2D-RA Map: Hyper-chaotic Encryption Core

• 2D-RA map is a two-dimensional hyper-chaotic discrete map combining key features of the Rastrigin and Ackley functions to drive image encryption.
• It utilizes quadratic, cosine, and exponential terms with modulo-one folding to ensure bounded evolutions, yielding high Lyapunov exponents and a correlation dimension near 2.
• The map is integrated into encryption schemes via self-adaptive diffusion and sorting-based permutation, offering strong sensitivity and uniform phase-space coverage.

Searching arXiv for papers relevant to “2D-RA map,” especially the image-encryption hyper-chaotic map definition and closely related interpretations of the term. The 2D-RA map is a two-dimensional hyper-chaotic discrete map introduced as the core dynamical component of an image-encryption scheme that combines a novel chaotic generator with a self-adaptive diffusion method. In the defining source, the label “RA” refers to the hybridization of the Rastrigin and Ackley functions, whose quadratic, oscillatory, and exponential terms are combined and then folded by a modulo-one operation to keep the iterates bounded. The map is proposed for generating high-complexity chaotic sequences with strong sensitivity, broad phase-space coverage, and empirical hyper-chaotic behavior for cryptographic use (Tang, 8 Sep 2025).

1. Definition and conceptual origin

The 2D-RA map is constructed by combining characteristic components of two classical multimodal benchmark functions from optimization.

The first parent function is the Rastrigin function. In $n$ dimensions it is given by

$$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$

with the typical choice $A=10$. Its 2D form is

$f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$

The second parent function is the Ackley function. In $n$ dimensions it is

$$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$

Its 2D form is

$$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$

From these two functions, the map retains the quadratic terms $x^2$ and $y^2$, the oscillatory cosine terms $\cos(2\pi x)$ and $$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$0, the Ackley-type exponential radial term $$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$1, and the Ackley-type exponential cosine-average term $$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$2. The resulting iterates are reduced modulo one. The stated purpose of this construction is to obtain stronger sensitivity, more uniform phase-space coverage, and robust hyper-chaotic behavior than existing advanced chaotic functions (Tang, 8 Sep 2025).

2. Mathematical formulation

The paper defines the 2D-RA map by the coupled recurrence

$$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$3

where $$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$4 and $$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$5 are the state variables, $$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$6 and $$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$7 are control parameters, $$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$8, and $$f(x)=A\cdot n+\sum_{i=1}^{n}\left[x_i^2-A\cos(2\pi x_i)\right],$$9 is Euler’s constant (Tang, 8 Sep 2025).

The paper also presents the generic notation

$A=10$0

with

$A=10$1

and

$A=10$2

The control parameters are described in two ways. The paper states that $A=10$3 are “in the set of non-negative integers,” and, in the encryption instantiation derived from the SHA-256 hash, reports

$A=10$4

The initial state is therefore

$A=10$5

Because the map applies $A=10$6, the iterates remain in $A=10$7. A plausible implication is that the large additive coefficients produced by $A=10$8 amplify nonlinear mixing before reduction modulo one, but the paper does not provide a separate analytical derivation of that effect (Tang, 8 Sep 2025).

3. Dynamical characterization

The paper evaluates the map by bifurcation diagrams, Lyapunov exponents, initial value sensitivity, 0–1 test, correlation dimension, and Kolmogorov entropy (Tang, 8 Sep 2025).

For a generic 2D map

$A=10$9

the Jacobian is written as

$f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$0

If $f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$1 are the eigenvalues referred to in the paper, the Lyapunov exponents are given as

$f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$2

The reported criterion is standard within the paper: one positive Lyapunov exponent indicates chaos, and at least two positive Lyapunov exponents in a 2D-or-higher system indicate hyper-chaos. Under the tested parameter scans,

$f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$3

and

$f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$4

the paper states that both Lyapunov exponents remain large and positive. It therefore characterizes the map as having “extremely robust hyper-chaotic properties.”

The attractor phase diagram is reported for

$f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$5

The paper describes the resulting cloud as spread across the whole phase space, uniform and stochastic, without obvious holes or dense clusters.

Initial-value sensitivity is tested with

$f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$6

$f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$7

$f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$8

The reported outcome is substantial divergence from the first iteration onward.

The 0–1 test is given by

$f(x,y)=2A+x^2-A\cos(2\pi x)+y^2-A\cos(2\pi y).$9

with

$n$0

and

$n$1

$n$2

where $n$3 is the $n$4-th sample and $n$5 is random. In the paper’s interpretation, $n$6 indicates chaos and $n$7 indicates regularity; for the 2D-RA map, $n$8 stays close to 1 over the tested ranges.

The correlation dimension is defined as

$n$9

where

$$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$0

The paper reports $$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$1, with average value

$$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$2

Kolmogorov entropy is written as

$$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$3

with

$$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$4

The map is reported to have $$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$5, with benchmark value

$$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$6

4. Role in the encryption system

The 2D-RA map is the sequence generator for an image-encryption pipeline that combines self-adaptive diffusion with confusion/permutation (Tang, 8 Sep 2025).

