Cryptanalysis of an image encryption algorithm based on PWLCM and inertial delayed neural network

Abstract

Recently, a color image encryption algorithm is suggested by Lakshmanan et al. in [IEEE Transactions on Neural Networks and Learning Systems 2016, doi: 10.1109/TNNLS.2016.2619345]. In encryption algorithm, a piece-wise linear chaotic map is adopted to create a permutation matrix to perform shuffling of pixels of plain-image. The encryption of shuffled image is proceeded by extracting keystreams from chaotic inertial delayed neural network. This paper evaluates the security of encryption algorithm to unveil its inherent defects and proposes to present a complete cryptanalysis. To demonstrate the break of algorithm, we apply proposed chosen-plaintext attack that successfully recovers the exact plain-image from encrypted image without secret key. The proposed cryptanalysis shows that the encryption algorithm is inapt to realize a secure communication, the security claims by authors are not valid as the algorithm has serious security flaws and susceptible to proposed attack.

Keywords

Cryptanalysis image encryption chaotic inertial neural network permutation matrix chosen-plaintext attack

1 Introduction

Drastic growth of Information technology led to imminent creation of huge amount of image data. Development of contemporary image cryptosystems became crucial as a result of increasing needs for better security relevant to applications in different fields like network communications, business, defense mechanisms, etc. Image data carry certain innate features viz. substantial correlation among image pixels and their considerably huge size [1]. The repercussions of such above reasons led to the need of appropriate encryption algorithms so as to realize quality encryption strength. In order to meet the Shannon requirements [2], an appropriate encryption scheme is characterized by confusion and diffusion, along with substantial pseudo-randomness. As a result, the encrypted image gets substantially uncorrelated to plain-image. This led to recent development of many image cryptosystems [3 –7], to meet the demands of improved encryption quality and attacks resistance ability.

Chaotic dynamical phenomenon being an interesting and popular property of chaotic systems is used to design dynamic image cryptosystems. The chaotic systems are sensitively dependent on its system parameters, initial conditions, exhibit long periodicity and randomness [8, 9]. Due to these dynamic properties, chaotic systems prove to be an effective possibility for designing appropriate image encryption schemes [10 –15].

The provision of security regarding the transfer of data over certain kinds of open networks, mainly Internet, is one of the many cryptographic needs. Cryptanalysis being one of the vital components of modern day cryptology can be used to analyze the susceptibilities regarding the security with an aim to arrive at practically more secure system. Therefore, it led to cryptanalysis of many contemporary image encryption schemes [16 –20], by many research scholars and academicians. The security assessment of such systems is done against direct or indirect chosen plain-text attack (CPA), known plain-text attack (KPA), chosen cipher-text attack (CCA), etc, [21, 22] and proposals assure an equitable degree of security. Cryptanalysis has emerged laterally with cryptography, and the concours is quite evident through the history of cryptography- contemporary cryptanalytic algorithms invented to crack the security algorithm. Professional and practical cryptanalysts play a vital role in assessment, validation and substantiation of security of cryptosystem [22].

