Weighted Fractional-Order Transform Based on Periodic Matrix

Tao et al. proposed the definition of the linear summation of fractional-order matrices based on the theory of Yeh and Pei. This definition was further extended and applied to image encryption. In this paper, we propose a reformulation of the definitions of Yeh et al. and Tao et al. and analyze them theoretically. The results show that many weighted terms are invalid. Therefore, we use the proposed reformulation to prove that the effective weighted terms depend on the period of the matrix. This also shows that the image encryption methods based on the weighted fractional-order transform will lead to the security risk of key invalidation. Finally, our hypothesis is verified by the unified theoretical framework of multiple-parameter discrete fractional-order transforms.


Introduction
Fractional Fourier transform (FRFT) is widely used in quantum mechanics, optics, pattern recognition, time-frequency representation, signal processing, information security and other fields [1][2][3][4][5][6][7][8][9][10][11][12]. Therefore, discrete fractional Fourier transforms, mainly including weighted-type FRFTs [13,14], eigendecomposition-type FRFTs [15,16] and sampling-type FRFTs [17][18][19], have been proposed. In 2003, Yeh and Pei proposed a new computation method of the discrete FRFT [20]. This method is similar to Shih's weighted FRFT [13], with the difference being that the fractional power of the discrete Fourier transform (DFT) is used in the definition of Yeh and Pei. Then, Tao et al. presented the linear summation of fractional-order matrices based on the method proposed by Yeh and Pei. Therefore, the fractional power for any diagonalizable periodic matrix is defined, which provides a new idea for information processing [21]. Recently, Kang et al. extended the definition of Tao et al., proposed a computation method for the multiple-parameter discrete FRFT, and further extended the method to multiple parameter discrete fractional cosine, sine, Hartley, and Hadamard transforms. These definitions can be applied to signal processing and image encryption [22]. In this paper, our analysis results show that there are only four effective weighted terms in the definition of Yeh and Pei, which will lead to the security risk of key invalidation when applied to image encryption. Furthermore, our results also show that the effective weighting term in the definition of Tao et al. is related to the period of the matrix. Such extension methods based on that definition are applied to image encryption, which will lead to the security risk of key invalidation.
The remainder of this paper is organized as follows. Preliminary knowledge is described in Section 2. Section 3 analyzes the definition of Yeh et al. Section 4 analyzes the definition of Tao et al. Effective weighted terms and security are discussed in Section 5. Finally, conclusions are presented in Section 6.

Preliminaries
Tao et al. proposed the idea of the linear summation of fractional-order matrices [21]. If a matrix L satisfies L P = I, then L is a periodic matrix with period P. Assume matrix L is In fact, the computation method of the discrete fractional Fourier transform (DFRFT) of Yeh and Pei [20] can be regarded as a special case of the definition of Tao et al. Consider DFRFT matrices I, F b , F 2b , . . . , F (M−1)b , where b = 4/M. Denote the sum of these DFRFT matrices as with coefficients where n = 0, 1, 2, · · · , M − 1. Tao et al. discussed the correlation between the signal length and the period of the matrix to present C n,α and C n,α/b . Because the fractional-order α is a real number, there is no essential difference between the two. Moreover, Shih's research also shows that the signal length is independent of the period of the matrix [13]. However, our analysis shows that the effective weighting term of such a definition depends on the period of the matrix. Next, we will reanalyze the definitions of Yeh et al. and Tao et al.

Theoretical Analysis of the Definition of Yeh et al.
Equation (3) is the sum of geometric progression, and its common ratio is e 2πi(n−α)/M . Then, Equation (3) can also be expressed as where n = 0, 1, · · · , M − 1; and Equation (4) can be further expressed as where w = exp(2πi/M). Then, the definition (Equation (2)) of Yeh and Pei can be expressed as Mathematics 2021, 9,2073 3 of 20 Here, we let Definition 1. A new reformulation of the definition of Yeh and Pei as In ref. [20], I, F b , F 2b , . . . , F (M−1)b are the DFRFT; and the DFRFT has diversity. For Equation (7), we use the eigendecomposition type and the weighted type FRFT for verification.

