Image encryption through lucas sequence, s box and chaos theory

A rather thorough numerical analysis was carried out by the authors of [23], who proposed an image encryption scheme based on an LA-semi group for confusion, while a chaotic continuous s

Image Encryption Through Lucas Sequence,

S-Box and Chaos Theory

Wassim Alexan, SMIEEE, and Mohamed ElBeltagy

Faculty of IET The German University in Cairo

Cairo, Egypt wassim.alexan@ieee.org mohamed.elbeltagy@ieee.org

Amr Aboshousha Physics Department The German University in Cairo

Cairo, Egypt amr.aboshousha@guc.edu.eg

Abstract—The need for image encryption schemes that are

of low computation complexity is ever increasing Meanwhile,

the use of chaotic functions and mathematical sequences to

administer security has been shown to provide very good

results in recent literature This paper proposes a lightweight

image encryption scheme that is based on 3 stages The first

stage incorporates the use of the Lucas sequence, the second

stage incorporates the use of an S-box, while the third stage

makes use of the Sin Logistic map Performance evaluation of

the proposed encryption scheme indicates great resistivity to

various forms of attacks, attained at a low computation cost

This renders the proposed image encryption ideal for

real-time applications Furthermore, a comparison of the proposed

scheme with the state-of-the-art indicates its superiority

Keywords–Cryptography, image encryption, Lucas

num-bers, chaotic maps, S-box

I INTRODUCTION

The rapid developments in communications and Internet

technology have lead to digital images being nowadays

one of the most widely transmitted types of multimedia

data However, users transmitting and receiving such

data are constantly concerned with regards to their

security Digital images utilized in various applications

can depict sensitive military data, medical data, or novel

engineering blueprints [1] Thus, this brings the problem

of securely storing and transmitting them to the forefront

of problems that engineers and scientists are interested

in Most prominently, cryptography, steganography and

watermarking as scientific fields that attempt at securing

sensitive data have been evolving in accordance with

the aforementioned developments in other technologies

This has lead to huge investments in their R&D in recent

decades [2]–[5], resulting in various forms of symmetric,

asymmetric encryption algorithms [6], [7], hash functions

[8], as well as steganography and watermarking schemes

for different forms of multimedia [9]–[13]

Nevertheless, since the inherent properties of digital

images render them having very large data capacities, as

well as a significant redundancy and correlation between

adjacent pixels, this makes their security rather different

than that of traditional text This translates into 3DES

and AES, for example, no longer being the best-suited

encryption algorithms to cater for the security of digital

images To that end, various approaches have been

proposed for the specific case of image encryption Those

include technologies that rely on chaos theory [14], neural

networks [15], DNA coding [16], image filtering [17],

quantum theory [18], cellular automata [19] and many others

Chaos theory in particular presents the field of security with multiple advantages from its utilization These are derived from the characteristics of chaotic function as a random phenomenon in nonlinear systems [14] Namely, their pseudo-randomness, ergodicity, periodicity, control parameters and sensitivity to initial conditions Such characteristics have been exploited in multiple research works In [20], the authors present an encryption algorithm for grayscale images, employing a 2D logistic sine map, along with an Arnold map and a linear congruential generator In [21], the authors present an image encryption scheme that makes use of multiple chaotic maps with a minimum number of rounds of encryption In [22], the authors present a lossless quantum chaos-based image encryption technique based on an S-box, a mutation operation, as well as an Arnold transform Furthermore,

in generating the keys, they make use of SHA-256 Their proposed technique offers a very high level of security,

as evidenced by the various performance metrics they calculate and provide Most notably of their research work

is the very large key space employed, in the order of 2256,

as well as their computation speed (11.920875 Mbit/s)

A rather thorough numerical analysis was carried out by the authors of [23], who proposed an image encryption scheme based on an LA-semi group for confusion, while a chaotic continuous system was adopted for diffusion Each

of the schemes in [20]–[23] are rather recent, utilizing chaotic systems and their computed evaluation metrics reflect excellent performance This makes them ideal counterpart schemes suitable for a comparative analysis with our proposed one