Li et al. have proposed a general framework to show that the permutation-only image encryption schemes are not secure and susceptible to attacks proposed by them in [23]. The reason being, a permutation matrix can always be revealed which is equivalent to secret key to recover the plain-image from cipher-image. In [24], Zeng and Liu analyzed the security of a couple images encryption scheme which is based on DNA operations and Lorenz chaotic system, it is found that the algorithm has weakness in the generation of pseudo-random sequence used for encrypting the images. They show that encryption algorithm is susceptible to the attack proposed by them using some pair of chosen-plain-text and corresponding cipher-texts. A divide and conquer attack procedure has been proposed by Yap et al. in [25] to cryptanalyze successfully an image encryption algorithm involving multiple rounds of operation. Norouzi and Mirzakuchaki cryptanalyzed an image encryption proposal which composed of two-stage permutation and one-stage substitution. The proposal found to exhibits excellent encryption effect but the same has been broken using chosen-plaintext attack by Norouzi in [26] to prove the insecurity of encryption proposal. Arroyo et al. [27] scrutinized the security of cryptosystem suggested by Vidal et al. in [28] using chaotic systems and features of quantum cryptographic communication. In [27], the authors have unveiled certain limitations due to quantum properties of the cryptosystem and loopholes similar to in analog chaos-based cryptography along with some configuration problems inherent to the encryption system. A color image encryption proposal using chaotic tent map and 6-th order hyper-chaotic CNN system consisted of one-time confusion and many-times diffusion procedures was suggested in [29]. This encryption proposal is completely cryptanalyzed by Wen as it was found to be insecure against the cryptographic attack proposed by Wen in [30]. In [31], Chen at al. have proposed a procedure of differential cryptanalysis using codebook attack to break multi-round cryptosystem of [32] using a number of differential binary images. Ahmad et al. [33] have performed security analysis of an image encryption algorithm suggested in [11] and uncovered few internal security defects to indicate that the scheme is practically unsuitable to realize image-based secure communication. In [34], Zhang et al. have broken a choquet fuzzy integral-based color image encryption technique with shift-diffusion architecture. Zhang et al. shown that the encryption technique is irreversible and breakable under known plain/cipher image pair. An novel image cryptosystem based on 3D bit matrix using 3D chaotic system and 3D Cat map are adopted for double random permutation stage which is followed by confusion process using a chaotic Logistic map is given [35]. Recently, Wu et al. have cryptanalyzed the encryption algorithm in [36]. They found that the algorithm is not applicable for secure communication as it holds few security shortcomings and vulnerable to attack proposed by them.

According to Kerckhoff, “only secrecy of the key provides security” [37], which can be further reformulated as Shannon’s maxim: “the enemy knows the system” [2], which assure the fact that the security of the cryptosystem in most instances is dependent on the secrecy of key only. For most cryptanalysts’ objectives, it is figuratively pretended that the attacker knows almost all information about the encryption system. As an implication, if the secret key or the plaintext gets arbitrated, it leads to total break of the cryptosystem [38].

Of late, Lakshamanan et al. [39] suggested an image encryption algorithm based on piece-wise linear chaotic map and synchronized master-slave chaotic inertial neural network. The algorithm is based on the permutation-diffusion structure. The statistical performance of encryption algorithm showed that it holds excellent encryption strength with high entropies, flat histograms, large key-space, good PSNR, low correlation among adjacent pixels, etc. However, the careful security investigation uncovers some security flaws that make the encryption algorithm weak and vulnerable to attack. In this paper, we performed a careful security analysis of image cryptosystem in [39], highlights implicit susceptibilities and flaws. Henceforth, by exploiting the susceptibilities, the cryptosystem is broken by proposed cryptanalysis which completely retrieves the plain-image without any prior knowledge of secret key.

The outline of remaining paper is as follows. In the subsequent section, a brief description of image encryption algorithm under study is provided. The findings of security evaluation of encryption algorithm in terms of security defects are discussed in Section 3. The proposed attack to break the encryption algorithm totally is persuaded in Section 4 along with plain-image sensitivity analysis. The Section 5 is prepared to conclude the work.

2 Lakshmanan et al. image encryption algorithm

In [39], an image encryption algorithm based on PWLCM and chaotic inertial neural network is proposed. The algorithm has two stages, namely the shuffling stage and encryption stage. A PWLCM system defined by Equation (1) is utilized to carry out shuffling of plain-image through a permutation matrix T. The shuffled image is then passed through the encryption stage. In this stage, the shuffled image is ciphered through keystreams U_R, U_G, U_B extracted from chaotic inertial neural network whose state is governed by Equation (2). $x_{n + 1} = {\begin{matrix} \frac{x_{n}}{p} & 0 < x_{n} \leq p \\ \frac{1 - x_{n}}{1 - p} & p < x_{n} \leq 0.5 \\ 1 - x_{n} & 0.5 < x_{n} \leq 1 \end{matrix}$ (1)

The chaotic inertial neural network has the matrix-vector form which is given in Equation (2). The authors have discussed its synchronization and dynamical behaviour in [39].

The Equation (2) is given as: ${\begin{matrix} \frac{du (t)}{dt} = - Au (t) + v (t) \\ \frac{dv (t)}{dt} = - Bv (t) - Cu (t) + W_{1} f (u (t)) \\ + W_{2} f (u (t - τ (t))) + I \end{matrix}$

Where, u (t) = [u₁ (t) , u₂ (t) , …, u_n (t)] ^T, v (t) = [v₁ (t) , v₂ (t) , …, v_n (t)] ^T, u_i (t) denotes the state variable of i-th neuron at time t with a delay of τ (t). The matrices A, B, C, W₁, W₂ and I have to be decided as per the procedure discussed in [39].

