Security Analysis of an Image Encryption Scheme Based on a New

Secure Variant of Hill Cipher and 1D Chaotic Maps

George Tes¸eleanu

1,2 a

Advanced Technologies Institute, 10 Dinu Vintil

a, Bucharest, Romania

Simion Stoilow Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei, Bucharest, Romania

Keywords:

Image Encryption Scheme, Chaos Based Encryption, Cryptanalysis.

Abstract:

In 2019, Essaid et al. introduced a chaotic map-based encryption scheme for color images. Their approach

employs three improved chaotic maps to dynamically generate the key bytes and matrix required by the cryp-

tosystem. It should be noted that these parameters are dependent on the size of the source image. According to

the authors, their method offers adequate security (i.e. 279 bits) for transmitting color images over unsecured

channels. However, we show in this paper that this is not the case. Speciﬁcally, we present two cryptanalytic

attacks that undermine the security of Essaid et al.’s encryption scheme. In the case of the chosen plaintext

attack, we require only two chosen plaintexts to completely break the scheme. The second attack is a a chosen

ciphertext attack, which requires two chosen ciphertexts and compared to the ﬁrst one has a rough complexity

of 2

. The attacks are feasible due to the fact that the key bits and matrix generated by the algorithm remain

unaltered for distinct plaintext images.

1 INTRODUCTION

The exponential increase in social media usage has

led to a heightened concern for the security of digital

images, particularly with regards to theft and unau-

thorized distribution. Consequently, this issue has

gained signiﬁcant attention, prompting numerous re-

searchers to develop various image encryption tech-

niques. Chaotic maps have emerged as a favored

approach for encrypting images, largely due to their

high sensitivity to previous states, initial conditions,

or both. This desirable feature makes it challeng-

ing to anticipate their behavior or outputs, thus giv-

ing rise to numerous novel cryptographic algorithms

based on chaos. We refer the reader to (Zolfaghari

and Koshiba, 2022; Muthu and Murali, 2021; Hosny,

2020;

Ozkaynak, 2018) for some surveys of such pro-

posals. Regrettably, due to inadequate security analy-

sis and a lack of design guidelines, a signiﬁcant num-

ber of image encryption schemes based on chaos have

been found to contain critical security vulnerabilities.

To illustrate our point, we provide a list of compro-

mised schemes in Table 1. Please be aware that the

list is not exhaustive.

In (Essaid et al., 2019b) a chaos based encryp-

https://orcid.org/0000-0003-3953-2744

tion scheme is proposed. The authors use the En-

hanced Logistic Map (ELM), Enhanced Chebyshev

Map (ECM) and Enhanced Sine Map (ESM) as pseu-

dorandom number generators (PRNGs). Using these

three PRNGs, Essaid et al. randomly generate the

necessary key bytes. Then, the ELM PRNG is used to

generate a key matrix of size 2 × 2, such that the ﬁrst

element of the matrix is invertible modulo 256. Since

ELM, ECM and ESM are simply used as PRNGs

and the scheme’s weakness is independent of the em-

ployed generators, we omit their description and sim-

ply consider the key bytes and matrix as being ran-

domly generated.

This paper presents our security analysis of the Es-

said et al. scheme. Speciﬁcally, we describe a chosen

plaintext attack and a chosen ciphertext attack, which

enables an attacker to decrypt all images of a particu-

lar size. To accomplish this, it is necessary to obtain

the ciphertexts of two chosen plaintexts or the plain-

texts of two chosen ciphertexts. Note that in the cho-

sen plaintext scenario, we reduce the scheme’s secu-

rity from 279 bits to 0 bits, while in the chosen cipher-

text scenario we reduce it to roughly 24 bits.

Structure of the Paper. We provide the necessary

preliminaries in Section 2. An alternative mathemati-

cal description of Essaid et al.’s scheme is outlined

Te¸seleanu, G.

Security Analysis of an Image Encryption Scheme Based on a New Secure Variant of Hill Cipher and 1D Chaotic Maps.