Proposition 1.
Eigendecomposition type FRFT is used as the basis function, there are only four effective weighting terms for the definition of Yeh and Pei.
Proof. At present, the discrete definition [16] closest to the continuous FRFT is where v k (n) is an arbitrary orthonormal eigenvector set of the N × N DFT. Equation (9) can be written as where V = (v 0 , v 1 , · · · , v N−1 ), v k is the kth-order DFT Hermite eigenvector, and D α is a diagonal matrix defined as and We only prove that N is odd (when N is even, the proof process is the same). In [23,24], the eigenvalues of the DFT can be expressed as λ n = e nπi/2 . Then, the possible values of the eigenvalue are λ n = {1, −1, i, −i}. Therefore, Thus, Equation (7) can be written as Mathematics 2021, 9, 2073 4 of 20 where w = exp(2πi/M) and k = 0, 1, · · · , M − 1. When the eigendecomposition type FRFT is used, Equation (15) is obtained as Therefore, we obtain Equation (16) as From Equation (16), the diagonal matrix only contains S (1) (k), S (i) (k), S (−1) (k), and S (−i) (k). The multiplicities of the DFT eigenvalues are shown in Table 1. Therefore, n + 1 n n n − 1 4n + 1 n + 1 n n n 4n + 2 n + 1 n + 1 n n 4n + 3 n + 1 n + 1 n + 1 n When the eigenvalue is 1, S (1) (k) can be expressed as Therefore, we obtain When the eigenvalue is i, S (i) (k) can be expressed as Therefore, Mathematics 2021, 9, 2073 5 of 20 When the eigenvalue is −1, S (−1) (k) can be expressed as Then, we can obtain When the eigenvalue is −i, S (−i) (k) can be expressed as Therefore, From Equations (19), (21), (23) and (25), Equation (16) can be written as Thus, Equation (8) is expressed as (27), it is not difficult to find that when I, F b , F 2b , . . . , F (M−1)b are eigendecomposition-type FRFTs, there are only four effective weighting terms defined by Yeh and Pei.

Proposition 2.
Weighted type FRFT is used as the basis function, there are only four effective weighting terms for the definition of Yeh and Pei.

Weighted Type FRFT
) are the DFRFTs, which can be explained as shown in Figure 1. Therefore, the definition of Yeh and Pei is more accurate as the generalized form of Shih's FRFT [13]. For Equation (7), we introduce the weighted fractional Fourier transform (WFRFT).
Shih proposed the WFRFT [13]. Shih's WFRFT with a period of 4 is also called the 4weighted type fractional Fourier transform (4-WFRFT), which is defined as Shih proposed the WFRFT [13]. Shih's WFRFT with a period of 4 is also called the 4-weighted type fractional Fourier transform (4-WFRFT), which is defined as with where f l (t) = F l [ f (t)] and l = 0, 1, 2, 3 (F denotes the Fourier transform). Equation (29) can also be expressed as According to the definition of the weighting coefficient A α l [13], Equation (30) can be expressed as where B α k = exp 2πikα 4 and k = 0, 1, 2, 3. Here, we let Therefore, Shih's WFRFT can be represented as From Equations (7) and (33), we can obtain where k = 0, 1, · · · , M − 1 and w = exp(2πi/M). Therefore, Equation (35) is obtained as According to Equations (18), (20), (22) and (24), for k = 0, 1, · · · , M − 1, we can easily determine and Thus, Equation (35) is simplified as Therefore, Equation (8) can be expressed as From Equation (41), the result shows once again that there are only four effective weighted terms for the definition of Yeh and Pei. (41), the result shows once again that there are only four effective weighted terms for the definition of Yeh and Pei.

Theoretical Analysis of the Definition of Tao et al.
Tao et al. proposed the definition of the fractional power of the periodic matrix, which can be expressed as where b = P/M (P is the period of the matrix). Then, Therefore, Equation (42) can be expressed as Here, we let We know that Tao et al. define the fractional power of the periodic matrix. Next, we will analyze the DFT and the discrete Hartley transform as examples. Proof. The calculation of the fractional power of the matrix is applied to the eigenvalues, so eigenvalue decomposition of the matrix is required. Therefore, the eigendecomposition of the matrix can be expressed as

