Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

CHAOS BASED INDISCERNIBLE IMAGE

STEGANOGRAPHY SCHEME^∗

Borislav Stoyanov^†

Tsvetelina Ivanova^‡

Plamen Ribarski^¶

Dimitar Dobrev^§

Communicated by V. Dr˘agan

DOI

10.56082/annalsarscimath.2026.2.131

Abstract

In the current digital era, information security is a top priority and

outdated algorithms must be updated or replaced. This article presents

a novel method that combines encryption and steganography to conceal

secret text messages in color graphics. Using a chaotic pseudorandom

generator, the position and arrangement of the picture pixels used for

information embedding are chosen at pseudorandom. Another layer of

protection is to encrypt the secret message before embedding it. This

issue will deter attackers from looking for signs of steganography. The

suggested indiscernible stegoalgorithm is being evaluated. We perform

randomness tests, histograms, peak signal-to-noise ratio analysis, and

other tests using conventional statistical and empirical methods. The

novel results are presented and analyzed in the current article.

Keywords: chaotic functions, pseudorandom bytes, least signiﬁcant bit,

steganography.

MSC: 68P25, 11K45, 94A60.

^∗Accepted for publication on January 14, 2026

^†borislav.stoyanov@shu.bg, Department of Computer Informatics, Faculty of Math-

ematics and Informatics, University of Shumen, Shumen, Bulgaria

^‡ts.r.ivanova@shu.bg, Department of Computer Informatics, Faculty of Mathematics

and Informatics, University of Shumen, Shumen, Bulgaria

^§d.d.dobrev@shu.bg, Department of Computer Informatics, Faculty of Mathematics

and Informatics, University of Shumen, Shumen, Bulgaria

^¶p.ribarski@shu.bg, Department of Computer Informatics, Faculty of Mathematics

and Informatics, University of Shumen, Shumen, Bulgaria

131

Chaos based indiscernible image steganography algorithm

132

1 Introduction

Information security is crucial in today’s digital environment. The necessity

for eﬃcient data protection techniques is become ever more important as

Internet usage rises. The study and practice of hiding and analyzing infor-

mation is called steganology. Steganography and steganalysis are its two

main subﬁelds. The process of concealing a hidden message inside another

data ﬁle is known as steganography [6]. Data hiding serves as one of the

best methods for cover up and protecting sensitive information [14]. In this

work, we focus on steganography in bitmap ﬁles. Hiding user information

in an image ﬁle is known as image steganography.

The key contributions of this work are as follows:

• We propose a novel algorithm for generating pseudorandom byte ar-

rays based on a double Ikeda function, demonstrating favourable sta-

tistical characteristics.

• This pseudorandom generation algorithm is integrated into a new

steganographic scheme.

• Comprehensive evaluation of the proposed approach indicates close to

zero Mean Square Error (MSE) values, high peak signal-to-noise ratio

(PSNR), strong structural similarity, and very similar histograms.

2 Proposed scheme

2.1

Double Ikeda function-based pseudorandom byte

generation

The following equations deﬁne the Ikeda map [10]:

x_n+1= 6 + 0.9 (x_ncos ξ_n− y_nsin ξ_n) ,

(1)

(2)

y_n+1= 0.9 (x_nsin ξ_n+ y_ncos ξ_n) ,

where

6

1 + x²_n+ y_n²

ξ_n= 3.1 −

.

(3)

The parameter u ∈ (0, 1] controls the dissipation of the system.

The double Ikeda function is plotted in Figure 1 .

The following steps form the basis for the byte generation:

B. Stoyanov, T. Ivanova, D. Dobrev, P. Ribarski

133

2

1.5

1

0.5

0

-0.5

-1

-2

0

2

4

6

8

y

x

Figure 1: Double Ikeda function

1. The initial values x_1,0, y_1,0, x_2,0, y_2,0, x_3,0, y_3,0for three double Ikeda

functions are determined.

2. The values of x_1,k, y_1,k, x_2,k, y_2,k, x_3,k, y_3,kare computed, where k is a

ﬁxed constant.

3. For each i > k, six non-negative integer values a, b, c, d, e, f ∈ [0, 256)

are computed as follows:

ꢀ

10

ꢀ

a = b10 · x_1,ic mod 256,

b = b10 · y_1,ic mod 256,

(4)

(5)

(6)

ꢀ

10

c = b10 · x_2,ic mod 256,

d = b10 · y_2,ic mod 256,

ꢀ

10

e = b10 · x_3,ic mod 256,

f = b10 · y_3,ic mod 256.