The algorithmic description of Lakshamanan et al. image encryption stages are as follows. Assume that a color plain-image be P having size M × N × 3. The image P is decomposed to its red, green and blue components, each having size M × N. Each of the color components of P is independently processed through two algorithm stages to generate corresponding color cipher image C.

The shuffling stage involves the following steps:

Iterate PWLCM with initial condition x₀ and p.

Compute u = floor [x_n × M] +1.

Perform the iteration of PWLCM until all M different values of u (1 = u_i = M) are retrieved.

Formulate a 2D permutation matrix T as $\begin{matrix} for i = 1 to M \\ T (i, u_{i}) = 1 all other entries of row i are 0 \\ end \end{matrix}$

Perform shuffling of image P using T as $P_{S} = transpose (P \times T) \times T$

The encryption stage takes the color components of P_S (the shuffled image from previous stage) as P_SR, P_SG, P_SB, the component can be separated as: $\begin{matrix} P_{SR} = P_{S} (1 : M, 1 : N, 1) \\ P_{SG} = P_{S} (1 : M, 1 : N, 2) \\ P_{SB} = P_{S} (1 : M, 1 : N, 3) \end{matrix}$

The computation of C from P_S is as follows:

Apply RK-4 method (with step-size = 0.001 and initial conditions u₀, v₀, x₀, y₀) to iterate chaotic inertial neural network in Equation (2) for more than L = M × N times.

Iterate chaotic inertial neural network for L times. Four floating point number sequences U₁, U₂, U₃, U₄ each of length are obtained. $\begin{matrix} U_{1} = {u_{1} (1), u_{1} (2), \dots \dots, u_{1} (L)} \\ U_{2} = {u_{2} (1), u_{2} (2), \dots \dots, u_{2} (L)} \\ U_{3} = {u_{3} (1), u_{3} (2), \dots \dots, u_{3} (L)} \\ U_{4} = {u_{4} (1), u_{4} (2), \dots \dots, u_{4} (L)} \end{matrix}$

Based on floating point sequences U₁, U₂, U₃, U₄, the encryption keystreams U_R, U_G, U_B are extracted as: $\begin{matrix} U_{R} (i) & = & [(| U_{1} (i) | - ⌊ | U_{1} (i) | ⌋) \times 10^{14}] \\ \mod 256 \\ U_{G} (i) & = & [(| U_{2} (i) | - ⌊ | U_{2} (i) | ⌋) \times 10^{14}] \\ \mod 256 \\ U_{B} (i) & = & [(| U_{3} (i) + U_{4} (i) | - ⌊ | U_{3} (i) + U_{4} (i) | ⌋) \\ \times 10^{14}] \mod 256 \end{matrix}$

The components P_SR, P_SG, P_SB of image P_S are encrypted through U_R, U_G, U_B as: $\begin{matrix} C_{R} & = & P_{SR} \oplus U_{R} \\ C_{G} & = & P_{SG} \oplus U_{G} \\ C_{B} & = & P_{SB} \oplus U_{B} \end{matrix}$

Reshape C_R, C_G, C_B to 2D matrices each of size M × N and prepare C as $\begin{matrix} C_{R} = reshape (C_{R}, M, N) \\ C_{G} = reshape (C_{G}, M, N) \\ C_{B} = reshape (C_{B}, M, N) \\ C (1 : M, 1 : N, 1) = C_{R} \\ C (1 : M, 1 : N, 2) = C_{G} \\ C (1 : M, 1 : N, 3) = C_{B} \end{matrix}$

Where, the symbols used have their usual meanings. The decryption algorithm is the reverse of encryption process (involving the two stages).

3 Drawbacks of image encryption algorithm

The scrutiny of Lakshamanan et al. encryption algorithm finds its inherent security defects which are discussed in what follows.