DFT as Periodic Matrix
where F is the matrix of the DFT, V is the eigenvector, and D is the eigenvalue. In refs. [23,24], the eigenvalues of the DFT can be expressed as λ n = e nπi/2 . Then, the possible values of the eigenvalue are λ r = {1, −1, i, −i} and r = 1, 2, · · · , n. In this way, the eigenvalue matrix D can be expressed as Then, the fractional power operation of matrix F can be expressed as For L = F, Equation (45) can be expressed as Therefore, we can obtain Here, let Then, Equation (51) can be expressed as The multiplicities of the DFT eigenvalues are shown in Table 1. Therefore, from Equation (17), we obtain For the sake of simplicity, Equation (52) can be expressed as (55) r = 1, 2, · · · , n.
When the eigenvalues λ r = e 0πi/2 = 1 and w = e 2πi/M , Q r (k) can be expressed using Equation (55) as Therefore, we obtain When the eigenvalue λ r = e πi/2 = i, Q Therefore, When the eigenvalue λ r = e 2πi/2 = −1, Q (−1) r (k) can be expressed using Equation (55) as Therefore, Using Equations (57), (59), (61) and (63), we can formulate Equation (45) as In this way, the definition of Tao et al. can be expressed as  [25] can be expressed as The Hartley matrix has a period of 2, L = H and Equation (45) can be expressed as where k = 0, 1, · · · , M − 1 and w = exp(2πi/M). The fractional power of the Hartley matrix can be expressed as where l = 0, 1, · · · , M − 1; D is the eigenvalue matrix and V is the eigenvector. Therefore, Equation (67) can be expressed as The eigenvalues of the Hartley matrix are {1, −1} [25], and the weighted sum of the diagonal matrix of Equation (69) can be expressed as or where k = 0, 1, · · · , M − 1. From Equation (70), we can obtain and from Equation (71), we can obtain Then, Equation (67) is determined as From Equation (46), the definition of Tao et al. can be expressed as  (65) and (75), we judge that the effective weighting term defined by Tao et al. is related to the period of the matrix. In Section 5, we will prove and explain the security risk of key invalidation when this definition is applied to image encryption.

Effective Weighted Terms Analysis
The definition of Tao

Proof. The eigenvalue can be expressed as
where h = 0, 1, · · · , P − 1. Therefore, Equation (45) can be expressed as Equation (77) can be further expressed as where the eigenvalues λ n ∈ e 2πi0/P , e 2πi1/P , · · · , e 2πi(P−1)/P . Therefore, the weighted sum of the diagonal matrix in Equation (77) can be expressed as where λ h = e 2πih/P with h = 0, 1, · · · , P − 1. Then, we obtain where k = 0, 1, · · · , M − 1. For P < M, Equation (80) can be written as Then, Equation (77) is expressed as Remark 5. Equation (46) has only P effective weighting terms. Therefore, the effective weighting terms depend on the period of the matrix. This explains our analysis in Sections 4.1 and 4.2. Since the DFT has a period of 4, it explains that there are only four effective weighting terms in Section 3.

Security Analysis
Kang et al. extended the definitions of Yeh et al. and Tao et al., and proposed a unified framework for multiple-parameter discrete fractional-order transforms (MPDFRT) [22]. This undoubtedly provides ideas for the further application of the weighted fractional-order transform based on the periodic matrix, especially for the security of image encryption [26]. However, with the help of our research in Section 5.1, the results indicate that the theoretical framework of Kang et al. cannot provide better security.
Therefore, we refer to the theoretical framework of II MPDFRT, and assume that L is a periodic matrix satisfying L P = I and the type II MPDFRT operator is defined as where b = P/M. In Equation (83), the vector parameter where n = 0, 1, 2, · · · , M − 1. Equation (84) can be further expressed as . . .

Definition 3.
A new reformulation of the definition of Kang et al.
where G k is the same as Equation (82). Thus, Equation (88) has only P effective weighting terms. Because b = P/M, Equation (88) can be further expressed as where X k = e −2πiα k k/P .