4. A single output byte byte_iis then computed as follows:

(

c ⊕ d, if (a ⊕ b) mod 2 = 0,

byte_i=

e ⊕ f, otherwise.

5. Steps 3 and 4 are repeated until the desired length of the pseudoran-

dom byte sequence is reached.

Chaos based indiscernible image steganography algorithm

134

2.2

Proposed steganography scheme

The proposed steganography algorithm consists of the following steps:

1. The initial seed for the double Ikeda function-based pseudorandom

byte generator is determined.

2. A special end-of-text symbol is appended to the end of the plain mes-

sage.

3. The plain message is converted into an integer byte array T of length

N, where T_iis the ASCII code of the i-th character of the plain text.

4. Assuming the maximum possible value of N is MAXN = n · m ·

c, MAXN pseudorandom bytes b_iare generated using the proposed

generator. These bytes will be used to encrypt the message.

5. The array T is encrypted. The elements of the encrypted array are

computed as T⁰= T_i⊕ b_i.

i

6. Let A_n,m,cbe a 3-dimensional array representing a stego image, where

n and m are the dimensions of the image, and c is the number of color

channels.

7. The array T⁰is converted into a bit array B⁰of length len = 8 · N.

Each encrypted bit B_i⁰, for i = 1, 2, . . . , len, is embedded into the image

using the following procedure:

• A total of dlog₂ne + dlog₂me + dlog₂ce bits are generated using

the proposed pseudorandom byte generator.

• These bits are split and interpreted as integers to determine the

coordinates (x, y) and the channel index z, by computing each as

the result modulo n, m, and c, respectively.

• If the location (x, y, z) has already been used, repeat the step to

generate a new set of coordinates.

• The bit B_i⁰is embedded into the least signiﬁcant bit (LSB) of the

pixel located at (x, y) in channel z.

• The location is marked as used to prevent future overwriting.

The decoding scheme uses the following steps:

1. The initial seed for the double Ikeda function-based pseudorandom

byte generator and the end-of-text symbol are identiﬁed.

B. Stoyanov, T. Ivanova, D. Dobrev, P. Ribarski

135

2. Using the proposed pseudorandom byte generator, generate MAXN =

n · m · c pseudorandom bytes b_ito decrypt the embedded message.

3. Extract the embedded bits and decrypt them using the following pro-

cedure:

• Generate coordinates (x, y, z) according to the method described

in step 6 of the embedding algorithm.

• If the coordinates (x, y, z) are already marked as used, generate

a new set of coordinates.

• Mark (x, y, z) as used and extract the embedded bit from the

LSB of the element A_x,y,z

.

• Decrypt each byte (group of 8 bits) by performing the exclu-

sive OR (⊕) operation between the extracted byte and the corre-

sponding pseudorandom byte b_i. Append the resulting value to

an integer list extracted

numbers.

• Repeat the extraction and decryption process until the resulting

value matches the ASCII code of the special end-of-text symbol,

which signals the end of the message.

4. The list extracted

numbers is converted into the original message by

mapping each number to its corresponding ASCII character. The spe-

cial end-of-text symbol is then removed from the reconstructed mes-

sage.

The dimensions of the initial seed values, the linear complexity, and the

results of the statistical tests suggest that the pseudorandom algorithm is ca-

pable of ensuring a favourable pseudorandom-like quality and an acceptable

level of security [12 ], [9], [5], [8], [4].

3 Security analysis

As an illustration, Figure 2 displays the 4.2.01 Woman image in Figure 2(a),

along with its stego counterparts in Figures 2(b) to 2(f). Visual inspection

shows no apparent diﬀerences between the original and the stego images;

they look identical to the naked eye, with no visible traces of embedded

information.

The randomness of the proposed scheme is evaluated using two statistical

tools, NIST and ENT.

Chaos based indiscernible image steganography algorithm

136

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Illustrated comparison of the 4.2.01 Woman input image and the

stego images: (a) original image, (b) stego image with 100 symbols, (c)

stego image with 200 symbols, (d) stego image with 300 symbols, (e) stego

image with 400 symbols, and (f) stego image with 500 symbols.

