Processing math: 100%
Research article Special Issues

Cryptanalysis of hyperchaotic S-box generation and image encryption

  • Received: 20 November 2024 Revised: 09 December 2024 Accepted: 13 December 2024 Published: 26 December 2024
  • MSC : 94A60, 68P25

  • Cryptography serves as the cornerstone for safe communication and data security in today's digital environment. Because they feature substitution boxes, substitution-permutation networks (SPNs) are crucial for cryptographic algorithms such as the popular Advanced Encryption Standard (AES). The structure and properties of S-boxes have a significant impact on the overall security of cryptographic systems. This article aims to improve cryptographic security through unique S-box construction methodologies. The proposed S-boxes improve the security features by employing chaotic maps and Galois fields, which go beyond traditional design approaches. The S-boxes were analyzed and the weaknesses were removed to design strong candidate S-boxes. The efficiency of the proposed S-boxes in increasing cryptographic resilience is thoroughly explored thereby taking nonlinearity, strict avalanche requirements, bit independence constraints, linear approximation, and differential approximation into account. The dynamic S-boxes have average scores of nonlinearity, strict avalanche criteria(SAC), nonlinearity of Bit Independence Criteria (BIC Nonlinearity), SAC of Bit Independence Criteria (BIC SAC), Linear Approximation Probability (LAP) and Differential Approximation Probability (DAP) is 111.1025, 111.1022, 0.5014, 0.5024, 111.1082, 111.0964, 0.5024, 0.5022, 0.0726, 0.0729 and 0.0214, 0.0219, respectively. Furthermore, given the prevalence of images in modern communication and data storage, this work studies the seamless incorporation of advanced S-boxes into image encryption systems. With its thorough research, the paper contributes to the current discussion on cryptographic security by providing theoretical understandings and practical solutions to improve digital communication and data security in an era of rising cyber dangers and ubiquitous connectivity.

    Citation: Mohammad Mazyad Hazzazi, Gulraiz, Rashad Ali, Muhammad Kamran Jamil, Sameer Abdullah Nooh, Fahad Alblehai. Cryptanalysis of hyperchaotic S-box generation and image encryption[J]. AIMS Mathematics, 2024, 9(12): 36116-36139. doi: 10.3934/math.20241714

    Related Papers:

    [1] Nadiyah Hussain Alharthi, Abdon Atangana, Badr S. Alkahtani . Numerical analysis of some partial differential equations with fractal-fractional derivative. AIMS Mathematics, 2023, 8(1): 2240-2256. doi: 10.3934/math.2023116
    [2] Abdon Atangana, Ali Akgül . Analysis of a derivative with two variable orders. AIMS Mathematics, 2022, 7(5): 7274-7293. doi: 10.3934/math.2022406
    [3] Hasib Khan, Jehad Alzabut, Anwar Shah, Sina Etemad, Shahram Rezapour, Choonkil Park . A study on the fractal-fractional tobacco smoking model. AIMS Mathematics, 2022, 7(8): 13887-13909. doi: 10.3934/math.2022767
    [4] Khaled M. Saad, Manal Alqhtani . Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear. AIMS Mathematics, 2021, 6(4): 3788-3804. doi: 10.3934/math.2021225
    [5] Amir Ali, Abid Ullah Khan, Obaid Algahtani, Sayed Saifullah . Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels. AIMS Mathematics, 2022, 7(8): 14975-14990. doi: 10.3934/math.2022820
    [6] Abdon Atangana, Seda İğret Araz . Extension of Chaplygin's existence and uniqueness method for fractal-fractional nonlinear differential equations. AIMS Mathematics, 2024, 9(3): 5763-5793. doi: 10.3934/math.2024280
    [7] Rahat Zarin, Amir Khan, Pushpendra Kumar, Usa Wannasingha Humphries . Fractional-order dynamics of Chagas-HIV epidemic model with different fractional operators. AIMS Mathematics, 2022, 7(10): 18897-18924. doi: 10.3934/math.20221041
    [8] Muhammad Farman, Ali Akgül, Sameh Askar, Thongchai Botmart, Aqeel Ahmad, Hijaz Ahmad . Modeling and analysis of fractional order Zika model. AIMS Mathematics, 2022, 7(3): 3912-3938. doi: 10.3934/math.2022216
    [9] Muhammad Farman, Ali Akgül, Kottakkaran Sooppy Nisar, Dilshad Ahmad, Aqeel Ahmad, Sarfaraz Kamangar, C Ahamed Saleel . Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel. AIMS Mathematics, 2022, 7(1): 756-783. doi: 10.3934/math.2022046
    [10] Muhammad Farman, Aqeel Ahmad, Ali Akgül, Muhammad Umer Saleem, Kottakkaran Sooppy Nisar, Velusamy Vijayakumar . Dynamical behavior of tumor-immune system with fractal-fractional operator. AIMS Mathematics, 2022, 7(5): 8751-8773. doi: 10.3934/math.2022489
  • Cryptography serves as the cornerstone for safe communication and data security in today's digital environment. Because they feature substitution boxes, substitution-permutation networks (SPNs) are crucial for cryptographic algorithms such as the popular Advanced Encryption Standard (AES). The structure and properties of S-boxes have a significant impact on the overall security of cryptographic systems. This article aims to improve cryptographic security through unique S-box construction methodologies. The proposed S-boxes improve the security features by employing chaotic maps and Galois fields, which go beyond traditional design approaches. The S-boxes were analyzed and the weaknesses were removed to design strong candidate S-boxes. The efficiency of the proposed S-boxes in increasing cryptographic resilience is thoroughly explored thereby taking nonlinearity, strict avalanche requirements, bit independence constraints, linear approximation, and differential approximation into account. The dynamic S-boxes have average scores of nonlinearity, strict avalanche criteria(SAC), nonlinearity of Bit Independence Criteria (BIC Nonlinearity), SAC of Bit Independence Criteria (BIC SAC), Linear Approximation Probability (LAP) and Differential Approximation Probability (DAP) is 111.1025, 111.1022, 0.5014, 0.5024, 111.1082, 111.0964, 0.5024, 0.5022, 0.0726, 0.0729 and 0.0214, 0.0219, respectively. Furthermore, given the prevalence of images in modern communication and data storage, this work studies the seamless incorporation of advanced S-boxes into image encryption systems. With its thorough research, the paper contributes to the current discussion on cryptographic security by providing theoretical understandings and practical solutions to improve digital communication and data security in an era of rising cyber dangers and ubiquitous connectivity.



    We all know Fermat's last theorem: For an integer n>2, the equation xn+yn=zn for x,y,z has no positive integer solution. It took 356 years from when it was proposed in 1637, to 1993 when Wiles conquered it. This equation can also be extended to functional equations. In complex analysis, the researchers began to focus on meromorphic solutions of the equation fn(z)+gn(z)=hn(z). As far as we know, Montel [22] was the first scholar to study this problem, and later Gross and Baker carried out the follow-up research work [2,9,10]. Then, applying Nevanlinna theory to the study of this kind of functional equation became a hot topic. For example, Yang et al. [31] studied the transcendental meromorphic solution of the functional equation f(z)2+f(z)2=1 and found that the solution to this equation must have the form of f(z)=12(peλz+1peλz), where p,λ are nonzero constants. With the establishment of the Nevanlinna theory with the difference of meromorphic function [8,11], more attention was paid to Fermat-type functional equation with the difference of meromorphic function. Liu et al. [15,16,17] studied the finite order transcendental entire solutions of Fermat-type difference equations f(z+c)2+f(z)2=1 and f(z+c)2+f(z)2=1, and they obtained that the solutions of these two equations are sine functions. Subsequently, more attention has been paid to this area of research [14,18,19,20,21,23,24,25,28,29,30,33,34].

    In 2016, Liu and Yang [18] studied the existence of solutions to quadratic trinomial functional equations, as well as the entire function and its derivatives and differences, and they converted the equations in the following two theorems into Fermat- type equations by transformation.

    Theorem 1.1. ([18, Theorem 1.4]) If α0,±1, then the finite order transcendental entire solution of equation

    f(z)2+2αf(z)f(z+c)+f(z+c)2=1

    is of order one.

    Theorem 1.2. ([18, Theorem 1.6]) If α0,±1, then the equation

    f(z)2+2αf(z)f(z)+f(z)2=1

    has no transcendental meromorphic solutions.

    On the other hand, Han and Lü [12] studied the existence of solutions to the Fermat-type equation when the right side was an exponential function. Here we only list the n=2 case in their results.