3.1 Permutation matrix T and encryption keystreams U_k are fixed

The encryption algorithm discussed in previous section is solely dependent on initial values of PWLCM and chaotic inertial neural network. However, once the secret key is kept unaltered, the resulting permutation matrix T is unchanged i.e. every plain-image is shuffled by this fixed permutation matrix. Likewise, the encryption keystreams U₁, U₂, U₃, U₄ subsequently yielding U_R, U_G, U_B are also get fixed, meaning the each shuffled image has same kind of encryption effect and diffusion. This is a serious flaw from security point of view. It may facilitate the attacker to recover the matrix T and keystreams U_R, U_G, U_B which are equivalent to secret keys.

3.2 Weak secret keys

Any chaos-based strong cryptosystem, the utilized chaotic system must not lead to fixed points and low period sequence which is the case of set of weak secret keys [40]. The weak keys results to low encryption quality and facilitates the attacker to know full or partial information of plain-text data [41]. The authors of image encryption algorithm under scrutiny utilized PWLCM system as pseudo-random number generator to obtain permutation matrix T for shuffling plain-image P securely. The initial condition assigned to (x₀, p) derives the quality of shuffling through matrix T. The analysis of PWLCM depicts that it turn to fixed and non-random values for specified initial values of (x₀, p) given in Table 1. The key space of Lakshamanan et al. included all possible values of (x₀, p). However, the specified values of the two parameters cannot be used as secret keys as it lead to weak encryption and must be avoided while encryption/decryption. Thus, the encryption algorithm has set of weak keys which may result to weak and poor encryption quality.

Table 1
Fixed iterative values of PWLCM system

x₀ p x_i (i > 0)

(0, 1) 0.5 A non-random sequence with poor period

(0, 1) x₀ 1, 0, 0, 0, ... ... ...

<0.5 2×x₀ 0.5, 1, 0, 0, 0, ... ... ....

x₀	p	x_i (i > 0)
(0, 1)	0.5	A non-random sequence with poor period
(0, 1)	x₀	1, 0, 0, 0, ... ... ...
<0.5	2×x₀	0.5, 1, 0, 0, 0, ... ... ....

3.3 Plain-image size restriction

The shuffling procedure using permutation matrix in turn imposes a size restriction on pending plain-images. Due to matrix T, the encryption algorithm under analysis is applicable to square images only. The encryption algorithm fails to encrypt any plain-images having size M × N × 3 and M ≠ N.

3.4 Precision effect

In chaos-based image cryptosystems, the chaotic systems are adopted as source of pseudo-random sequence generation. These sequences, in particular from 1D chaotic systems like PWLCM, degrade to periodic sequences after iterations for computer’s finite precision [42]. The key space analysis in [39] reveals that the authors have worked in 14-digit precision. However, a care has to be taken while encrypting large sized messages and images. As a preventive measure, some perturbation needs to be applied to chaotic systems that can easily avoid its settling to periodic sequence.

4 Proposed cryptanalysis to recover plain-image

According to Bruce Schneier [38], it is needed to know the inherent security defect(s) in security system which can be exploited to break the system. Crytanalyzing an encryption system may require unrealistic images or storage, but the complexity of attacks should be less than the brute-force attack. Breaking a cryptosystem can be a certified defect showing that the system doesn’t work as desired. An encryption algorithm is deemed as unsafe and insecure if it is not able to withstand any cryptanalytic attacks [30]. Kerckhoff’s principle entails the attacker to know everything of encryption algorithm except the secret key [37]. It entails that the attacker may have access to encryption or decryption machine of cryptosystem under analysis. We, as cryptanalysts exploited the security defects uncovered in previous section to propose and execute the proposed attack targeted to break the Lakshamanan et al. encryption algorithm.

In shuffling-stage of [39], the permutation matrix T is completely dependent to initial condition x0 and p of PWLCM, and it has nothing to do with pending plain-color image P. As a result, same permutation matrix is generated every time when encrypting distinct plain-images. In encryption-stage, the shuffled is PS is encrypted with keystreams extracted from chaotic inertial neural network which also doesn’t have the plain-image dependency. The keystreams solely depends on initial condition u0, v0, x0, y0, of chaotic inertial neural network. Neither the shuffling process nor the encryption process depends on pending plain-image. Consequently, the same T and U_R, U_G, U_B gets generated by Lakshamanan et al. encryption algorithm when encrypting distinct plain-images once the secret key is unchanged. Consequently, this leads to a highly concerning susceptibility that the security of the complete cryptosystem is entirely dependent on the secrecy of the key and retrieval of it by the attacker will lead to easy decryption of the images being transferred over respective communication network. Hence, recovering the matrix T is equivalent to retrieving initial values of x0 and p of PWLCM and recovering U_R, U_G, U_B are equivalent to retrieving initial values of u0, v0, x0, y0 of chaotic inertial neural network.