DOI: 10.5220/0012356600003648

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 10th International Conference on Information Systems Security and Privacy (ICISSP 2024), pages 745-749

ISBN: 978-989-758-683-5; ISSN: 2184-4356

745

Table 1: Broken chaos based image encryption algorithms.

Scheme (Yen and Guo, 2000) (Matoba and Javidi, 2004) (Wang et al., 2012) (Huang et al., 2014) (Khan, 2015) (Song and Qiao, 2015) (Chen et al., 2015)

Broken by (Li and Zheng, 2002) (Wang et al., 2019) (Arroyo et al., 2013) (Wen et al., 2021) (Alanazi et al., 2021) (Wen et al., 2019) (Hu et al., 2017)

Scheme (Hu et al., 2017) (Niyat et al., 2017) (Hua and Zhou, 2017) (Pak and Huang, 2017) (Liu et al., 2018) (Shaﬁque and Shahid, 2018) (Sheela et al., 2018)

Broken by (Li et al., 2019a) (Li et al., 2018) (Yu et al., 2021) (Wang et al., 2018) (Ma et al., 2020) (Wen and Yu, 2019) (Zhou et al., 2019)

Scheme (Wu et al., 2018) (Yosefnezhad Irani et al., 2019) (Khan and Masood, 2019) (Pak et al., 2019) (Mondal et al., 2021) (Essaid et al., 2019a)

Broken by (Chen et al., 2020) (Liu et al., 2020) (Fan et al., 2021) (Li et al., 2019b) (Li et al., 2021) (Tes¸eleanu, 2023)

Algorithm 1: Encryption algorithm.

Input: A plaintext P, three secret keys k

, k

and k

, and a secret matrix h

Output: A ciphertext C

1 for i ∈ [0, HW ) do

2 if i = 0 then





←



a b

c d





1,0





2,0

3,0



mod 256

← T

+ P

mod 256

4 else





←



a b

c d





1,i





2,i

3,i



mod 256

i+1

← T

+ P

i+1

mod 256

6 return C

7 SetAlFnt

in Section 3. In Sections 4 and 5 we show how an at-

tacker can recover the secret values in a chosen plain-

text/ciphertext scenario. We conclude in Section 6.

2 PRELIMINARIES

Notations. In this paper, the subset {1, . . . , s − 1} ∈

N is denoted by [1, s). The action of selecting a ran-

dom element x from a sample space X is represented

by x

←− X, while x ← y indicates the assignment of

value y to variable x. By H and W we denote an im-

age’s height and width. Hexadecimal numbers will

always contain the preﬁx 0x.

2.1 Essaid et al. Image Encryption

Scheme

In this section we present Essaid et al.’s encryption

(Algorithm 1) and decryption (Algorithm 2) algo-

rithms as described in (Essaid et al., 2019b). Before

the encryption/decryption process starts, the image is

always converted into a vector of size H · W . At the

end, the resulting vector is translated back into an im-

age of size H ×W. Please note that both the key bytes

1,i

, k

2,i

, and k

2,i

, and the matrix h values a, b, c, and

d are generated randomly. Also, a is always invertible

modulo 256.

Algorithm 2: Decryption algorithm.

Input: A ciphertext C, three secret keys k

and k

, and a secret matrix h

Output: A plaintext P

1 for i ∈ [0, HW ) do

2 if i = 0 then

← a

−1

· (C

− b · k

1,0

− k

2,0

) mod 256

tmp ← c · P

+ d · k

1,0

+ k

3,0

mod 256

4 else

← a

−1

· (C

− b · k

1,i

− k

2,i

) mod 256

← P

−tmp mod 256

tmp ← c · a

−1

· (C

− b · k

1,i

− k

2,i

) mod 256

tmp ← temp + d · k

1,i

+ k

3,i

mod 256

6 return P

7 SetAlFnt

3 A NEW LOOK AT ESSAID et al.’s

SCHEME

In this section we provide an equivalent description

of the scheme presented in (Essaid et al., 2019b). We

ﬁrst start with studying Algorithm 1.