    Theorem 1.3. ([12, Theorem 1.1]) The meromorphic solutions of f of the following differential equation

    f(z)2+f(z)2=eαz+β

    are

    f(z)=eβ2sin(z+b)

    if α=0, and

    f(z)=deαz+β2

    if α0 with d2(1+(α2)2)=1.

    In the same article, they also studied the case of replacing f(z) with f(z+c) in the above equation, and found that the solution of equation

    f(z)2+f(z+c)2=eαz+β

    is f(z)=deαz+β2 with d2(1+eac)=1.

    Combining these conclusions above, Luo et al. [20] studied the following three equations with finite order transcendental entire solutions. These three equations are

    f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), (1.1)
    f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z) (1.2)

    and

    f(z)2+2αf(z)f(z)+f(z)2=eg(z), (1.3)

    where α0,±1, c are constants and g(z) is a polynomial. If all these equations admit finite order transcendental entire solutions, g(z) must be a polynomial with the degree of one, and the solutions f of these equations are all exponential functions or the sum of two exponential functions whose exponents are polynomials of the degree of one, as seen in Theorems 2.1–2.3 [20]. Each of these three equations contains only two terms of f(z),f(z) or f(z+c), so can we consider a quadratic equation that contains all three of these terms?

    Inspired by this, we shall study the problem of finite order transcendental entire solutions of functional equations involving the quadratic of f, its derivative and its difference. In fact, we studied the finite order transcendental entire solution for (1.4) below.

    Theorem 1.4. Suppose that α0,±1, β0, γ0 and c0 are four constants such that α2+β2+γ21+2αβγ, and g(z) is a nonconstant polynomial. If the complex equation

    [f(z)]2+[f(z+c)]2+f2(z)+2αf(z)f(z+c)+2βf(z)f(z+c)+2γf(z)f(z)=eg(z) (1.4)

    admits a transcendental entire solution f(z) of finite order, then for

    δ=1α2β2γ2+2αβγ1α2,

    the solution has two forms:

    (1)

    f(z)=deaz+b22iδ,

    where a0 and b is an arbitrary constant. Moreover, g(z)=az+b, and d is a constant with

    1+a24+eac+(aα+2β)eac/2+aγ=4δd2.

    (2)

    f(z)=ea1z+b1ea2z+b22iδ,

    where a1,a2(a1a2) are nonzero constants satisfying (1.5), and b1,b2 are arbitrary constants, g(z)=(a1+a2)z+b1+b2.

    {a21+2γa1+e2a1c+(2αa1+2β)ea1c+1=0;a22+2γa2+e2a2c+(2αa2+2β)ea2c+1=0;[ea1cea2c+α(a1a2)]2+(1α2)(a1a2)2+4δ=0. (1.5)

    Let's give two examples to show that Theorem 1.4 is true.

    Example 1.5. Suppose α=1/3, β=γ=1 and c=2 in (1.4), then δ=1/2. Set a=2,b=2, then by the relationship of a and d, we have

    d=24+83e2+e4.

    We can verify that the entire function

    f(z)=±14+83e2+e4ez+1

    is a solution of

    [f(z)]2+[f(z+2)]2+f2(z)+23f(z)f(z+2)+2f(z)f(z+2)+2f(z)f(z)=e2z+2.

    Example 1.6. Suppose

    α=3πi4,β=1+3π22,γ=5πi4

    and c=1 in (1.4), then δ=π2. a1=πi,a2=3πi, b1,b2 are arbitrary constants. We can verify that the entire function

    f(z)=eπiz+b1e3πiz+b2±2πi

    is a solution of

    [f(z)]2+[f(z+1)]2+f2(z)+3πi2f(z)f(z+1)+(2+3π2)f(z)f(z+1)5πi2f(z)f(z)=e4πiz+b1+b2.

    From Theorem 1.4 we have the following corollary.

    Corollary 1.7. Under the assumption of Theorem 1.4, if the degree of polynomial g(z) is greater than one, then (1.4) does not have a transcendental solution with finite order.

    If q(0,1) is a constant, then f(qz) is called the q-difference of meromorphic function f(z). The q-difference is also an important research content in the value distribution theory, and the research on it can be traced back to the early 20th century [5,13].

    In recent decades, with the establishment of Nevanlinna theory related to it [3], the research on q-difference has been vigorously developed, and this theory has been applied to many q-difference equations to get a lot of results [4,6,7,16,26,27]. Therefore, we considered to replace f(z+c) in (1.4) by f(qz) as to get a q-difference functional equation, and then studied the finite order transcendental entire solution of this equation. Through the complicated discussion and calculation of different cases, we came to the following conclusion.

    Theorem 1.8. Suppose that α0,±1, β0, γ0,±1 and q0,1 are four constants such that α2+β2+γ21+2αβγ, and g(z) is a nonconstant polynomial. If the complex equation

    [f(z)]2+[f(qz)]2+f2(z)+2αf(z)f(qz)+2βf(z)f(qz)+2γf(z)f(z)=eg(z) (1.6)

    admits a transcendental entire solution f(z) of finite order, then

    f(z)=±eaz+b2

    and g(z)=aqz+b, b is an arbitrary constant, a0 and γ21 such that

    {a24+γa+1=0,αa+2β=0. (1.7)

    Here is an example to test the truth of the Theorem 1.8.

    Example 1.9. Suppose α=1/2,β=1,γ=5/4 and q is any constant except 0,1 in (1.6), then we can verify that the entire function f(z)=±e2z+1 is a solution of

    [f(z)]2+[f(qz)]2+f2(z)+f(z)f(qz)2f(z)f(qz)52f(z)f(z)=e4qz+2.

    A corollary also can be obtained from Theorem 1.8.

    Corollary 1.10. Under the assumption of Theorem 1.8, if the degree of polynomial g(z) is greater than one, then (1.6) does not admit transcendental entire solution with finite order.

    Remark 1.11. Equations (1.4) and (1.6) can be transformed into three term quadratic equations by linear transformation. The purpose of restrictions α21 and α2+β2+γ21+2αβγ in Theorems 1.4 and 1.8 is to not allow these three term quadratic equations to degenerate into quadratic equations with two or one terms, which have been studied in previous literatures. This can be seen easily from (3.2) in the proof below.

    Remark 1.12. From the proof of Theorems 1.4 and 1.8 and the above three examples, we can find that if the two equations have finite order transcendental entire solutions, then the solutions of both equations are exponential functions and their exponents are polynomials with the degree of one. For (1.4), after the solution was substituted into the equation, the terms of the equation contained the common factor eg(z). After dividing both sides of the equation by eg(z), the relationship between the coefficients of the equation and the coefficients of the exponent was obtained. For (1.6), when one substitutes the solution into it, the term eg(z) in the right side of the equation is equal to [f(qz)]2 in the left side, which can be subtracted from both sides of the equation. The signs of the other two mixed terms containing f(qz) are opposite to each other, so these two mixed terms were canceled out.

    The following lemma played a key role in the proofs of this paper. It is about the factorization of an entire function. In particular, if f(z) was a finite order entire function without zero, then f(z)=eh(z) where h(z) was a nonconstant polynomial, as seen in Theorems 1.42 and 1.44 [32].

    Lemma 2.1. (Hadamard's factorization theorem) [32, Theorem 2.5] Let f be an entire function of finite order λ(f) with zeros {a1,a2,}C{0} and a k-fold zero at the origin. Then,

    f(z)=zkP(z)eQ(z)

    where P(z) is the canonical product of f formed with the non-null zeros of f,

    P(z)=n=1(1zan)ezan+12(zan)2++1h(zan)h,

    and h is the smallest integer for which this series converges, called the genus of the canonical product. Q(z) is a polynomial of degree λ(f) and hλ.

    The second lemma belongs to Borel. It's about the combination of entire functions, and we'll use it repeatedly in the proofs in Sections 3 and 4. When using it, the key is to verify the second condition below.

    Lemma 2.2. [32, Theorem 1.52] If fj(z)(j=1,2,,n) and gj(z)(j=1,2,,n)(n2) are entire functions satisfying

    (1) nj=1fj(z)egj(z)0,

    (2) the orders of fj are less than that of egh(z)gk(z) for 1jn,1h<kn.

    Then, fj0,(j=1,2,,n).