Key material is derived from the SHA-256 hash of the flattened image. The 256-bit hash is split into eight 32-bit blocks,

$$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$7

and then reduced to four operational parameters: $$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$8

$$f(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) -\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right)+20+e.$$9

$$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$0

$$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$1

Starting from

$$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$2

the map is iterated and the output is serialized as

$$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$3

In the self-adaptive diffusion stage, the image is scanned row by row. The first pixel of the first row is unchanged, and each later pixel is diffused based on the previous processed pixel. A histogram table of grayscale frequencies is maintained. The paper states that grayscale values with smaller current counts are placed nearer the predicted grayscale value to balance frequencies. The 2D-RA sequence is used to initialize the grayscale occurrence frequencies with chaotic values in $$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$4 instead of zeros. This changes tie-breaking among equally counted grayscale levels while preserving the adaptive balancing logic.

In the confusion stage, the chaotic sequence is filled into a matrix of the same size as the image, ranked by magnitude, and converted into a permutation mapping. If two chaotic values are equal, the earlier position receives the lower rank. The ranking matrix is then used to rearrange pixel positions.

This division of labor is important. The map is not presented merely as a generic keystream source; it is specifically used to randomize the initialization of the diffusion mechanism and to construct the sorting-based permutation for confusion (Tang, 8 Sep 2025).

5. Empirical comparison and encryption performance

The paper compares the 2D-RA map against advanced benchmark chaotic maps using average values of $$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$5, $$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$6, $$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$7, and $$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$8 (Tang, 8 Sep 2025).

For the 2D-RA map, the reported values are

$$f(x,y)= -20\exp\left(-0.2\sqrt{\frac{1}{2}(x^2+y^2)}\right) -\exp\left(\frac{1}{2}\big[\cos(2\pi x)+\cos(2\pi y)\big]\right)+20+e.$$9

The strongest listed baseline is Toktas et al. (2024), with

$x^2$0

Other listed examples include Gao (2021),

$x^2$1

Nan et al. (2022),

$x^2$2

and Lai and Liu (2023),

$x^2$3

Within this benchmark table, the 2D-RA map exceeds the compared maps on all reported summary indicators.

The full encryption scheme is evaluated on standard benchmark images including Pepper and Baboon. The reported metrics are histogram analysis, correlation distribution, correlation coefficient, NPCR, and information entropy. The correlation coefficient is defined as

$x^2$4

with

$x^2$5

$x^2$6

$x^2$7

NPCR is defined as

$x^2$8

where

$x^2$9

For one test image, the paper reports original-image statistics

$y^2$0

and encrypted-image statistics

$y^2$1

with

$y^2$2

For another test image, the reported original-image statistics are

$y^2$3

and encrypted-image statistics are

$y^2$4

with

$y^2$5

The paper also compares the full encryption scheme against methods by Panwar et al. (2024), Afify et al. (2024), Liu et al. (2024), Hosny et al. (2021), Mohamed et al. (2021), Li et al. (2017), and Enayatifar et al. (2017). In that comparison, the proposed method reports

$y^2$6

with near-zero correlation coefficients and NPCR values

$y^2$7

6. Scope, limitations, and terminological context

Within the supplied literature, the exact term “2D-RA map” is explicitly defined in the image-encryption paper as the Rastrigin–Ackley-based hyper-chaotic map described above (Tang, 8 Sep 2025). Other papers in the supplied corpus use nearby abbreviations or related phrases in different senses, but do not define the same object.

For example, one paper discusses a “2D-Gray map” for codon organization rather than a chaotic dynamical system (Oliveira et al., 2015). Another uses “RA” in “2-D RARMA” to mean Rayleigh autoregressive moving average, a statistical model for positive-valued 2D SAR imagery rather than a modulo-one chaotic map (Palm et al., 2022). Several papers describe 2D radio maps or radio map estimation, but those are spatial propagation fields, not the 2D-RA hyper-chaotic map (Santos et al., 2023, Wang et al., 7 Feb 2025, Huang et al., 26 Apr 2026). This suggests that “2D-RA map” is terminologically specific in the encryption paper and potentially ambiguous outside that context.

The defining paper also notes limitations, though these mainly concern the full encryption scheme rather than the map alone. The self-adaptive diffusion design is reported to be vulnerable to cropping attacks, because local ciphertext damage propagates widely during decryption. The method is also said to be weak on image classes where adjacent pixels are intentionally dissimilar, such as chessboard-like patterns. The paper does not specify transient-length removal before using the generated sequence, and it does not analyze finite-precision effects despite the use of $y^2$8, exponentials, cosine terms, and modulo-one folding. A plausible implication is that implementation details may matter for exact reproducibility, but that point is not developed analytically in the source.

In the literature represented here, the 2D-RA map is therefore best understood as a two-dimensional hyper-chaotic iterative map for image encryption, defined by hybridization of Rastrigin and Ackley components, empirically characterized by large positive Lyapunov exponents and strong sequence complexity, and operationally integrated into diffusion and permutation stages of a hash-dependent encryption pipeline (Tang, 8 Sep 2025).