characterized dynamical systems, explicit post-generation filtering, and hybrid evaluation that includes classical cryptanalysis, side-channel criteria, and implementation constraints.

In that sense, chaos-based adaptive S-box generation is no longer a single technique but a research program spanning dynamical systems, finite-field constructions, heuristic optimization, hardware design, and application-specific cryptography. Its central question is not whether chaos can generate permutations, but under what evaluation regime, threat model, and integration architecture such adaptive substitution genuinely improves security over fixed S-box design.