    According to the linear algebra, any quadratic form can be reduced to the standard form by a non-degenerate linear transformation. So, setting

    {f(z)=w,f(z)=uαv+αβγ1α2w,f(z+c)=vβαγ1α2w (3.1)

    and substituting it into (1.4), we obtain that

    u2+(1α2)v2+1α2β2γ2+2αβγ1α2w2=eg(z). (3.2)

    For simplicity and convenience, let's denote

    δ:=1α2β2γ2+2αβγ1α2,

    then (3.2) can convert into

    (u2+(1α2)v2eg(z)2)2+(δweg(z)2)2=1. (3.3)

    Consequently, we have

    (u2+(1α2)v2eg(z)2+iδweg(z)2)(u2+(1α2)v2eg(z)2iδweg(z)2)=1. (3.4)

    By Hadamard's factorization theorem, if the multiplicities of any zeros of the entire function u2+(1α2)v2 is even number, then u2+(1α2)v2 is also an entire function. The following (3.5) holds in the complex plane, where p(z) is a polynomial. If u2+(1α2)v2 have some zeros with odd number multiplicities, then u2+(1α2)v2 has branch points in the complex plane. Branches are obtained by connecting finite branch points and infinity points appropriately by line segments. These segments are called branch cuts, and u2+(1α2)v2 is analytic and univalent in every branch [1,35]. Since the two analytical factors on the left side of (3.4) have no zeros in each branch, there exists an analytical function p(z) such that the equations

    {u2+(1α2)v2eg(z)2+iδweg(z)2=ep(z),u2+(1α2)v2eg(z)2iδweg(z)2=ep(z) (3.5)

    hold in every branch. Denote

    λ1(z):=p(z)+g(z)2,   λ2(z):=p(z)+g(z)2,

    then,

    w=eλ1(z)eλ2(z)2iδ (3.6)

    hold in every branch. Moreover, noting that w=f(z) is an entire function with finite order, the righthand side of (3.6) can be extended to the whole complex plane. Therefore, one can supplement the definition of function p(z) at the points on branch cuts by the limiting values, and it is still called p(z) after supplementary definition. Thus, p(z) is analytic on the whole complex plane, so it's an entire function. Because ep(z) is of finite order, p(z) is actually a polynomial, so we get

    u2+(1α2)v2=(eλ1(z)+eλ2(z)2)2. (3.7)

    Noting that f(z)=w, we have

    {f(z)=eλ1(z)eλ2(z)2iδ,f(z)=λ1(z)eλ1(z)λ2(z)eλ2(z)2iδ,f(z+c)=eλ1(z+c)eλ2(z+c)2iδ. (3.8)

    From (3.1), we know that

    u=f(z)+αf(z+c)+γf(z)

    and

    v=f(z+c)+βαγ1α2f(z).

    Substituting the above u,v into (3.7) we get

    [f(z)]2+[f(z+c)]2+β2+γ22αβγ1α2f2(z)+2αf(z)f(z+c)+2βf(z+c)f(z)+2γf(z)f(z)=e2λ1(z)+e2λ2(z)+2eλ1(z)+λ2(z)4. (3.9)

    For simplicity and convenience, we give the following notation:

    λ1:=λ1(z),λ2:=λ2(z),¯λ1:=λ1(z+c),¯λ2:=λ2(z+c).

    Substituting (3.8) into (3.9) we obtain

    λ21e2λ1+λ22e2λ22λ1λ2eλ1+λ24δ+e2¯λ1+e2¯λ22e¯λ1+¯λ24δ+β2+γ22αβγ1α2e2λ1+e2λ22eλ1+λ24δ+2αλ1eλ1+¯λ1λ1eλ1+¯λ2λ2e¯λ1+λ2+λ2eλ2+¯λ24δ+2βeλ1+¯λ1eλ1+¯λ2e¯λ1+λ2+eλ2+¯λ24δ+2γλ1e2λ1λ1eλ1+λ2λ2eλ1+λ2+λ2e2λ24δ=e2λ1+e2λ2+2eλ1+λ24. (3.10)

    The transcendental terms appearing in the above equation have exponents

    2λ1,2λ2,λ1+λ2,2¯λ1,2¯λ2,¯λ1+¯λ2,λ1+¯λ1,λ1+¯λ2,¯λ1+λ2andλ2+¯λ2.

    In order to apply Lemma 2.2 to (3.10), we checked whether the pairwise difference between these exponents was constant. If λ1λ2, then f(z)0 this was impossible, so λ1λ2. The following two cases are discussed.

    Case 1. If λ1λ2 is a nonzero constant, then p(z) is a constant, denoted by p in the following. Consequently, for pkπi(kZ),

    f(z)=w=(epep)eg(z)/22iδ. (3.11)

    Then,

    f(z)=(epep)eg(z)/22iδg(z)2 (3.12)

    and

    f(z+c)=(epep)eg(z+c)/22iδ. (3.13)

    Substituting (3.11)–(3.13) into (1.4), the terms in the left side of (1.4) can be expressed respectively as

    {[f(z)]2=d2eg(z)4δ(g(z))24,[f(z+c)]2=d2eg(z+c)4δ,f2(z)=d2eg(z)4δ,2αf(z)f(z+c)=2αd2eg(z)+g(z+c)24δg(z)2,2βf(z)f(z+c)=2βd2eg(z)+g(z+c)24δ,2γf(z)f(z)=2γd2eg(z)4δg(z)2, (3.14)

    where d:=epep. If the degree of polynomial g(z) is greater than one, the three exponents g(z),g(z+c) and g(z)+g(z+c)2 are pairwise distinct. By Lemma 2.2, we obtained that after combining like terms, the coefficients of these three exponential terms eg(z),eg(z+c) and eg(z)+g(z+c)2 are zero. In particular, d24δ=0 since it's the coefficient of the sole term eg(z+c). This is impossible, because that means f0, so the degree of g(z) is one. Therefore, suppose g(z)=az+b, a(0),b are constants. Substitute it into (3.14), then into (1.4), and eliminate eg(z) from both sides of this equation. Then, we get

    1+a24+eac+(aα+2β)eac/2+aγ=4δd2. (3.15)

    This means if the constants α,β,γ,c in the original (1.4) are known, then the solution is

    f(z)=deaz+b22iδ,

    where constants a,d should satisfy the relationship of (3.15), and b is an arbitrary constant.

    Case 2. If λ1λ2 is not a constant, then p(z) is not a constant; instead, it is a nonconstant polynomial. For (3.10) we multiply 4δ on both sides, combine like terms and move all the terms to the left side of this equation, then the right side is just zero. Thus, the coefficients of the distinct transcendental terms can be listed in Table 1.

    Table 1.  Transcendental terms and corresponding coefficients.
    Transcendental terms Corresponding coefficients
    e2λ1 λ21+2γλ1+1
    e2λ2 λ22+2γλ2+1
    eλ1+λ2 2λ1λ22γ(λ1+λ2)+4δ2
    e2¯λ1 1
    e2¯λ2 1
    e¯λ1+¯λ2 2
    eλ1+¯λ1 2αλ1+2β
    eλ1+¯λ2 2αλ12β
    e¯λ1+λ2 2αλ22β
    eλ2+¯λ2 2αλ2+2β

     | Show Table
    DownLoad: CSV

    Because the difference between any two of 2λ1,2λ2,λ1+λ2 is not constant, the term containing e2λ1 cannot combine with terms containing e2λ2 or eλ1+λ2.

    Suppose that deg(λ1)=m>1 and deg(λ2)=n>1. If the term containing e2λ1 cannot combine with any other transcendental terms, then its coefficient has to be zero for any zC by Lemma 2.2. This is impossible, since its coefficients are nonconstant polynomials. The only term in the coefficient that may cancel out with λ21 is the term that contains λ2. They must have the same degree, so we have 2(m1)=n1. By the same arguments, we have m1=2(n1) by considering λ22 with λ1. Then, we get a contradiction, and it yields that deg(λ1),deg(λ2) are both at most one.

    Therefore, we can assume that λ1=a1z+b1 and λ2=a2z+b2 where a1a2 are constants, and b1,b2 are arbitrary constants. The transcendental terms and the corresponding coefficients in Table 1 can convert into those in Table 2. Since the three transcendental terms e2λ1, e2λ2 and eλ1+λ2 cannot be combined with each other, we get the following system of (3.16) with respect to the coefficients by Lemma 2.2.