The complete proposed break procedure is prepared as chosen_plaintext_attack(). The proposed attack needs the specially designed images Z₂, Z₃ and J₂, J₃ as follows to reveal the matrix T and keystreams U_R, U_G, U_B. $\begin{matrix} Z_{2} & = & {(\begin{matrix} 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & \dots & ⋮ \\ 0 & 0 & 0 & \dots & 0 \end{matrix})}_{M \times N} \\ J_{2} & = & {(\begin{matrix} 1 & 2 & 3 & \dots & 255 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 0 & 0 & \dots & 0 \end{matrix})}_{M \times N} \end{matrix}$

The Z₃ = [z_i,j,k] _M×N×3 is 3D zero image where the pixels z_i,j,k = 0 for all 1 ≤ i ≤ M, 1 ≤ j ≤ N, 1 ≤ k ≤ 3, and J₃ is also a RGB image whose red component is J₂ and green and blue components both are Z₂.

The method y = Lakshamanan_encrypt(x) denotes the Lakshamanan et al. encryption algorithm as black box and generates encrypted color image y when encrypting color image x, z = Ex - OR (x, y) performs the bit-wise exclusive operation on inputs x and y to generate output z, reshape (x, y, z) changes the dimensions of matrix x to y × z, find_index (x, y) returns the index of element y in vector x, and rotate_CR (x, y) rotates circularly the 1D vector x in right direction by y positions.

P = chosen_plaintext_attack()
Input : Images Z₃, J₂ and received encrypted color image C
Output: Recovered color image P
begin
C_R = C (1 : M, 1 : N, 1)
C_G = C (1 : M, 1 : N, 2)
C_B = C (1 : M, 1 : N, 3)
[U_R, U_G, U_B] = get _ enc _ key (Z₃)
PSR = Ex - OR (C_R, U_R) ; PSG = Ex - OR (C_G, U_G) ; P_SB = Ex - OR (C_B, U_B)
P_SR = reshape (P_SR, 1, L) ; P_SG = reshape (P_SG, 1, L) ; P_SB = reshape (P_SB, 1, L)
T = get _ eqv _ T (Z₃, J₂)
for i = 1 to L
P₁ (i) = P_SR (t_i)
P₂ (i) = P_SG (t_i)
P₃ (i) = P_SB (t_i)
end
P₁ = reshape (P₁, M, N) ;
P₂ = reshape (P₂, M, N) ;
P₃ = reshape (P₃, M, N) ;
P (1 : M, 1 : N, 1) = P₁
P (1 : M, 1 : N, 2) = P₂
P (1 : M, 1 : N, 3) = P₃
return recoveredcolorimage P
end
[K_R, K_G, K_B] = get_enc_key ()
Input : Chosen zero image Z₃
Output: Keystreams K_R, K_G, K_B used in encryption-stage
begin
K = Lakshamanan_encrypt (Z₃)
K_R = K (1 : M, 1 : N, 1)
K_G = K (1 : M, 1 : N, 2)
K_B = K (1 : M, 1 : N, 3)
returnK_R, K_G, K_B
end
T = get_eqv_T()
Input : Chosen image Z₃ and J₂
Output: Permutation séquence équivalent to T = {t₁, t₂, t₃, …, t_L}
begin
[U_R, U_G, U_B] = get _ enc _ key (Z₃)
$for n = 1 to ⌈ \frac{M \times N}{255} ⌉$
J₃ (1 : M, 1 : N, 1) = J₂
J₃ (1 : M, 1 : N, 2) =0
J₃ (1 : M, 1 : N, 3) =0
H = Lakshamanan_encrypt(J₃)
H_R = H (1 : M, 1 : N, 1)
D = Ex - OR (H_R, U_R)
D = reshape (D, 1, L)
for j = 1 to 255
i = 255 × (n - 1) + j
t_i = find _ index (D, j)
end
J₂ = reshape (J₂, 1, L)
J₂ = rotate _ CR (J₂, 255)
J₂ = reshape (J₂, M, N)
end
return T
end

An exemplification of proposed attack on an image of size 4 × 4 ×3 is provided. The simulation of anticipated attack on a benchmark image of peppers is shown in Fig. 1.

Fig.1

Simulation of successful cryptanalysis under proposed chosen-plaintext attack: (a) color image P, (b) encrypted image C (known to attacker) of plain-image P (unknown and to be recovered), (c) zero image of size M×N×3, (d)-(f) recovered encryption keystreams, (g)-(i) recovered components of plain-image P, and (j) recovered color image P.

An example of proposed attack on a color image P (i, j) = (p_r (i, j) , p_g (i, j) , p_b (i, j)) of size 4 × 4 ×3 is as follows. $\begin{matrix} P & = & [\begin{matrix} 162, 84, 240 & 77, 158, 194 & 43, 124, 190 & 155, 123, 24 \\ 44, 41, 194 & 24, 44, 203 & 31, 173, 207 & 191, 98, 0 \\ 180, 25, 234, & 181, 23, 255 & 175, 146, 25 & 19, 5, 224 \\ 116, 253, 47 & 223, 65, 107 & 133, 124, 55 & 73, 104, 47 \end{matrix}] T = [\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}] \\ P_{S} & = & [\begin{matrix} 175, 146, 25 & 31, 173, 207 & 133, 124, 55 & 43, 124, 190 \\ 181, 23, 255 & 24, 44, 203 & 223, 65, 107 & 77, 158, 194 \\ 19, 5, 224 & 191, 98, 0 & 73, 104, 47 & 155, 123, 24 \\ 180, 25, 234 & 44, 41, 194 & 116, 253, 47 & 162, 84, 240 \end{matrix}] \\ C & = & [\begin{matrix} 86, 151, 148 & 131, 92, 10 & 122, 97, 240 & 164, 38, 255 \\ 214, 111, 25 & 98, 251, 156 & 182, 66, 45 & 31, 241, 90 \\ 129, 202, 38 & 205, 91, 84 & 252, 83, 137 & 32, 138, 7 \\ 9, 233, 118 & 220, 43, 252 & 121, 233, 141 & 140, 159, 72 \end{matrix}] \end{matrix}$

As a cryptanalyst, the encrypted image C is known to us rest are unknown $\begin{matrix} Z_{2} & = & (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) \end{matrix}$ $\begin{matrix} U_{R} & = & (\begin{matrix} 249 & 156 & 255 & 143 \\ 99 & 122 & 105 & 82 \\ 146 & 114 & 181 & 187 \\ 189 & 240 & 13 & 46 \end{matrix}) U_{G} = (\begin{matrix} 5 & 241 & 29 & 90 \\ 120 & 215 & 3 & 111 \\ 207 & 57 & 59 & 241 \\ 240 & 2 & 20 & 203 \end{matrix}) U_{B} = (\begin{matrix} 141 & 197 & 199 & 65 \\ 230 & 87 & 70 & 152 \\ 198 & 84 & 166 & 31 \\ 156 & 62 & 162 & 184 \end{matrix}) \\ P_{SR} & = & (\begin{matrix} 175 & 31 & 133 & 43 \\ 181 & 24 & 223 & 77 \\ 19 & 191 & 73 & 155 \\ 180 & 44 & 116 & 162 \end{matrix}) P_{SG} = (\begin{matrix} 146 & 173 & 124 & 124 \\ 23 & 44 & 65 & 158 \\ 5 & 98 & 104 & 123 \\ 25 & 41 & 253 & 84 \end{matrix}) P_{SB} = (\begin{matrix} 25 & 207 & 55 & 190 \\ 255 & 203 & 107 & 194 \\ 224 & 0 & 47 & 24 \\ 234 & 194 & 47 & 240 \end{matrix}) \\ J_{2} & = & (\begin{matrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{matrix}) \end{matrix}$

Retrieved permutation sequence equivalent to matrix T is [16 , 11].