In cryptography, a substitution box (S-box) is considered

to be a basic component of symmetric key algorithms which perform substitution In block ciphers, they are usually used to obscure the relationship between the key and the ciphertext, thus ensuring Shannon’s property

of confusion One of the employments of S-box was used by the authors of [24], where they presented a Gray S-box for AES A more recent work for secure image encryption algorithm design uses a chaos based S-box [25], where a novel S-box design algorithm was introduced to create the chaos based S-box to be used

in the encryption scheme and performance evaluation

examinations were carried out The authors of [26]

propose an image encryption scheme that is based on

elliptic curve cryptography In their scheme, the authors

utilize a dynamic S-box which is based on the Henon map

Mathematical sequences are also commonly used in

cryptography There are mainly 4 types of sequences:

Arithmetic, Geometric, Harmonic and the Fibonacci

sequences Each of those 4 types is different and has

unique relations among their terms The Lucas sequence

has the same recursive relationship as the Fibonacci

sequence, where each term is the sum of the two previous

terms, but with different starting values This produces a

sequence where the ratios of successive terms approach

the golden ratio, and in fact, the terms themselves

are roundings of integer powers of the golden ratio

This sequence also has a variety of relationships with

the Fibonacci numbers, like the fact that adding any

2 Fibonacci numbers 2 terms apart in the Fibonacci

sequence results in the Lucas number in between [27],

as in Fig 1 Since many cryptographic algorithms

are based on random number generators (RNGs), the

Lucas sequence is a great fit, as it can be utilized as a

pseudo-RNG (PRNG) One of the earliest employments

of the Lucas sequence in cryptography was proposed in

[28] Soon enough, scientists and engineers adopted the

idea of utilizing it for encryption purposes Achieved

numerical results clearly showcase that the use of Lucas

sequence provides good defense against various attacks

[29] Furthermore, the authors of [30] proposed encryption

of images by employing the Lucas sequence at different

iterations of scrambled images of the Arnold transform

A more recent work for digital image confidentiality

scheme based on pseudo-quantum chaos and the Lucas

sequence is presented in [29], where the authors showcase

that their encryption technique possesses an excellent

key space and attains significant confidentiality Their

paper indicates that employing a chaotic system, along

with an S-box, exhibits additional complicated dynamical

behavior, sufficient arbitrariness, and uncertainty over

other research works that focus solely on chaotic functions

In this paper, we propose an image encryption scheme

that is based on 3 stages The first stage incorporates

the use of the Lucas sequence, while the second stage

incorporates the use of an S-box and the third stage

incorporates the use of Chaos Theory in the form of the

Sine Logistic map This paper is organized as follows

Section II briefly presents the Lucas sequence, followed

by the adopted S-box and the chaotic map used for the

proposed image encryption scheme Section III outlines

the numerical results of the computations and testing and

provides appropriate commentary on them Section IV

finally draws the conclusions of the paper and suggests a

future work

II THEPROPOSEDIMAGEENCRYPTIONSCHEME

The proposed image encryption scheme is composed

of three stages The first stage makes use of the Lucas

Fig 1: A comparison between Fibonacci and Lucas se-quences

Fig 2: Lucas numbers’ graphical representation

sequence, while second stages makes use of an S-box and

in the third stage chaotic map was used The next few sections introduce each of those three concepts

A The Lucas Sequence The Lucas numbers form an integer sequence named after the mathematician Franc¸ois ´Edouard Anatole Lu-cas (1842–91), who studied both that sequence and the closely related Fibonacci numbers Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences Fig 2 shows the pattern for the Lucas sequence, calculated using the iterative expression:

∞

X

n=1

For n ∈ Z and n ≥ 3, the attained values are {1, 3, 4, 7, 11, 18, 29, 47, 76, 123, } as in Fig 2 Note that here, we use L1= 1 and L2= 3 in (1) as the initial values

It was suggested in [19] that a Lucas sequence can be considered as a PRNG, since the Lucas numbers binary equivalence satisfies the characteristics of a randomly gen-erated bit stream For example, examining the first 200 numbers in the sequence reveals that the resulting bit stream is {1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, } After examining this sample which has a total of 14054 1s and 0s, the number of 1s is equal to 7075 and the number of 0s is equal to 6979 This satisfies that the Lucas sequence can give almost equally distributed stream of bits (50% are 1s and 50% are 0s) Furthermore, in Section III, we test the generated Lucas sequence with a NIST analysis and the result is that all the items of the NIST analysis are successfully passed

B S-Box

An S-box is a main part of modern-day block ciphers that helps in the generation of a disordered ciphertext for a specific plaintext Through the addition of an S-box, a nonlinear mapping among the input and output data is resolved to create confusion [31] The more

confusion an S-box can form in the output data, the more

secure a block cipher is As an outcome, the provision

of security by a block cipher employing one or more

S-boxes directly depends on the solidity of the S-box

Block ciphers consist of many components in addition to

one or more S-boxes Contrary to other components, an

S-box is the sole nonlinear component of block ciphers

that supports the improvement of data protection [32], [33]

The latest symmetric ciphers designed recently usually

use S-boxes that create more confusion for the attackers

S-boxes help in the provision of data security by creating

jumbled ciphertext An S-box design establishes a

nonlinear relation between the input and output data

in such a way that an invader is unable to deduce

input data from the output data through any analysis

Scientists have broadly explored such nonlinear mappings

to create stronger S-boxes The process of constructing

an S-box should be simple and efficient The construction

process of most of the S-boxes presented in the literature

consumes vast amount of time and are complex As an

example, S-boxes generated with the help of a linear

fractional transformation (LFT) depend heavily on the

use of the Galois field LFT, also known as the Mobius

transformation, is one of the many mappings that have

been extensively used for the creation of S-boxes [34], [35]

A straightforward and efficient method for S-box

construction using the idea of novel transformation,

modular inverse and permutation was inherited from the

authors of [36], where An example S-box was tested

and analyzed to verify its cryptographic strength using

standard criteria This was carried out by comparing it

with other recently projected S-boxes The investigation

outcomes seem to be in synchronization with the current

benchmarks that validate a technique and the performance

of their proposed S-box depicted good results when

compared to other S-boxes Table IV shows their S-box

C The Sine Logistic Map

The 2D Logistic Sine Map (LSM) is a 2 dimensional

chaotic map that exhibits a good chaotic performance as

it is derived from both the Logistic map and the Sine

map Fig 3 is the 2D graphical representation for the first

250 iterations which were used in our proposed encryption

scheme to generate a key of randoms bits It is defined as:

xn+1= sin(πa(yn+ 3)xn(1 − xn)), (2)

and

yn+1= sin(πa(xn+ 1 + 3)yn(1 − yn)) (3)

The output are the two sequences xn and yn while a

is the control parameter and a ∈ [0, 1] The following

parameters were used to generate the chaotic sequence:

x(1) = 0.3, y(1) = 0.3, and a = 0.9 as in [37]

D Image Encryption and Decryption Processes

The proposed image encryption scheme is implemented

as follows First, an image of appropriate dimensions is

Fig 3: The 2D shape of our proposed usage of the Sin Logistic map

chosen and its pixels are converted into a 1D stream of bytes Next, these bytes are converted into a bit stream

d Second, the mean intensity of the image pixels is calculated The resulting value is a rather small number, which we multiply by a magnifying factor fM Let us denote the resulting value by µ Next, we cyclically shift

d to the right by µ places and the resulting bit stream, now denoted dµ, is then XORed with kCA kCA is the first key, a bit stream of the same length as d and dµ, that

is made up of a repetition of the first NCA bits resulting from the binary representation of the first 256 elements of the Lucas sequence generated bit stream Let us denote the resulting bit stream as C1 This concludes the first step

of encryption Next, the S-box is used for substituting the decimal representation for each 8 bits from the bit stream acquired after the first step, as in Table IV Next, we change those decimal representations to a bit stream C2 At this point, we take the x and y coordinates of each of the points

of the resulting Sine logistic map equations and flatten them into a single 1D array Next, we list plot those values into 2D, as shown in Fig 3 Examining the plot in Fig 3, it

is clear that there are more positive values than there are negative ones So, we choose a threshold value λ, such that

if any of the values are above this threshold, they would be accounted as 1s, otherwise, they would be accounted as 0s This newly obtained bit stream of length NL would make

up the seed of our Sine Logistic Map based key We repeat those NL bits until it is of the same length as d and C1, thus forming the second key Let us denote it kL Next, we XOR kL with C2 obtaining C3 This concludes the third step of encryption Finally, C3 is reshaped back into an image of the same dimensions as those of the plain image, obtaining the encrypted image Fig 4 provides a graphical illustration of the proposed image encryption scheme The decryption process is implemented in a reverse manner as

to that of the encryption process

III NUMERICALRESULTS ANDPERFORMANCE

EVALUATION

This section outlines the numerical results and analysis of the proposed lightweight image encryption scheme Performance is evaluated and compared to counterpart algorithms found in the literature The metrics employed for evaluation are presented and well expressed mathematically in [38], [39] The proposed scheme is implemented using the computer algebra system

Fig 4: Simplified flowchart of the proposed image

encryp-tion scheme

Wolfram Mathematica® on a machine running Windows

10 Enterprise, equipped with a 2.3 GHz 8-Core Intel®

CoreTM i7 processor and 32 GB of 2400 MHz DDR4

of memory The utilized keys are assigned the following

values: NCA = 100, NL = 50, fM = 106 and λ = 0.68

Four images that are commonly used in image processing

are utilized in this section These are Lena, Mandrill,

Peppers and House, all of dimensions 256 × 256

Examining Table II indicates that no information could

be discerned from any of the encrypted images Moreover,

decrypting any of the encrypted images results back in

the plain images Table II also shows the corresponding

histograms for each of the images A histogram of an

image shows the characteristics of its pixel distribution

One can clearly see that the histograms for each of

the plain and decrypted images have varying pixel

characteristics This is not the case for the histograms of

the encrypted images, which are of uniform nature The

more uniform the histogram distribution of an encrypted

image, the better its performance in resisting attacks of

statistical nature

Fig 6 shows the correlation coefficient diagrams of the

plain and encrypted Lena image It is clearly seen that the

horizontal, vertical and diagonal correlation coefficients of

the adjacent pixels for the plain image are linear While on

inspecting the plots generated from the encrypted image,

it is clear that the plots are uniform and have a scatter-like

distribution This signifies a resistance of the proposed

scheme to statistical analyses or attacks

To test the suitability of the proposed scheme to

real time applications, we measure the encryption

and decryption time for a range of image dimensions

{128, 256, 512, 1024, 2048} Fig 5 and Table I provide the

time taken for encryption, decryption and their total For

an image of dimensions 128 × 128, both the encryption

TABLE I: Processing time for various dimensions of the Lena image

Image Dimensions Time [s]

Encryption Decryption Total

128 × 128 0.660872 0.462961 1.123833

256 × 256 2.333844 1.727118 4.060962

512 × 512 9.062353 6.875539 15.937892

1024 × 1024 35.129869 26.615498 61.745367

2048 × 2048 142.455636 113.529309 255.984945

and decryption times are close to half a second This means that the proposed scheme is suitable for real-time applications

Table III lists the computed values of MSE and PSNR

of our proposed scheme, as well as those of 2 of its counterparts from the literature, specifically [21] and [23]

A larger value of the MSE signifies an improved level

of security Our proposed schemes seems to outperform the MSE values of [23], while it is achieving a lower performance than that achieved in [21] Since the PSNR

as a metric is inversely proportional to the MSE, the comparison among those 3 schemes in terms of PSNR still holds the same significance as aforementioned

An encryption scheme should maximize the randomness

of an encrypted image One metric employed to evaluate the randomness in a encrypted image is the entropy Table

V shows the computed entropy values of the proposed scheme as well as 3 of its counterparts, namely [21]–[23] Our proposed scheme exhibits very comparable values to its counterparts

The NIST analysis is a statistical computational suite that is used for testing random numbers and PRNGs used for cryptographic modelling and simulation It tests the di-vergence of randomness in a bit stream This test is applied

on the binary sequence data stream of the encrypted image Values resulting from the NIST suite must be greater than 0.01, to render an encryption scheme successful By successful here, we mean that a scheme is resistant against any cryptographic attacks Table VI shows the computed values and the success of our proposed scheme at passing all the tests

IV CONCLUSIONS ANDFUTUREWORKS

In this paper, we proposed an image encryption scheme that is based on 3 stages The first stage incorporated the use of the Lucas sequence, while the second stage incorpo-rated the use of an S-box and the final stage incorpoincorpo-rated the use of the Sine Logistic Map Performance evaluation

of the proposed scheme was carried out utilizing a number

of appropriate metrics and analyses Those included visual inspection of both plain and encrypted images, a histogram analysis, a cross correlation analysis, entropy values, com-putation of the MSE and the PSNR values A comparison with counterpart schemes from the literature was carried out and the proposed scheme exhibited comparable security

TABLE II: Numerical results of the achieved values for various metrics.

Image data Plain image/histogram Encrypted image/histogram Decrypted image/histogram Lena

d = 256 × 256

Mandrill

d = 256 × 256

Peppers

d = 256 × 256

House

d = 256 × 256

TABLE III: A comparison of MSE and PSNR values among the proposed scheme and its counterparts from the literature.

Image Proposed Scheme [21] [23]

MSE PSNR [dB] MSE PSNR [dB] MSE PSNR [dB] Lena 8926.96 8.6237 10869.73 7.7677 4859.03 11.3 Mandrill 8290.84 8.9448 10930.33 7.7447 7274.44 9.55 Peppers 10045.1 8.11128 N/A N/A 6399.05 10.10 House 8351.64 8.91309 N/A N/A N/A N/A

TABLE IV: S-box values from [40]

203 153 138 245 187 130 186 167 144 40 131 250 202 47 244 136

141 166 91 116 121 13 210 55 7 126 217 113 90 71 127 70

12 119 104 54 190 88 184 32 42 248 112 158 89 11 209 154

229 30 207 220 195 23 216 128 118 102 109 255 249 4 53 1

211 74 197 206 235 198 18 193 81 149 19 117 115 31 5 147

231 25 182 242 163 14 177 180 254 24 208 123 111 84 224 178

161 201 157 133 175 236 218 241 106 165 137 213 36 162 38 230

10 205 107 69 97 251 159 222 191 65 57 93 179 212 17 72

76 20 214 194 61 125 114 101 34 152 171 122 228 68 85 199

170 83 0 174 87 58 172 189 29 135 86 105 223 156 143 132

196 63 43 237 181 185 240 45 78 164 200 192 66 35 98 6

160 188 150 52 247 27 219 95 221 44 120 92 151 16 39 21

82 124 100 56 96 79 33 173 146 134 49 233 3 77 80 243

94 15 75 232 26 110 252 226 142 140 238 108 176 64 239 59

22 51 60 183 46 67 204 253 8 2 148 155 139 129 41 234

62 37 50 227 28 103 48 246 168 99 145 9 215 225 73 169

Fig 5: Processing time for encryption and decryption,

depicted for a range of image dimensions Dashed red

depicting encryption and solid blue depicting decryption

TABLE V: Entropy values for encrypted images

Image Proposed [21] [22] [23]

Lena 7.9990 7.9990 7.9978 7.9968

Mandrill 7.9990 7.9991 7.9993 N/A

Peppers 7.9990 N/A N/A N/A

House 7.9989 N/A N/A N/A

performance Finally, the processing time was computed

and shown to be rather low, signifying the suitability of its

use in real-time applications A future work that is expected

to result in even better performance could include the use

of different S-boxes or chaos-based functions

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TABLE VI: NIST analysis on Lena encrypted image.

Test name Value Remarks

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Block Frequency 0.718621 Success

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