    {a21+2γa1+e2a1c+(2αa1+2β)ea1c+1=0,a22+2γa2+e2a2c+(2αa2+2β)ea2c+1=0,2a1a22γ(a1+a2)2e(a1+a2)c(2αa1+2β)ea2c(2αa2+2β)ea1c+4δ2=0. (3.16)
    Table 2.  The transcendental term after the change and the corresponding coefficients.
    Before After Corresponding coefficients
    e2λ1 e2λ1 a21+2γa1+1
    e2¯λ1 e2λ1 e2a1c
    eλ1+¯λ1 e2λ1 (2αa1+2β)ea1c
    e2λ2 e2λ2 a22+2γa2+1
    e2¯λ2 e2λ2 e2a2c
    eλ2+¯λ2 e2λ2 (2αa2+2β)ea2c
    eλ1+λ2 eλ1+λ2 2a1a22γ(a1+a2)+4δ2
    e¯λ1+¯λ2 eλ1+λ2 2e(a1+a2)c
    eλ1+¯λ2 eλ1+λ2 (2αa12β)ea2c
    e¯λ1+λ2 eλ1+λ2 (2αa22β)ea1c

     | Show Table
    DownLoad: CSV

    Adding the three equations in (3.16) together, they convert into

    {a21+2γa1+e2a1c+(2αa1+2β)ea1c+1=0,a22+2γa2+e2a2c+(2αa2+2β)ea2c+1=0,[ea1cea2c+α(a1a2)]2+(1α2)(a1a2)2+4δ=0. (3.17)

    Therefore, the original (1.4) has solutions of the form

    f(z)=ea1z+b1ea2z+b22iδ,

    where a1,a2(a1a2) are nonzero constants satisfying (3.17), and b1,b2 are arbitrary constants.

    For an exponential polynomial f(z) with finite order, the exponents for each exponential terms of f(z) are the same as those of f(z), but the exponents of f(qz) are not the same as the exponents of f(z) for q0,1. The term eg(z) in the right side of (1.6) with coefficient one must combine with one of these two kinds of exponential terms, transcendental terms in f(z) or f(qz), by Lemma 2.2. The following are divided into two cases for discussion.

    Case 1. If eg(z) can combine with the exponential terms in f(z), then replacing f(z+c) by f(qz) in Section 3 and using the same methods in it we get

    {f(z)=eλ1(z)eλ2(z)2iδ,f(z)=λ1(z)eλ1(z)λ2(z)eλ2(z)2iδ,f(qz)=eλ1(qz)eλ2(qz)2iδ, (4.1)

    where λ1(z),λ2(z),δ are the same as in Section 3, and we also have

    u=f(z)+αf(qz)+γf(z),v=f(qz)+βαγ1α2f(z).

    Substituting the above u,v into

    u2+(1α2)v2=(eλ1(z)+eλ2(z)2)2, (4.2)

    it yields

    [f(z)]2+[f(qz)]2+β2+γ22αβγ1α2f2(z)+2αf(z)f(qz)+2βf(qz)f(z)+2γf(z)f(z)=e2λ1(z)+e2λ2(z)+2eλ1(z)+λ2(z)4. (4.3)

    For simplicity and convenience, we denote

    λ1:=λ1(z),λ2:=λ2(z),~λ1:=λ1(qz),~λ2:=λ2(qz).

    Substituting (4.1) into (4.3) we get

    λ21e2λ1+λ22e2λ22λ1λ2eλ1+λ24δ+e2~λ1+e2~λ22e~λ1+~λ24δ+β2+γ22αβγ1α2e2λ1+e2λ22eλ1+λ24δ+2αλ1eλ1+~λ1λ1eλ1+~λ2λ2e~λ1+λ2+λ2eλ2+~λ24δ+2βeλ1+~λ1eλ1+~λ2e~λ1+λ2+eλ2+~λ24δ+2γλ1e2λ1λ1eλ1+λ2λ2eλ1+λ2+λ2e2λ24δ=e2λ1+e2λ2+2eλ1+λ24. (4.4)

    The transcendental terms appearing in (4.4) have exponents

    2λ1,2λ2,λ1+λ2,2~λ1,2~λ2,~λ1+~λ2,λ1+~λ1,λ1+~λ2,~λ1+λ2andλ2+~λ2.

    In order to apply Lemma 2.2 to (4.4), we checked whether the pairwise difference between these exponents was constant. If λ1λ2, then f(z)0. This is impossible, so λ1λ2.

    Subcase 1.1. If λ1(z)λ2(z) is a nonzero constant, then p(z) is a nonzero constant, denoted by p in the following for simplicity. Consequently,

    f(z)=w=(epep)eg(z)/22iδ. (4.5)

    Then,

    f(z)=(epep)eg(z)/22iδg(z)2 (4.6)

    and

    f(qz)=(epep)eg(qz)/22iδ. (4.7)

    Substituting (4.5)–(4.7) into (1.6), the terms in the left side of this equation can be expressed as

    {[f(z)]2=d2eg(z)4δ(g(z))24,[f(qz)]2=d2eg(qz)4δ,f2(z)=d2eg(z)4δ,2αf(z)f(qz)=2αd2eg(z)+g(qz)24δg(z)2,2βf(z)f(qz)=2βd2eg(z)+g(qz)24δ,2γf(z)f(z)=2γd2eg(z)4δg(z)2, (4.8)

    where d:=epep.

    If polynomial g(z) contains at least two nonconstant terms, without loss of generality, we set

    g(z)=anzn++amzm++a0,n>m,

    then,

    g(qz)=an(qz)n++am(qz)m++a0

    and

    g(qz)g(z)=an(qn1)zn++am(qm1)zm+.

    If g(qz)g(z) is a constant and one has q=1, then this contradicts the assumption and g(qz)g(z) is not a constant. By the same argument, g(qz)g(z)+g(qz)2 is also not a constant. Therefore, the three exponential terms, eg(z),eg(qz) and eg(z)+g(qz)2, are pairwise distinct, even if we don't consider their constant coefficients. Substituting (4.8) into (1.6) and applying Lemma 2.2 to the obtained equation, we get that the coefficients of the three exponential terms are zero. In particular, d24δ=0 since it's the coefficient of the sole term eg(qz). This is impossible, because that means that f0.

    So, g(z) is the form of g(z)=anzn+b, where an(0),b are constants. Substitute this into (4.8) and we obtain that

    {[f(z)]2=d2eanzn+b4δ(nanzn1)24,[f(qz)]2=d2eanqnzn+b4δ,f2(z)=d2eanzn+b4δ,2αf(z)f(qz)=2αnanzn12d2ean(1+qn)zn2+b4δ,2βf(z)f(qz)=2βd2ean(1+qn)zn2+b4δ,2γf(z)f(z)=2γd2eanzn+b4δnanzn12, (4.9)

    then take the above expressions into (1.6). If qn1, the expression of [f(qz)]2 has a zero coefficient by Lemma 2.2, that is d24δ=0, which is impossible, so qn=1. Then, for all zC we have

    (nanzn1)24+2+(α+β)nanzn1+2β4δd2

    by eliminating eg(z) from both sides of (1.6), so n has to be one, and q=qn=1, which contradicts the assumption.

    Subcase 1.2. If λ1λ2 is not a constant, then p(z) is not a constant; instead, it is a nonconstant polynomial. We multiply 4δ, combine like terms in (4.4), and move all the terms to the left side of the equation, then the right side is just zero. Thus, the coefficients of the distinct transcendental terms are listed in Table 3.

    Table 3.  Transcendental terms and corresponding coefficients.
    Transcendental terms Corresponding coefficients
    e2λ1 λ21+2γλ1+1
    e2λ2 λ22+2γλ2+1
    eλ1+λ2 2λ1λ22γ(λ1+λ2)+4δ2
    e2~λ1 1
    e2~λ2 1
    e~λ1+~λ2 2
    eλ1+~λ1 2αλ1+2β
    eλ1+~λ2 2αλ12β
    e~λ1+λ2 2αλ22β
    eλ2+~λ2 2αλ2+2β

     | Show Table
    DownLoad: CSV

    By the same method in Case 2 of Section 3 (proof of Theorem 1.4), the degree of λ1(z) and λ2(z) both are at most one. Set λ1(z)=a1z+b1 and λ2(z)=a2z+b2, where a1a2, b1,b2 are arbitrary constants, then ~λ1(z)=a1qz+b1 and ~λ2(z)=a2qz+b2. Substituting these into Table 3, we get the results in Table 4.

    Table 4.  Transcendental terms and corresponding coefficients.
    No. Before After Corresponding coefficients
    e2λ1 e2a1z+2b1 a21+2γa1+1
    e2λ2 e2a2z+2b2 a22+2γa2+1
    eλ1+λ2 e(a1+a2)z+b1+b2 2a1a22γ(a1+a2)+4δ2
    e2~λ1 e2a1qz+2b1 1
    e2~λ2 e2a2qz+2b2 1
    e~λ1+~λ2 e(a1+a2)qz+b1+b2 2
    eλ1+~λ1 ea1(1+q)z+2b1 2αa1+2β
    eλ1+~λ2 e(a1+a2q)z+b1+b2 2αa12β
    e~λ1+λ2 e(a1q+a2)z+b1+b2 2αa22β
    eλ2+~λ2 ea2(1+q)z+2b2 2αa2+2β

     | Show Table
    DownLoad: CSV

    The coefficient of term ④ in Table 4 is one and it must combine with some like terms by Lemma 2.2. 2a1q in term ④ may be equal to 2a2 in term ②, a1+a2 in term ③, a1+a2q in term ⑧ and a2+a2q in term ⑩ since q1,a1a2. Then, we also considered that terms ⑤ and ⑥ must combine with some other like terms, respectively, since their coefficients are both nonzero, so there are many cases that have to be discussed. Through discussion for all possible cases, it is impossible to have a finite order entire solution (see the Appendix).