(a) MPDFRFT
The MPDFRFT is proposed in Ref. [22]. According to our reformulation process, the MPDFRFT can be redefined. Here, L = F (F denotes the Fourier matrix, and F 4 = I), and period P = 4, we let Therefore, the MPDFRFT is redefined as For Equation (87), when L = F, G F k is obtained.
The MPDFRCT is proposed in ref. [22]. Ref. [27] presents four types of discrete cosine transform (DCT) kernel matrices, where the DCT-I (C I N = 2 N−1 k m k n cos mnπ N−1 ) kernel is a symmetric-structured periodic matrix with period 2. Here m, n = 0, 1, · · · , N − 1, and k m and k n are defined as we let X k = exp(−jπα k k).
Then, the MPDFRCT is redefined: For Equation (87), when L = C I N , G C k is obtained. (c) MPDFRST Like the DCT, the discrete sine transform (DST) has four definitions [27]. The DST-I kernel is a symmetric-structured periodic matrix with period 2. Therefore, L = S I N with period 2. X k is the same as Equation (93), so the MPDFRST is redefined as For Equation (87), MPDFRHT is defined in ref. [22], where the discrete Hartley transform (DHT) [25] is Here, L = H with period 2. Here, X k is the same as Equation (93), and the MPDFRHT is redefined as For Equation (87), when L = H, G H k is obtained. (e) MPDFRHaT A Hadamard matrix is a symmetric matrix whose elements are the real numbers 1 and −1. The rows (and columns) of a Hadamard matrix are mutually orthogonal [28]. The normalized Hadamard matrices of order 2 n , denoted by Ha n , can be defined recursively: Ha n Ha n Ha n Ha n ; n ≥ 1 We make L = Ha n with period 2, and X k is the same as Equation (93), therefore the MPDFRHaT is redefined: For Equation (87), when L = Ha, G Ha k is obtained.

Remark 6.
The transforms a-e involved here are proposed in [22]. However, definitions a-e are easy to present with the help of our reformulation. In our new reformulation, the effective weighting terms depend on the period of the matrix. For example, the base matrix L = F of MPDFRFT with period 4, so there are only four effective weighting terms. The parameter α = (α 0 , α 1 , · · · , α M−1 ) is the main key of the system. From Equation (91), it is not difficult to find that the valid keys are only (α 0 , α M−3 , α M−2 , α M−1 ). Furthermore, when k = 0, α 0 is also invalid, and only (α M−3 , α M−2 , α M−1 ) are valid. MPDFRCT, MPDFRST, MPDFRHT and MPDFRHaT have the base matrix with period 2, which has only two effective weighting terms. It is not difficult to find that the valid keys are only (α 0 , α M−1 ). Furthermore, when k = 0, α 0 is also invalid, and only α M−1 are valid.
Therefore, we take MPDFRHaT as an example for numerical verification analysis, and the code is shown in the Appendix A. M is a positive integer greater than or equal to 4. For example, if M = 6, there are 6 weighting terms and the vector parameters α = (α 0 , α 1 , · · · , α 5 ). The size of the image is selected as 256 × 256, so N = 256.
The image encryption/decryption based on MPDFRHaT is shown in Figure 2, and Figure 2a is the original image (plaintext). We set the encryption keys as: encryption/decryption process, regardless of the value. The wrong decryption key is selected again, A numerical simulation also verified that many keys are invalid for the image encryption based on MPDFRHaT, which again supports our hypothesis. This security risk comes from the periodicity of the basis matrix. Strictly speaking, the period of the matrix determines the number of weighted terms of the weighted fractional-order transform. This discovery provides an important reference for future research.

Conclusions
In this paper, we propose a reformulation of the definition of Yeh et al., and the eigendecomposition type FRFT and the weighted type FRFT are verified. The results show that there are only four effective weighting terms. Furthermore, we determine that the

Conclusions
In this paper, we propose a reformulation of the definition of Yeh et al., and the eigendecomposition type FRFT and the weighted type FRFT are verified. The results show that there are only four effective weighting terms. Furthermore, we determine that the definition of Tao et al. is an extended definition for that of Yeh et al., and propose a reformulation of the former. The fractional power of the DFT and the fractional power of the discrete Hartley transform are verified, and the results show that the effective weighting terms are defined as four terms and two terms, respectively. We perform a further analysis, and the results show that the effective weighting terms depend on the period of the matrix, which will lead to the security risk of key invalidation. Therefore, we propose a reformulation of the unified framework for MPDFRT, and determine that many keys are invalid for image encryption. Finally, we take MPDFRHaT as an example to verify that there is only one valid key, with other keys being invalid. Our observations prove once again that the effective weighting terms of the weighted fractional-order transform based on the periodic matrix depend on the period of the matrix, which will lead to the security risk of key invalidation for image encryption.