Re-arranging above P_SR, P_SG, P_SB to get recovered components of plain-image P as: $P_{1} = (\begin{matrix} 162 & 77 & 43 & 155 \\ 44 & 24 & 31 & 191 \\ 180 & 181 & 175 & 19 \\ 116 & 223 & 133 & 73 \end{matrix}) P_{2} = (\begin{matrix} 84 & 158 & 124 & 123 \\ 41 & 44 & 173 & 98 \\ 25 & 23 & 146 & 5 \\ 253 & 65 & 124 & 104 \end{matrix}) P_{3} = (\begin{matrix} 240 & 194 & 190 & 24 \\ 194 & 203 & 207 & 0 \\ 234 & 255 & 25 & 224 \\ 47 & 107 & 55 & 47 \end{matrix})$

4.1 Lack of sensitivity to plain-image

The Lakshamanan et al. encryption algorithm exhibits considerably no sensitivity or responsiveness towards any (minor) alterations in plain-images. In order to arbitrate the sensitivity of encryption algorithm, we take two plain-images namely - P₁ and P₂ having only one (middle) pixel difference. Let C₁ and C₂ being their respective cipher images independently obtained by encryption algorithm under study. The difference image D between images C₁ and C₂ is obtained as D = bit-exor(C₁, C₂). Apparently, if the cipher-text images C₁ and C₂ are identical then their difference image D has almost all zero or near to zero gray value pixels, consequently D will be similar to a black image. Whereas, if they are completely specific and different in comparison to one another, up to a considerable extension, then the difference image D will be quite similar to random-image or noise-like image. The algorithm’s responsiveness to plain-image is simulated in Fig. 2 to demonstrate the low sensitivity to plain-image. From the figure, it is quite noticeable that the content of image D is like a black image except that middle pixel, implicating that the cipher-text images C₁ and C₂ are almost exact and give merely one pixel difference out of M×N pixels.

Fig.2

Plain-image sensitivity analysis: (a) plain-image P₁, (b) plain-image P₂, (c) encrypted image C₁ of P₁, (d) encrypted image C₂ of P₂, and (e) difference between images C₁ and C₂.

A strong image cryptosystem should be able to bring substantial difference between C₁ and C₂ even if their respective plain-images differ minutely [1, 43]. Therefore, it can be stated that the Lakshamanan et al. encryption algorithm possesses no plain-image sensitivity contrary to the fact that it is an indispensable need of strong image cryptosystems.

Hence, the security flaws presented in Section 3, cryptanalysis under proposed attack in Section 4 validate that the encryption algorithm doesn’t deemed secure. The authors claim, refer to conclusion section in [39] that their algorithm is robust and resistant against different security attacks is completely invalid. To contradict the authors claim, made in abstract section in [39], we claim that their encryption algorithm highly lacks security and unreliable for use in secure communication applications.

5 Conclusion

In this paper, we evaluate the security of a recently suggested chaotic systems based image encryption algorithm. The encryption algorithm is exclusively dependent on secret key and has considerably no dependency to pending plain images. This implicit flaw and others make it susceptibility to cryptanalysis proceeded in this paper as proposed attack. The proposed security analysis of algorithm uncovers its certain implicit security defects The exploitation of these susceptibilities are manifested and it has been shown through simulation that the plain-image can be recovered from encrypted one under proposed cryptanalysis without any prior knowledge of secret key or initial values of chaotic systems utilized. Henceforth, we can allegedly proclaim that the encryption process suggested by Lakshamanan et al. completely lacks security; it is totally unreliable, insecure and unfeasible to use it for secure transmission of images over communication channels.

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Cryptanalysis of an image encryption algorithm based on PWLCM and inertial delayed neural network

Abstract

Keywords

1 Introduction

2 Lakshmanan et al. image encryption algorithm

3.1 Permutation matrix T and encryption keystreams Uk are fixed

3.2 Weak secret keys

Table 1 Fixed iterative values of PWLCM system x0 p x i (i > 0) (0, 1) 0.5 A non-random sequence with poor period (0, 1) x0 1, 0, 0, 0, ... ... ... <0.5 2×x0 0.5, 1, 0, 0, 0, ... ... ....

3.4 Precision effect

4 Proposed cryptanalysis to recover plain-image

References

3.1 Permutation matrix T and encryption keystreams U_k are fixed

Table 1
Fixed iterative values of PWLCM system

x₀ p x_i (i > 0)

(0, 1) 0.5 A non-random sequence with poor period

(0, 1) x₀ 1, 0, 0, 0, ... ... ...

<0.5 2×x₀ 0.5, 1, 0, 0, 0, ... ... ....