    Case 2. If eg(z) can combine with the exponential terms in f(qz), then

    {u=f(z)+αf(qz)+γf(z),v=f(z)+βαγ1γ2f(qz),w=f(qz), (4.10)

    and (1.6) can convert into

    u2+(1γ2)v2+δw2=eg(z), (4.11)

    where

    δ:=1α2β2γ2+2αβγ1γ2.

    By the same method in Section 3, we have

    {f(qz)=eλ1(z)eλ2(z)2iδ,f(z)=eλ1(z/q)eλ2(z/q)2iδ,f(z)=λ1eλ1(z/q)λ2eλ2(z/q)2qiδ, (4.12)

    where

    λ1(z)=p(z)+g(z)/2,  λ2(z)=p(z)+g(z)/2,

    p(z) is a polynomial. Here, λ1(z/q), λ2(z/q) are composite functions with respect to z, and according to the chain rule for derivatives, λ1 and λ2 represent the derivative of the outer function. By the same method in Case 1, we can divide into two subcases.

    Subcase 2.1. If λ1(z)λ2(z) is a constant, then (4.12) can be rewritten as

    {f(z)=deg(z/q)22iδ,f(z)=deg(z/q)22iδg2q,f(qz)=deg(z)22iδ, (4.13)

    where d=epep. Here, g(z/q) is a composite function with respect to z, and according to the chain rule for derivatives, g here represents the derivative of the outer function. Substituting (4.13) into each term in the right side of (1.6), we have

    {[f(z)]2=d2eg(z/q)4δ(g)24q2,[f(qz)]2=d2eg(z)4δ,f2(z)=d2eg(z/q)4δ,2αf(z)f(qz)=2αd2eg(z)+g(z/q)24δg2q,2βf(z)f(qz)=2βd2eg(z)+g(z/q)24δ,2γf(z)f(z)=2γd2eg(z/q)4δg2q. (4.14)

    If polynomial g(z) contains at least two nonconstant terms, without loss of generality we set

    g(z)=anzn++amzm++a0,   n>m,

    then

    g(z/q)=an(z/q)n++am(z/q)m++a0

    and

    g(z/q)g(z)=an(1qn1)zn++am(1qm1)zm+.

    If g(z/q)g(z) is a constant and one has q=1, this contradicts the assumption and g(z/q)g(z) is not a constant. By the same argument, g(z/q)g(z)+g(z/q)2 is also not a constant. Therefore, the three exponential terms, eg(z),eg(qz) and eg(z)+g(qz)2, are distinct and can't combine like terms. Substituting (4.14) into (1.6) and applying Lemma 2.2 to the obtained equation, we get that the coefficients of the three exponential terms are zeroes after combining like terms:

    {d24δ10,d24δ(g)24q2+d24δ+2γd24δg2q0,2αd24δg2q+2βd24δ0. (4.15)

    Since the three equations hold for all zC, g(z/q) has a degree of one, and so does g(z).

    Therefore, we can set g(z/q)=az+b, then g(z)=aqz+b, where a0 and b is an arbitrary constant. Noting that g represents the derivative of the outer function, so g=aq, then the above equations convert into

    {d24δ=1,a24+1+2γa0,αa+2β0. (4.16)

    Thus, it yields α2+β22αβγ=0 and γ21.

    In other words, (1.6) admits a solution in this case with the form f(z)=eaz+b2 and g(z)=aqz+b such that

    {a24+1+2γa0,αa+2β0, (4.17)

    then α2+β22αβγ=0 and γ21.

    Subcase 2.2. If λ1(z)λ2(z) is nonconstant, then from (4.11) we have

    u2+(1γ2)v2=(eλ1(z)+eλ2(z)2)2. (4.18)

    Substituting (4.10) into (4.18) we deduce that

    [f(z)]2+[f(z)]2+α2+β22αβγ1γ2[f(qz)]2+2αf(z)f(qz)+2βf(qz)f(z)+2γf(z)f(z)=e2λ1(z)+e2λ2(z)+2eλ1(z)+λ2(z)4. (4.19)

    For simplicity and convenience, we denote

    λ1:=λ1(z),λ2:=λ2(z),^λ1:=λ1(z/q),^λ2:=λ2(z/q).

    Substituting (4.12) into (4.19) we obtain

    1q2λ21e2^λ1+λ22e2^λ22λ1λ2e^λ1+^λ24δ+e2^λ1+e2^λ22e^λ1+^λ24δ+α2+β22αβγ1γ2e2λ1+e2λ22eλ1+λ24δ+2α1qλ1eλ1+^λ1λ1e^λ1+λ2λ2eλ1+^λ2+λ2eλ2+^λ24δ+2βeλ1+^λ1eλ1+^λ2e^λ1+λ2+eλ2+^λ24δ+2γ1qλ1e2^λ1λ1e^λ1+^λ2λ2e^λ1+^λ2+λ2e2^λ24δ=e2λ1+e2λ2+2eλ1+λ24. (4.20)

    Let's multiply both sides of (4.20) by 4δ and move all terms to the left side of the equation. After combining the terms of the same kind, we get the results in Table 5. Using the method in Subcase 1.2, we can also obtain that there are no suitable finite order transcendental entire solutions for (1.6) in this case. The details are omitted here.

    Table 5.  Transcendental terms and corresponding coefficients.
    Transcendental terms Corresponding coefficients
    e2λ1 1
    e2λ2 1
    eλ1+λ2 4δ2
    e2^λ1 1q2λ21+2γqλ1+1
    e2^λ2 1q2λ22+2γqλ2+1
    e^λ1+^λ2 2q2λ1λ22γq(λ1+λ2)2
    eλ1+^λ1 2αqλ1+2β
    eλ1+^λ2 2αqλ22β
    e^λ1+λ2 2αqλ12β
    eλ2+^λ2 2αqλ2+2β

     | Show Table
    DownLoad: CSV

    In this paper we proved two theorems (Theorems 1.4 and 1.8), studied the finite order entire solutions of (1.4) and (1.6), respectively and found the concrete forms of solutions of these two equations, both of which were exponential functions. Examples 1.5 and 1.6 verified the two cases of solutions of the equation in Theorem 1.4, and Example 1.9 verified the truth of Theorem 1.8. The equations studied in this paper can be transformed into the Fermat-type equation with three quadratic terms by linear transformation, which improves the previous Fermat-type equations with only two quadratic terms, so it is very novel.

    The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by Natural Science Foundation of Henan (No. 232300420129) and the Key Scientific Research Project of Colleges and Universities in Henan Province (No. 22A110004), China. The author would like to thank the referee for a careful reading of the manuscript and valuable comments.

    The author declares no conflict of interest.

    We divided Table 4 into the following four cases with 10 subcases, but in each case there is no finite order transcendental entire solution. The classification is based on the fact that terms ④, ⑤ and ⑥ in Table 4 must be combined with other terms, since their coefficients are nonzero. Otherwise, they contradict with Lemma 2.2.

    Case A1. Term ④ can combine with term ② and ⑨ in Table 4, that is, a1q=a2. Since term ⑤ should also combine with other terms, we split into some subcases.

    Subcase A1.1. Term ⑤ may combine with term ① in Table 4. Thus, we get the results in Table A1.

    Table A1.  q=a2a1=1,a2=a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e2a1z e2b2(a212γa1+1)
    e0 eb1+b2(2a21+4δ2)
    e2a1z e2b1
    e2a1z e2b2
    e0 2eb1+b2
    e0 e2b1(2αa1+2β)
    e2a1z eb1+b2(2αa12β)
    e2a1z eb1+b2(2αa12β)
    e0 e2b2(2αa1+2β)

     | Show Table
    DownLoad: CSV

    After combining terms of the same kind in Table A1, according to Lemma 2.2, we know that the coefficients must always be zero, and the following equations were obtained

    {e2b1(a21+2γa1+1)+e2b2+eb1+b2(2αa12β)=0,e2b2(a212γa1+1)+e2b1+eb1+b2(2αa12β)=0,eb1+b2(a21+2δ2)+e2b1(αa1+β)+e2b2(αa1+β)=0.