Lemma 3.1. Let C

and T

be the variables from Al-

gorithm 1. Then we can rewrite them as follows

≡ a

∑

j=0

i− j

+ α

mod 256,

≡ c

∑

j=0

i− j

+ β

mod 256,

where β

−1

= 0 and

≡ aβ

i−1

+ bk

1,i

+ k

2,i

mod 256,

≡ cβ

i−1

+ dk

1,i

+ k

3,i

mod 256.

Proof. We will prove our assertion using induction.

When i = 0 we have that

≡ aP

+ bk

1,0

+ k

2,0

= aP

+ α

mod 256,

≡ cP

+ dk

1,0

+ k

3,0

= cP

+ β

mod 256.

We assume that the assertion is true for i and we prove

ICISSP 2024 - 10th International Conference on Information Systems Security and Privacy

746

it for i + 1. Therefore, we have

i+1

≡ aS

i+1

+ bk

1,i+1

+ k

2,i+1

≡ a(T

+ P

i+1

) + bk

1,i+1

+ k

2,i+1

≡ ac

∑

j=0

i− j

+ aP

i+1

+ aβ

+ bk

1,i+1

+ k

2,i+1

≡ a

i+1

∑

j=0

(i+1)− j

+ α

i+1

mod 256

and

i+1

≡ cS

i+1

+ dk

1,i+1

+ k

3,i+1

≡ c(T

+ P

i+1

) + dk

1,i+1

+ k

2,i+1

≡ c

∑

j=0

i− j

+ cP

i+1

+ cβ

+ dk

1,i+1

+ k

3,i+1

≡ c

i+1

∑

j=0

(i+1)− j

+ β

i+1

mod 256,

as desired.

According to Lemma 3.1, in order to encrypt an

image using Essaid et al.’s scheme is enough to know

the secret values a, c and α

, for i ∈ [0, HW ). As a

consequence, we can also decrypt using these values.

Corollary 3.1.1. We can recover P

using

≡ a

−1

− a

i−1

∑

j=0

i− j

− α

) mod 256.

A more efﬁcient method for decrypting is given in

the following lemma.

Corollary 3.1.2. We can recover P

using

≡ a

−1

− γ

− α

) mod 256,

where γ

= 0 and

≡ acP

i−1

+ cγ

i−1

mod 256.

4 CHOSEN PLAINTEXT ATTACK

A chosen plaintext attack (CPA) is a scenario in which

the attacker A brieﬂy gains access to the encryption

machine O

enc

and is permitted to query it with vari-

ous inputs. In this way, A generates speciﬁc plaintexts

that can facilitate his attack and uses O

enc

to obtain

the corresponding ciphertexts. We demonstrate in this

paper that Essaid et al.’s image encryption scheme is

vulnerable to such attacks.

Lets assume that we query O

enc

with two plain-

texts P and P

′

and receive C and C

′

, respectively. Ac-

cording to Lemma 3.1 we have

≡ aP

+ α

mod 256,

′

≡ aP

′

+ α

mod 256.

Therefore, if gcd(P

− P

′

, 256) = 1 then we can re-

cover a using

a ≡ (C

−C

′

)(P

− P

′

)

−1

mod 256,

and α

from

≡ C

′

− aP

′

mod 256. (1)

Using Lemma 3.1 we also obtain

≡ aP

+ acP

+ α

mod 256,

′

≡ aP

′

+ acP

′

+ α

mod 256,

and since we already computed a we can rewrite the

equations as

− aP

≡ acP

+ α

mod 256,

′

− aP

′

≡ acP

′

+ α

mod 256.

Therefore, we can recover c using

c ≡ (C

− aP

−C

′

+ aP

′

) · a

−1

− P

′

)

−1

mod 256,

since gcd(a, 256) = gcd(P

− P

′

, 256) = 1. Also, α

is computed as follows

≡ C

′

− aP

′

− acP

′

mod 256. (2)

Once a and c are computed, the remaining α

are com-

puted from

≡ C

′

− a

∑

j=0

i− j

′

mod 256. (3)

In order to optimize the recovery of the secret val-

ues, we choose two plaintexts such that P

= 1 and

= . . . = P

HW −1

= P

′

= . . . = P

′

HW −1

= 0. There-

fore, we obtain the following relations

a ≡ C

−C

′

mod 256,

c ≡ a

−1

−C

′

) mod 256,

≡ C

′

mod 256, for i ∈ [0, HW ).

We can easily see that the complexity of our attack

is constant and is dominated by computing an inverse

and a multiplication modulo 256. Therefore, it is very

efﬁcient.

5 CHOSEN CIPHERTEXT

ATTACK

In contrast to a chosen plaintext attack, a chosen ci-

phertext attack (CCA) assumes that the attacker A

brieﬂy gains access to the decryption machine O

dec

A then generates speciﬁc ciphertexts that can assist

his attack and uses O

dec

to obtain the corresponding

Security Analysis of an Image Encryption Scheme Based on a New Secure Variant of Hill Cipher and 1D Chaotic Maps

747

plaintexts. In this scenario, we describe an attack on

Essaid et al.’s cryptosystem.

Lets assume that we query O

dec

with two cipher-

texts C and C

′

and receive P and P

′

, respectively. Us-

ing Corollary 3.1.1 we obtain

≡ a

−1

− α

) mod 256,

′

≡ a

−1

′

− α

) mod 256.

Therefore, if gcd(C

−C

′

, 256) = 1 then we can re-

cover a

−1

using

−1

≡ (P

− P

′

)(C

−C

′

)

−1

mod 256.

Applying Corollary 3.1.1 to the second byte we obtain

≡ a

−1

− acP

− α

) mod 256,

′

≡ a

−1

′

− acP

′

− α

) mod 256,

and since we already computed a

−1

we can rewrite

the equations as

−1

− P

≡ cP

+ a

−1

mod 256,

−1

′

− P

′

≡ cP

′

+ a

−1

mod 256.

Note that since gcd(a, 256) = gcd(C

−C

′

, 256) = 1,

we obtain that gcd(P

− P

′

, 256) = 1. Therefore, we

can recover c using

c ≡ (a

−1

− P

− a

−1

′

+ P

′

) · (P

− P

′

)

−1

mod 256.

Once a and c are computed, the α

values are com-

puted using Equations (1) to (3).

In order to optimize the recovery of the secret val-

ues, we choose two ciphertexts such that C

= 1 and

= . . . = C

HW −1

= C

′

= . . . = C

′

HW −1

= 0. There-

fore, we obtain the following relations

a ≡ (P

− P

′

)

−1

mod 256,

c ≡ a(P

′

− P

) mod 256,

≡ −a

∑

j=0

i− j

′

mod 256, for i ∈ [0, HW ). (4)

Note that Equation (4) can be rewritten as

≡ −aP

′

mod 256,

≡ −aP

′

+ cα

i−1

mod 256, for i ∈ [1, HW ).

The complexity of our attack dominated by two

inverses and 2HW multiplications modulo 256. Us-

ing the fact that an inverse and a multiplication mod-

ulo 256 has constant complexity O(1), we obtain that

our attack has a complexity of O(2HW ). For exam-

ple, if we encrypt 2 megapixels

images we obtain a

complexity of O (2

21.87

). In the case of 12 megapix-

els

, we obtain O(2

24.51

W × H = 1600 × 1200

W × H = 4000 × 3000

6 CONCLUSIONS

The authors of (Essaid et al., 2019b) presented an im-

age encryption scheme that they claimed to have a se-

curity strength of 279 bits. However, our research

in this paper demonstrated that the actual security

strength of Essaid et al.’s scheme is essentially 0 bits.

To establish our security bound, we designed a cho-

sen plaintext attack that requires only 2 queries to the

encryption oracle. Furthermore, we outline a chosen

ciphertext attack that requires 2 queries to the decryp-

tion oracle and has a complexity of roughly O(2

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