B. Stoyanov, T. Ivanova, D. Dobrev, P. Ribarski

137

Table 1: ENT test results.

ENT test

Bit stream

Byte stream

Entropy

1.000000 bits/bit

7.999998 bits/byte

Optimum compression

decrease the size

with 0%

Chi

2.53

233.17

increase with 11.17%

increase with 83.30%

π value

3.141079566

Correlation coeﬃcient

0.000040

0.000018

The ENT package [13 ] comprises six diﬀerent statistical tests. We eval-

uated the output generated by the proposed pseudorandom byte scheme

based on the double Ikeda function using a data array of 100,000,000 bytes.

The results in Table 1 met all the criteria of the ENT tests.

A total of 15 statistical tests form the basis of the NIST program [2].

For a sample of 1000 binary arrays, the minimum acceptable pass rate for

each statistical test — except the random excursion variant test is around

980. In the case of the random excursion variant test, the minimum pass

rate is approximately 603 out of 617 binary arrays. The output numbers are

presented in Table 2. The NIST test is successfully passed, with all p-values

uniformly distributed across the 10 subintervals and the pass rate falling

within the acceptable range.

PSNR quantiﬁes the ratio between the maximum achievable power of a

signal and the power of the corrupting noise that degrades its representation.

It is formally expressed as:

(2^c− 1)²

PSNR = 10 log₁₀

(dB),

(7)

MSE

here, c denotes the pixel bit depth, and MSE refers to the mean squared error

between the input image and the stego image. The MSE is mathematically

declared as follows:

1

MSE =

n

X X

(q[i, j] − q⁰[i, j])²,

(8)

nn

i=1 j=1

where q[i, j], q⁰[i, j] is the ith-row jth-column pixel in the input and stego

images, respectively. Tables 3 and 4 present the calculated MSE and PSNR

values for the proposed steganographic scheme, with data embedded in vary-

ing payload sizes - 100, 200, 300, 400 and 500 characters (equivalent to

Chaos based indiscernible image steganography algorithm

138

Table 2: NIST test results.

NIST test

P-value

Pass rate

frequency

0.235589

989/1000

block frequency

0.083526

990/1000

cumulative sums

0.881625

990/1000

runs

0.920383

989/1000

longest run

0.763677

988/1000

rank

0.884671

994/1000

ﬀt

0.348869

989/1000

non-overlapping template

0.585863

990/1000

overlapping template

0.043087

981/1000

universal

0.429923

989/1000

approximate entropy

0.821937

991/1000

random excursion

0.430269

611/617

random excursion variant

0.584246

611/617

serial

0.453050

993/1000

linear complexity

0.807412

987/1000

800, 1600, 2400, 3200, and 4000 bits, respectively). The results indicate

consistently low, close to 0.0 MSE values, and repeatedly high PSNR val-

ues, all exceeding 67 dB, demonstrating that the proposed chaos-based LSB

steganography scheme maintains strong imperceptibility and a high level of

security.

The Structural Similarity Index (SSIM) is a metric used to evaluate the

visual similarity between the input and the stego images [7]. The SSIM

between two image patches x and y is deﬁned as:

(2µ_xµ_y+ C₁)(2σ_xy+ C₂)

SSIM(x, y) =

(9)

(µ²_x+ µ²_y+ C₁)(σ_x²+ σ_y²+ C₂)

where: µ_xis the average (mean) of image x, µ_yis the average (mean) of

image y, σ_x²is the variance of image x, σ_y²is the variance of image y, σ_xyis

the covariance between images x and y, C₁= (K₁L)²and C₂= (K₂L)²are

constants to stabilize the division with weak denominators, L is the dynamic

range of the pixel values (typically 255 for 8-bit grayscale images), and K₁ꢀ

1, K₂ꢀ 1 are small constants (e.g., K₁= 0.01, K₂= 0.03). The SSIM score

ranges from -1 to 1, where a value of 1 denotes perfect structural similarity.

In contrast to PSNR, which measures pixel-wise intensity diﬀerences, SSIM

assesses structural variations between two images, oﬀering a metric that

aligns more closely with human visual perception. In Table 5 , we provide

B. Stoyanov, T. Ivanova, D. Dobrev, P. Ribarski

139

Table 3: MSE for images with 100, 200, 300, 400, and 500 characters em-

bedded.