    Thus, we have

    {eb1=±eb2,α=±γ,β=±1,

    it yields δ=0, which is impossible.

    Subcase A1.2. Term ⑤ may combine with term ③ in Table 4. Then, we obtained the results in Table A2 as follows.

    Table A2.  q=a2a1=1/2,a2=a1/2.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    ea1z e2b2(a214γa1+1)
    ea12z eb1+b2(a21γa1+4δ2)
    ea1z e2b1
    ea12z e2b2
    ea14z 2eb1+b2
    ea12z e2b1(2αa1+2β)=0
    e5a14z eb1+b2(2αa12β)=0
    ea1z eb1+b2(αa12β)=eb1+b2(3β)
    ea14z e2b2(αa1+2β)

     | Show Table
    DownLoad: CSV

    Combining terms of the same kind and according to Lemma 2.2, we know that the coefficients must always be zero, and the following equations are obtained

    {e2b1(a21+2γa1+1)=0,e2b2(a214γa1+1)+e2b1+eb1+b2(αa12β)=0,eb1+b2(a21γa1+4δ2)+e2b2+e2b1(2αa1+2β)=0,2eb1+b2+e2b2(αa1+2β)=0,eb1+b2(2αa12β)=0.

    For the above equation system, there is no suitable solution a1.

    Case A2. Term ④ can combine with term ③ in Table 4, that is, 2a1q=a1+a2. Since term ⑤ should also combine with other terms, we split it into three subcases.

    Subcase A2.1. Term ⑤ may combine with term ① in Table 4. We get the results in Table A3.

    Table A3.  q=a1+a22a1=1/2,a2=2a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e4a1z e2b2(4a214γa1+1)
    ea1z eb1+b2(4a21+2γa1+4δ2)
    ea1z e2b1
    e2a1z e2b2
    ea12z 2eb1+b2
    ea12z e2b1(2αa1+2β)
    e2a1z eb1+b2(2αa12β)
    e5a12z eb1+b2(4αa12β)
    ea1z e2b2(4αa1+2β)

     | Show Table
    DownLoad: CSV

    From Table A3, after combining terms of the same kind, according to Lemma 2.2, we know that the coefficients must always be zero, and the following equations are obtained

    {e2b1(a21+2γa1+1)+e2b2+eb1+b2(2αa12β)=0,e2b2(4a214γa1+1)=0,eb1+b2(4a21+2γa1+4δ2)+e2b1+e2b2(4αa1+2β)=0,2eb1+b2+e2b1(2αa1+2β)=0,eb1+b2(4αa12β)=0.

    For the above equation system, there is no suitable solution a1.

    Subcase A2.2. Term ⑤ may combine with term ⑦ in Table 4. Then, we get the results in Table A4.

    Table A4.  q=a1+a22a1=14,a2=32a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e3a1z e2b2(94a213γa1+1)
    e12a1z eb1+b2(3a21+γa1+4δ2)
    e12a1z e2b1
    e34a1z e2b2
    e18a1z 2eb1+b2
    e34a1z e2b1(2αa1+2β)
    e118a1z eb1+b2(2αa12β)
    e7a14z eb1+b2(3αa12β)
    e98a1z e2b2(3αa1+2β)

     | Show Table
    DownLoad: CSV

    In Table A4, the term ⑥ cannot combine with other transcendental terms; it's impossible.

    Subcase A2.3. Term ⑤ may combine with term ⑨ in Table 4. Then, we deduce the results in Table A5.

    Table A5.  q=a1+a22a1=14,a2=12a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    ea1z e2b2(a22+2γa2+1)
    e12a1z eb1+b2(2a1a22γ(a1+a2)+4δ2)
    e12a1z e2b1
    e14a1z e2b2
    e18a1z 2eb1+b2
    e54a1z e2b1(2αa1+2β)
    e78a1z eb1+b2(2αa12β)
    e14a1z eb1+b2(2αa22β)
    e58a1z e2b2(2αa2+2β)

     | Show Table
    DownLoad: CSV

    In Table A5, the term ⑥ cannot combine with other transcendental terms; it's impossible.

    Case A3. Term ④ can combine with term ⑧ in Table 4, that is, 2a1q=a1+a2q. Since term ⑤ should also combine with other terms, we split it into two subcases.

    Subcase A3.1. Term ⑤ may combine with term ③ in Table 4. We get the results in Table A6.

    Table A6.  q=a12a1a2=14,a2=2a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e4a1z e2b2(a22+2γa2+1)
    ea1z eb1+b2(2a1a22γ(a1+a2)+4δ2)
    e12a1z e2b1
    ea1z e2b2
    e14a1z 2eb1+b2
    e54a1z e2b1(2αa1+2β)
    e12a1z eb1+b2(2αa12β)
    e74a1z eb1+b2(2αa22β)
    e52a1z e2b2(2αa2+2β)

     | Show Table
    DownLoad: CSV

    The term ⑥ cannot combine with other transcendental terms; this is impossible.

    Subcase A3.2. Term ⑤ may combine with term ⑨ in Table 4. We get the results in Table A7.

    Table A7.  q=a12a1a2=13,a2=a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e2a1z e2b2(a22+2γa2+1)
    e0 eb1+b2(2a1a22γ(a1+a2)+4δ2)
    e23a1z e2b1
    e23a1z e2b2
    e0 2eb1+b2
    e43a1z e2b1(2αa1+2β)
    e23a1z eb1+b2(2αa12β)
    e23a1z eb1+b2(2αa22β)
    e43a1z e2b2(2αa2+2β)

     | Show Table
    DownLoad: CSV

    By ⑦ and ⑩ in Table A7, we have a1=a2, which is a contradiction.

    Case A4. Term ④ can combine with term ⑩ in Table 4, that is, 2a1q=a2+a2q. Since term ⑤ should also combine with other terms, we split it into three subcases.

    Subcase A4.1. Term ⑤ may combine with term ① in Table 4. We get the results in Table A8.

    Table A8.  q=a22a1a2=12,a2=2a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e4a1z e2b2(4a214γa1+1)
    ea1z eb1+b2(4a21+2γa1+4δ2)
    ea1z e2b1
    e2a1z e2b2
    e12a1 2eb1+b2
    e12a1z e2b1(2αa1+2β)
    e2a1z eb1+b2(2αa12β)
    e52a1z eb1+b2(4αa12β)
    ea1z e2b2(4αa1+2β)

     | Show Table
    DownLoad: CSV

    Subcase A4.2. Term ⑤ may combine with term ③ in Table 4. Then, we have the results in Table A9.

    Table A9.  q=a22a1a2=14,a2=23a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e43a1z e2b2(a22+2γa2+1)
    e13a1z eb1+b2(2a1a22γ(a1+a2)+4δ2)
    e12a1z e2b1
    e13a1z e2b2
    e112a1 2eb1+b2
    e34a1z e2b1(2αa1+2β)
    e76a1z eb1+b2(2αa12β)
    e1112a1z eb1+b2(2αa22β)
    e12a1z e2b2(2αa2+2β)

     | Show Table
    DownLoad: CSV

    The term ⑥ cannot combine with other transcendental terms; it's impossible.

    Subcase A4.3. Term ⑤ may combine with term ⑦ in Table 4. Then, we get the results in Table A10.

    Table A10.  q=a22a1a2=13,a2=a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e2a1z e2b2(a22+2γa2+1)
    e0 eb1+b2(2a1a22γ(a1+a2)+4δ2)
    e23a1z e2b1
    e23a1z e2b2
    e0 2eb1+b2
    e23a1z e2b1(2αa1+2β)
    e43a1z eb1+b2(2αa12β)
    e43a1z eb1+b2(2αa22β)
    e23a1z e2b2(2αa2+2β)

     | Show Table
    DownLoad: CSV

    From ⑧ and ⑨ in Table A10, we have a1=a2; it's impossible.



