mathematics Article Weighted Fractional-Order Transform Based on Periodic Matrix Tieyu Zhao * and Yingying Chi Information Science Teaching and Research Section, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China; [email protected] * Correspondence: [email protected] Abstract: 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. Keywords: fractional-order matrix; fractional Fourier transform; eigenvalue; image encryption 1. Introduction Fractional Fourier transform (FRFT) is widely used in quantum mechanics, optics, pattern recognition, time-frequency representation, signal processing, information security Citation: Zhao, T.; Chi, Y. Weighted and other fields [1–12]. Therefore, discrete fractional Fourier transforms, mainly including Fractional-Order Transform Based on weighted-type FRFTs [13,14], eigendecomposition-type FRFTs [15,16] and sampling-type Periodic Matrix. Mathematics 2021, 9, FRFTs [17–19], have been proposed. In 2003, Yeh and Pei proposed a new computation 2073. https://doi.org/10.3390/ method of the discrete FRFT [20]. This method is similar to Shih’s weighted FRFT [13], math9172073 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 Academic Editors: Theodore E. Simos of fractional-order matrices based on the method proposed by Yeh and Pei. Therefore, and Charampos Tsitouras 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 Received: 30 July 2021 of Tao et al., proposed a computation method for the multiple-parameter discrete FRFT, Accepted: 25 August 2021 and further extended the method to multiple parameter discrete fractional cosine, sine, Published: 27 August 2021 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 Publisher’s Note: MDPI stays neutral effective weighted terms in the definition of Yeh and Pei, which will lead to the security risk with regard to jurisdictional claims in of key invalidation when applied to image encryption. Furthermore, our results also show published maps and institutional affil- that the effective weighting term in the definition of Tao et al. is related to the period of the iations. 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 Section2. Section3 analyzes the definition of Yeh et al. Section4 analyzes the Copyright: © 2021 by the authors. definition of Tao et al. Effective weighted terms and security are discussed in Section5. Licensee MDPI, Basel, Switzerland. Finally, conclusions are presented in Section6. This article is an open access article distributed under the terms and 2. Preliminaries conditions of the Creative Commons Tao et al. proposed the idea of the linear summation of fractional-order matrices [21]. Attribution (CC BY) license (https:// If a matrix L satisfies LP = I, then L is a periodic matrix with period P. Assume matrix L is creativecommons.org/licenses/by/ 4.0/). Mathematics 2021, 9, 2073. https://doi.org/10.3390/math9172073 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 2073 2 of 20 a matrix satisfying LP = I and its eigendecomposition form is L = VDVH. Let b = P/M and Lb = LP/M = VDP/MVH. Then, La can be computed as M−1 a nb L = ∑ Cn,a/b L . (1) n=0 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, Fb, F2b, ... , F(M−1)b, where b = 4/M. Denote the sum of these DFRFT matrices as M−1 a nb F = ∑ Cn,aF , (2) n=0 with coefficients 1 1 − e2pi(n−a) Cn,a = , (3) M 1 − e(2pi/M)(n−a) 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 Cn,a and Cn,a/b. Because the fractional-order a 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. 3. Theoretical Analysis of the Definition of Yeh et al. Equation (3) is the sum of geometric progression, and its common ratio is e2pi(n−a)/M. Then, Equation (3) can also be expressed as M−1 1 2pi(n−a)k/M Cn,a = M ∑ e k=0 M−1 1 (−2piak/M) (2pink/M) (4) = M ∑ e e k=0h i = IDFT e(−2piak/M) k=0,1,2,······ ,M−1 where n = 0, 1, ··· , M − 1; and Equation (4) can be further expressed as 0 1 0 0×0 0×1 0×(M−1) 10 (−2pia0/M) 1 C0,a w w ··· w e B 1×0 1×1 1×(M−1) CB (−2pia1/M) C B C1,a C 1 B w w ··· w CB e C B . C = B CB C, (5) B . C M B . .. CB . C @ . A @ . A@ . A (M−1)×0 (M−1)×1 (M−1)×(M−1) (−2pia(M−1)/M) CM−1,a w w ··· w e where w = exp(2pi/M). Then, the definition (Equation (2)) of Yeh and Pei can be expressed as M−1 a 4n/M F = ∑ Cn,aF n=0 0 1 C0,a B C1,a C 0 4/M 4(M−1)/M B C = F , F , ··· , F B . C @ . A (6) CM− a 0 1, 10 1 w0×0 w0×1 ··· w0×(M−1) e(−2pia0/M) B 1×0 1×1 1×(M−1) CB (−2pia1/M) C B w w ··· w CB e C = 1 F0, F4/M, ··· , F4(M−1)/M B CB C. M B . .. CB . C @ . A@ . A w(M−1)×0 w(M−1)×1 ··· w(M−1)×(M−1) e(−2pia(M−1)/M) Mathematics 2021, 9, 2073 3 of 20 Here, we let 8 4(M−1) 0×0 0 1×0 4 (M−1)×0 > W0 = w F + w F M + ··· + w F M > 4(M−1) > 0×1 0 1×1 4 (M−1)×1 > W1 = w F + w F M + ··· + w F M <> 4(M−1) 0×2 0 1×2 4 (M−1)×2 W2 = w F + w F M + ··· + w F M (7) > > . > . > 4(M−1) > 0×(M−1) 0 1×(M−1) 4 (M−1)×(M−1) : WM−1 = w F + w F M + ··· + w F M Definition 1. A new reformulation of the definition of Yeh and Pei as 0 1 e(−2pia0/M) B e(−2pia1/M) C a 1 B C F = (W , W , ··· , W − )B C M 0 1 M 1 B . C @ . A (8) e(−2pia(M−1)/M) M−1 1 (−2piak/M) = M ∑ Wke . k=0 In ref. [20], I, Fb, F2b, ... , 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. 3.1. Eigendecomposition Type FRFT 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 N−1 a −i p ka F (m, n) = ∑ vk(m)e 2 vk(n), (9) k=0 where vk(n) is an arbitrary orthonormal eigenvector set of the N × N DFT. Equation (9) can be written as Fa = VDaVH, (10) a where V = (v0, v1, ··· , vN−1), vk is the kth-order DFT Hermite eigenvector, and D is a diagonal matrix defined as a −i p a −i p (N−2)a −i p (N−1)a D = diag 1, e 2 , ··· , e 2 , e 2 , f or N odd, (11) and a −i p a −i p (N−2)a −i p (N)a D = diag 1, e 2 , ··· , e 2 , e 2 , f or N even. (12) We only prove that N is odd (when N is even, the proof process is the same). In [23,24], npi/2 the eigenvalues of the DFT can be expressed as ln = e . Then, the possible values of the eigenvalue are ln = f1, −1, i, −ig. Therefore, Da = diag(1)a, (−i)a, (−1)a, (i)a, (1)a, (−i)a, (−1)a, (i)a, ······ , (1 or − 1)a. (13) Thus, Equation (7) can be written as ( − ) 0×k 1×k 4 (M−1)×k 4 M 1 Wk = w × I + w × F M + ··· + w × F M , (14) Mathematics 2021, 9, 2073 4 of 20 where w = exp(2pi/M) and k = 0, 1, ··· , M − 1.