Images

100 chars

200 chars

300 chars

400 chars

500 chars

4.1.01

0.0019

0.0042

0.0061

0.0080

0.0104

4.1.02

0.0022

0.0040

0.0061

0.0080

0.0103

4.1.03

0.0020

0.0040

0.0062

0.0083

0.0102

4.1.04

0.0021

0.0042

0.0061

0.0082

0.0100

4.1.05

0.0020

0.0041

0.0060

0.0082

0.0104

4.1.06

0.0021

0.0041

0.0060

0.0082

0.0103

4.1.07

0.0020

0.0041

0.0060

0.0082

0.0103

4.1.08

0.0020

0.0040

0.0061

0.0081

0.0103

Table 4: PSNR for images with 100, 200, 300, 400, and 500 characters

embedded.

Images

100 chars

200 chars

300 chars

400 chars

500 chars

4.1.01

75.2348

71.9499

70.2533

69.0995

67.9641

4.1.02

74.7321

72.0905

70.2895

69.0775

68.0069

4.1.03

75.0680

72.1292

70.2282

68.9610

68.0262

4.1.04

74.9178

71.8760

70.2569

68.9663

68.1332

4.1.05

75.0246

71.9659

70.3664

68.9851

67.9535

4.1.06

74.9602

72.0143

70.3370

68.9690

67.9940

4.1.07

75.0680

71.9927

70.3296

68.9743

67.9876

4.1.08

75.0354

72.0631

70.2931

69.0283

68.0198

Chaos based indiscernible image steganography algorithm

140

Table 5: SSIM for images with 100, 200, 300, 400, and 500 characters em-

bedded.

Images

100 chars

200 chars

300 chars

400 chars

500 chars

4.1.01

1.0000

0.9999

4.1.02

1.0000

0.9999

4.1.03

1.0000

0.9999

4.1.04

1.0000

0.9999

4.1.05

1.0000

0.9999

4.1.06

1.0000

0.9999

4.1.07

1.0000

0.9999

4.1.08

1.0000

0.9999

Table 6: Comparison of our steganography scheme with other techniques.

Scheme

MSE

PSNR

SSIM

Proposed

0.0019

75.2348

1.000000

Ref. [7]

0.0016

76.1253

0.999985

Ref. [1]

0.3300

52.9530

-

Ref. [3]

0.2705

53.4623

-

Ref. [11 ]

0.0191

113.5238

0.999979

the calculated values for SSIM for the presented steganography scheme.

The presented results show that the SSIM scores are close to 1.0. These

ﬁndings suggest that the novel scheme ensures high image quality and strong

structural similarity.

The image histograms graphically represent the gray-level distribution

within digital images. In this test, the histograms of the input and stego

images are compared. Furthermore, histogram analysis was performed using

ImageJ (https://imagej.net/ij/) to assess the gray value distribution in the

original and stego versions of image 4.2.01 Woman, as shown in Figure 3.

The analysis reveals that the stego image histograms closely match those of

the original, with no discernible anomalies or evidence of embedded data.

A selection of computed values for the proposed scheme and various

existing algorithms is outlined in Table 6. The ﬁndings indicate that the

proposed scheme produces results that are on par with or superior to those

of closely related techniques.

B. Stoyanov, T. Ivanova, D. Dobrev, P. Ribarski

141

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Histogram analysis of the 4.2.01 Woman input image and the

stego images: (a) original image, (b) stego image with 100 symbols, (c)

stego image with 200 symbols, (d) stego image with 300 symbols, (e) stego

image with 400 symbols, and (f) stego image with 500 symbols.

4 Conclusions

We proposed a novel indiscernible steganographic scheme based on spread

spectrum image processing, utilizing chaos-driven pseudorandom insertion

of least signiﬁcant bits. This technique enables the embedding of a digital

message within an input image without altering its dimensions. Importantly,

retrieval of the hidden data does not require the original image, and security

is ensured through shared secret keys between sender and receiver. Even if

the embedding method is known, unauthorized parties cannot extract the

message without the correct keys. Moreover, the embedded signal’s power

is minimal relative to the cover image, resulting in a low probability of

detection and making the presence of hidden data virtually imperceptible.

Acknowledgements. This work is supported by the Scientiﬁc research

fund of Konstantin Preslavsky University of Shumen under the grant No.

RD-08-75/31.01.2025.

Chaos based indiscernible image steganography algorithm

142

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