    [1] L. Zhang, Y. Lin, X. Yang, T. Chen, X. Cheng, W. Cheng, From sample poverty to rich feature learning: A new metric learning method for few-shot classification, IEEE Access, 12 (2024), 124990–125002. https://doi.org/10.1109/ACCESS.2024.3444483 doi: 10.1109/ACCESS.2024.3444483
    [2] Y. Lin, Z. Xie, T. Chen, X. Cheng, H. Wen, Image privacy protection scheme based on high-quality reconstruction DCT compression and nonlinear dynamics, Expert Syst. Appl., 257 (2024), 124891. https://doi.org/10.1016/j.eswa.2024.124891 doi: 10.1016/j.eswa.2024.124891
    [3] L. Cui, Y. Cao, A new S-box structure named affine-power-affine, Int. J Innov. Comput. Info. Ctrl, 3 (2007), 751–759. Available from: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=8044cda70fa8d0a18ff4708df185476bb92f3f7a
    [4] H. Liu, A. Kadir, P. Gong, A fast color image encryption scheme using one-time S-Boxes based on complex chaotic system and random noise, Optics commun., 338 (2015), 340–347. https://doi.org/10.1016/j.optcom.2014.10.021 doi: 10.1016/j.optcom.2014.10.021
    [5] X. Wang, Q. Wang, A novel image encryption algorithm based on dynamic S-boxes constructed by chaos, Nonlinear Dyn., 75 (2014), 567–576. https://doi.org/10.1007/s11071-013-1086-2 doi: 10.1007/s11071-013-1086-2
    [6] I. Hussain, T. Shah, H. Mahmood, A new algorithm to construct secure keys for AES, Int. J. Contemp. Math. Sci., 5 (2010), 1263. Available from: https://m-hikari.com/ijcms-2010/25-28-2010/hussainIJCMS25-28-2010.pdf
    [7] X. Zhang, Y. Mao, Z. Zhao, An efficient chaotic image encryption based on alternate circular S-boxes, Nonlinear Dyn., 78 (2014), 359–369. https://doi.org/10.1007/s11071-014-1445-7 doi: 10.1007/s11071-014-1445-7
    [8] I. Hussain, T. Shah, H. Mahmood, A projective general linear group based algorithm for the construction of substitution box for block ciphers, Neural Comput. Appl., 22 (2013), 1085–1093. https://doi.org/10.1007/s00521-012-0870-0 doi: 10.1007/s00521-012-0870-0
    [9] R. Ali, M. K. Jamil, A. S. Alali, J. Ali, G. Afzal, A robust S-box design using cyclic groups and image encryption, IEEE Access, 11 (2023), 135880–135890. https://doi.org/10.1109/ACCESS.2023.3337443 doi: 10.1109/ACCESS.2023.3337443
    [10] R. Liu, H. Liu, M. Zhao, Reveal the correlation between randomness and Lyapunov exponent of n-dimensional non-degenerate hyper chaotic map, Integration, 93 (2023), 102071. https://doi.org/10.1016/j.vlsi.2023.102071 doi: 10.1016/j.vlsi.2023.102071
    [11] C. Luo, Y. Wang, Y. Fu, P. Zhou, M. Wang, Constructing dynamic S-boxes based on chaos and irreducible polynomials for image encryption, Nonlinear Dyn., 112 (2024), 1–19. https://doi.org/10.1007/s11071-024-09353-w doi: 10.1007/s11071-024-09353-w
    [12] B. M. Savadkouhi, A. M. Tootkaboni, S-Boxes design based on the Lu-Chen system and their application in image encryption, Soft Comput., 28 (2024), 1–22. https://doi.org/10.1007/s00500-024-09912-8 doi: 10.1007/s00500-024-09912-8
    [13] R. Ali, J. Ali, P. Ping, M. K. Jamil, A novel S-box generator using Frobenius automorphism and its applications in image encryption, Nonlinear Dyn., 1 (2024), 1–24. https://doi.org/10.1007/s11071-024-10003-4 doi: 10.1007/s11071-024-10003-4
    [14] D. Ustun, S. Sahinkaya, N. Atli, Developing a secure image encryption technique using a novel S-box constructed through real-coded genetic algorithm's crossover and mutation operators, Expert Syst. Appl., 256 (2024), 124904. https://doi.org/10.1016/j.eswa.2024.124904 doi: 10.1016/j.eswa.2024.124904
    [15] Q. Lai, G. Hu, A nonuniform pixel split encryption scheme integrated with compressive sensing and its application in IoMT, IEEE Trans. Ind. Electron., 20 (2024), 11262–11272. https://doi.org/10.1109/TII.2024.3403266 doi: 10.1109/TII.2024.3403266
    [16] S. Gao, H. H. C. Iu, U. Erkan, C. Şimşek, J. Mou, A. Toktas, Design, dynamical analysis, and hardware implementation of a novel memcapacitive hyperchaotic logistic map, IEEE Internet Things J., 11 (2024), 30368–30375. Available from: https://ieeexplore.ieee.org/abstract/document/10552354
    [17] Z. Xie, Y. Lin, T. Liu, H. Wen, Face privacy protection scheme by security-enhanced encryption structure and nonlinear dynamics, IScience, 27 (2024), 110768. https://doi.org/10.1016/j.isci.2024.110768 doi: 10.1016/j.isci.2024.110768
    [18] Y. Wang, Z. Zhang, L. Y. Zhang, J. Feng, J. Gao, P. Lei, A genetic algorithm for constructing bijective substitution boxes with high nonlinearity, Info. Sci., 523 (2020), 152–166. https://doi.org/10.1016/j.ins.2020.03.025 doi: 10.1016/j.ins.2020.03.025
    [19] Q. Lai, L. Yang, G. Chen, Two-dimensional discrete memristive oscillatory hyperchaotic maps with diverse dynamics, IEEE Trans. Ind. Electron., 72 (2024), 969–979. https://doi.org/10.1109/TIE.2024.3417974 doi: 10.1109/TIE.2024.3417974
    [20] M. Wang, H. Liu, M. Zhao, Construction of a non-degeneracy 3D chaotic map and application to image encryption with keyed S-box, Multimed. Tools Appl., 82 (2023), 34541–34563. https://doi.org/10.1007/s11042-023-14988-9 doi: 10.1007/s11042-023-14988-9
    [21] R. Liu, H. Liu, M. Zhao, Cryptanalysis and construction of keyed strong S-Box based on random affine transformation matrix and 2D hyper chaotic map, Expert Syst. Appl., 252 (2024), 124238. https://doi.org/10.1016/j.eswa.2024.124238 doi: 10.1016/j.eswa.2024.124238
    [22] S. Yuanyuan, H. Liu, M. Zhao, Constructing keyed strong S-Box with higher nonlinearity based on 2D hyper chaotic map and algebraic operation, Integration, 88 (2023), 269–277. https://doi.org/10.1016/j.vlsi.2022.10.011 doi: 10.1016/j.vlsi.2022.10.011
    [23] M. Zhao, H. Liu, Y. Niu, Batch generating keyed strong S-Boxes with high nonlinearity using 2D hyper chaotic map, Integration, 92 (2023), 91–98. https://doi.org/10.1016/j.vlsi.2023.05.006 doi: 10.1016/j.vlsi.2023.05.006
    [24] J. Pieprzyk, G. Finkelstein, Towards effective nonlinear cryptosystem design, IEEE Proc.-E: Comput. Digit. Tech., 135 (1988), 325–335.
    [25] J. Ali, M. K. Jamil, A. S. Alali, R. Ali, A medical image encryption scheme based on Mobius transformation and Galois field, Heliyon, 10 (2024), e23652. https://doi.org/10.1016/j.heliyon.2023.e23652 doi: 10.1016/j.heliyon.2023.e23652
    [26] Y. Ma, Y. Tian, L. Zhang, P. Zuo, Two-dimensional hyperchaotic effect coupled mapping lattice and its application in dynamic S-box generation, Nonlinear Dyn., 112 (2024), 1–32. https://doi.org/10.1007/s11071-024-09907-y doi: 10.1007/s11071-024-09907-y
    [27] F. Artuger, F. Ozkaynak, A new chaotic system and its practical applications in substitution box and random number generator, Multimed. Tools Appl., 2024, 1–15. https://doi.org/10.1007/s11042-024-19053-7
    [28] M. Vijayakumar, A. Ahilan, An optimized chaotic S-box for real-time image encryption scheme based on 4-dimensional memristive hyperchaotic map, Ain Shams Eng. J., 1 (2024), 102620. https://doi.org/10.1016/j.asej.2023.102620 doi: 10.1016/j.asej.2023.102620
    [29] S. Ullah, X. Liu, A. Waheed, S. Zhang, An efficient construction of S-box based on the fractional order Rabinovich Fabrikant chaotic system, Integration, 94 (2024), 102099. https://doi.org/10.1016/j.vlsi.2023.102099 doi: 10.1016/j.vlsi.2023.102099
    [30] A. S. Alali, R. Ali, M. K. Jamil, J. Ali, Gulraiz, Dynamic S-Box construction using mordell elliptic curves over galois field and its applications in image encryption, Mathematics, 12 (2024), 587. https://doi.org/10.3390/math12040587 doi: 10.3390/math12040587
    [31] A. Waheed, F. Subhan, S-box design based on logistic skewed chaotic map and modified Rabin-Karp algorithm: Applications to multimedia security, Phys. Scr., 99 (2024), 055236. 10.1088/1402-4896/ad3991 doi: 10.1088/1402-4896/ad3991
    [32] F. Artuger, Strong S-box construction approach based on Josephus problem, Soft Comput., 28 (2024), 1–13. https://doi.org/10.1007/s00500-024-09751-7 doi: 10.1007/s00500-024-09751-7
    [33] T. Shah, A. Elmoasry, S. I. Batool, M. Khan, Quantum harmonic oscillator and Schrödinger paradox based nonlinear confusion component, Int. J. Theor. Phys., 59 (2020), 3558–3573. https://doi.org/10.1007/s10773-020-04616-9 doi: 10.1007/s10773-020-04616-9
    [34] F. Artuger, F. Ozkaynak, A new algorithm to generate AES-like substitution boxes based on sine cosine optimization algorithm, Multimed. Tools Appl., 83 (2024), 38949–38964. https://doi.org/10.1007/s11042-023-17200-0 doi: 10.1007/s11042-023-17200-0
    [35] S. Ibrahim, A. M. Abbas, Efficient key-dependent dynamic S-boxes based on permuted elliptic curves, Info. Sci, 558 (2021), 246–264. https://doi.org/10.1016/j.ins.2021.01.014 doi: 10.1016/j.ins.2021.01.014
    [36] T. Haider, N. A. Azam, U. Hayat, Substitution box generator with enhanced cryptographic properties and minimal computation time, Expert Syst. Appl., 241 (2024), 122779. https://doi.org/10.1016/j.eswa.2023.122779 doi: 10.1016/j.eswa.2023.122779
    [37] A. F. Weister, S. E. Tavares, On the design of S-boxes, Adv. Crypt.-CRYPTO'85, 1 (1986), 1–15. https://doi.org/10.1007/3-540-39799-X\_41 doi: 10.1007/3-540-39799-X\_41
    [38] P. T. Akkasaligar, S. Biradar, Selective medical image encryption using DNA cryptography, Inf. Secur. J. Glob. Perspect., 29 (2020), 91–101. https://doi.org/10.1080/19393555.2020.1718248 doi: 10.1080/19393555.2020.1718248
    [39] W. Cao, Y. Zhou, C. L. P. Chen, L. Xia, Medical image encryption using edge maps, Signal Process., 132 (2017), 96–109. https://doi.org/10.1016/j.sigpro.2016.10.003 doi: 10.1016/j.sigpro.2016.10.003
    [40] A. H. Zahid, A. M. Iliyasu, M. Ahmad, M. M. U. Shaban, M. J. Arshad, H. S. Alhadawi, et al., A novel construction of dynamic S-box with high nonlinearity using heuristic evolution, IEEE Access, 9 (2021), 67797–67812. https://doi.org/10.1109/ACCESS.2021.3077194 doi: 10.1109/ACCESS.2021.3077194
    [41] B. Idrees, S. Zafar, T. Rashid, W. Gao, Image encryption algorithm using S-box and dynamic Hénon bit level permutation, Multimed. Tools Appl., 79 (2020), 6135–6162. https://doi.org/10.1007/s11042-019-08282-w doi: 10.1007/s11042-019-08282-w
    [42] P. Wang, Y. Wang, J. Xiang, X. Xiao, Fast image encryption algorithm for logistics-sine-cosine mapping, Sensors, 22 (2022), 9929. https://doi.org/10.3390/s22249929 doi: 10.3390/s22249929
    [43] A. Ur Rehman, X. Liaa, H. Wang, An innovative technique for image encryption using tri-partite graph and chaotic maps, Multimed. Tools Appl., 80 (2021), 21979–22005. https://doi.org/10.1007/s11042-021-10692-8 doi: 10.1007/s11042-021-10692-8
    [44] X. Chai, X. Fu, Z. Gan, A color image cryptosystem based on dynamic DNA encryption and chaos, Sign. Process., 155 (2019), 44–62. https://doi.org/10.1016/j.sigpro.2018.09.029 doi: 10.1016/j.sigpro.2018.09.029
  • This article has been cited by:

    1. Hira Soomro, Nooraini Zainuddin, Hanita Daud, Joshua Sunday, Noraini Jamaludin, Abdullah Abdullah, Mulono Apriyanto, Evizal Abdul Kadir, Variable Step Block Hybrid Method for Stiff Chemical Kinetics Problems, 2022, 12, 2076-3417, 4484, 10.3390/app12094484
    2. Zeeshan Ali, Faranak Rabiei, Kamyar Hosseini, A fractal–fractional-order modified Predator–Prey mathematical model with immigrations, 2023, 207, 03784754, 466, 10.1016/j.matcom.2023.01.006
    3. Sümeyra Uçar, Analysis of hepatitis B disease with fractal–fractional Caputo derivative using real data from Turkey, 2023, 419, 03770427, 114692, 10.1016/j.cam.2022.114692
    4. Mohammad Partohaghighi, Ali Akgül, Rubayyi T. Alqahtani, New Type Modelling of the Circumscribed Self-Excited Spherical Attractor, 2022, 10, 2227-7390, 732, 10.3390/math10050732
    5. G.M. Vijayalakshmi, Roselyn Besi. P, A fractal fractional order vaccination model of COVID-19 pandemic using Adam’s moulton analysis, 2022, 8, 26667207, 100144, 10.1016/j.rico.2022.100144
    6. SHAIMAA A. M. ABDELMOHSEN, SHABIR AHMAD, MANSOUR F. YASSEN, SAEED AHMED ASIRI, ABDELBACKI M. M. ASHRAF, SAYED SAIFULLAH, FAHD JARAD, NUMERICAL ANALYSIS FOR HIDDEN CHAOTIC BEHAVIOR OF A COUPLED MEMRISTIVE DYNAMICAL SYSTEM VIA FRACTAL–FRACTIONAL OPERATOR BASED ON NEWTON POLYNOMIAL INTERPOLATION, 2023, 31, 0218-348X, 10.1142/S0218348X2340087X
    7. D.A. Tverdyi, R.I. Parovik, A.R. Hayotov, A.K. Boltaev, Распараллеливание численного алгоритма решения задачи Коши для нелинейного дифференциального уравнения дробного переменного порядка с помощью технологии OpenMP, 2023, 20796641, 87, 10.26117/2079-6641-2023-43-2-87-110
    8. Khalid Hattaf, A New Class of Generalized Fractal and Fractal-Fractional Derivatives with Non-Singular Kernels, 2023, 7, 2504-3110, 395, 10.3390/fractalfract7050395
    9. Ali Raza, Ovidiu V. Stadoleanu, Ahmed M. Abed, Ali Hasan Ali, Mohammed Sallah, Heat transfer model analysis of fractional Jeffery-type hybrid nanofluid dripping through a poured microchannel, 2024, 22, 26662027, 100656, 10.1016/j.ijft.2024.100656
    10. Mubashir Qayyum, Efaza Ahmad, Hijaz Ahmad, Bandar Almohsen, New solutions of time-space fractional coupled Schrödinger systems, 2023, 8, 2473-6988, 27033, 10.3934/math.20231383
    11. Muhammad Shahzad, Soma Mustafa, Sarbaz H A Khoshnaw, Inference of complex reaction mechanisms applying model reduction techniques, 2024, 99, 0031-8949, 045242, 10.1088/1402-4896/ad3291
    12. B. El Ansari, E. H. El Kinani, A. Ouhadan, Symmetry analysis of the time fractional potential-KdV equation, 2025, 44, 2238-3603, 10.1007/s40314-024-02991-1
    13. Muhammad Farman, Changjin Xu, Perwasha Abbas, Aceng Sambas, Faisal Sultan, Kottakkaran Sooppy Nisar, Stability and chemical modeling of quantifying disparities in atmospheric analysis with sustainable fractal fractional approach, 2025, 142, 10075704, 108525, 10.1016/j.cnsns.2024.108525
    14. Hira Khan, Gauhar Rahman, Muhammad Samraiz, Kamal Shah, Thabet Abdeljawad, On Generalized Fractal-Fractional Derivative and Integral Operators Associated with Generalized Mittag-Leffler Function, 2025, 24058440, e42144, 10.1016/j.heliyon.2025.e42144
    15. Emmanuel Kengne, Ahmed Lakhssassi, Dynamics of stochastic nonlinear waves in fractional complex media, 2025, 542, 03759601, 130423, 10.1016/j.physleta.2025.130423
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(790) PDF downloads(29) Cited by(2)

Figures and Tables

Figures(23)  /  Tables(5)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog