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1-2010 On the Eigenstructures of Functional K-Potent Matrices and Their nI tegral Forms Yan Wu Georgia Southern University

Daniel F. Linder Georgia Southern University, [email protected]

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Recommended Citation Wu, Yan, Daniel F. Linder. 2010. "On the Eigenstructures of Functional K-Potent Matrices and Their nI tegral Forms." WSEAS Transactions on , 9 (1): 244-253. https://digitalcommons.georgiasouthern.edu/biostat-facpubs/2

This article is brought to you for free and open access by the Biostatistics, Department of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Biostatistics Faculty Publications by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. WSEAS TRANSACTIONS on MATHEMATICS Yan Wu, Daniel F. Linder

On the Eigenstructures of Functional K-Potent Matrices and Their Integral Forms

Yan Wu1 and Daniel F. Linder2 1Department of Mathematical Sciences Georgia Southern University Statesboro, GA 30460 [email protected]

2Department of Biostatistics Medical College of Georgia Augusta, GA 30901 USA

Abstract: - In this paper, a functional k-potent satisfies the equation AIAk =α + β r , where k and r are positive integers, α and β are real numbers. This class of matrices includes idempotent, Nilpotent, and involutary matrices, and more. It turns out that the matrices in this group are best distinguished by their associated eigen-structures. The spectral properties of the matrices are exploited to construct integral k-potent matrices, which have special roles in digital image encryption.

Key-Words: Nilpotent, Idempotent, Involutary, , Skewed k-potent Matrix, Diagonalizability, Image Encryption

1 Introduction n× n With (2), the between the subset of periodic Let AC∈ be an n by n complex matrix and it is matrices and unipotent matrices are readily seen, so are said to be idempotent if AA2 = . This definition can be the skew-periodic and skew-unipotent matrices. It is not k generalized to a higher power on A, if AA= for some difficult to verify that <1,0 > ⊂ < 0,1 > and positive integer k ≥ 2 . With the same condition on A, <−1,0 > ⊂ < 0, − 1 > . The results can be made stronger if if Ak = 0 , a , for some positive integer k, the we impose that the matrices in <0, ± 1 > be invertible. In matrix is called a . Another important 2 that case, those subsets are identical, i.e. class of matrices is called involutary, i.e. AI= , the = and = . Next, we . We define the unipotent matrix as a <1,0 > < 0,1 > <−1,0 > < 0, − 1 > natural extension of the involutary matrix as follows: a introduce an index number for a matrix in <α, β > . It matrix A is unipotent if it satisfies AIk = , for some turns out that such an index number is closely related to positive integer k. A skew-periodic matrix the eigen-structure of the matrix. The index number of satisfies AAk = − , while a skew unipotent matrix is a k-potent matrix, I<α, β > , is defined as defined as AIk = − . All the above mentioned special matrices can be unified by a single equation: I<α, β > = min k (3) k k AIA=α + β AIA=α + β , (1) k ≥2 where αβ=0, α , β ∈{ − 1,0,1,and} k ≥ 2. A matrix A is which is understood as the smallest positive integer that said to be k-potent if it satisfies (1). Consequently, we k introduce the class of k-potent matrices, satisfies AIA=α + β . Some of the k-potent matrices, for instance, the k nilpotent matrix, are mentioned occasionally in linear =AAIA| =α + β, αβ = 0, α , β ∈{} − 1,0,1 , k ≥ 2 (2) <α, β > { } algebra textbooks, such as [1] and [2], in the context of

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eigenvalue problems of a matrix. However, they are with strong perplexing characteristics, such as non- not studied in details in those books. It is also hard to periodic, non-convergent, randomness, and ergodic to find relevant discussions over such matrices, let alone the visual data. The most common nonlinear chaotic the more generalized the k-potent matrices defined in maps inherit properties as discrete cryptographic (1), in those more research-oriented handbooks in systems. Such systems are hybrids between matrix theory, such as [3], [4], and [5]. The eigen- permutation and substitution ciphers with specific structures of rational functions of tridiagonal matrices properties. Scharinger [18] was the first to apply a with closed form expressions were obtained in [6], class of nonlinear maps known as Kolmogorov flows where the is raised to a positive for the digital encryption purpose. More papers on integral power. Steinberg, et. al. [7] studied the chaotic encryption followed, such as the chaotic key- solvability of differential algebraic equations with a based algorithms [19], chaotic systems for permutation nilpotent matrix as the descriptor. Bakasalary, et. al. transformation in images [20], and high-dimensional published a series of papers [8-11] on the idempotency Arnold and Fibonacci-Q maps [21]. However, some of linear combinations of idempotent and tripotent chaotic cryptosystems have been identified susceptible matrices. However, a thorough discussion on the eigen- to cryptanalysis due to the design disfigurement of their structures of the various matrices of interest is missing part-linear characters. Some attack algorithms have from the literature. The purpose of this paper is to been developed in [22-23]. A common concern of the characterize these k-potent matrices via the canonical aforementioned encryption methods arises from the forms associated with the matrices. The result turns out decryption site, where the data is unscrambled. In to be useful for systematic construction of such many occasions, the perfect decryption is impossible matrices via a similarity transformation over the integer due to slight disparity of the encryption/decryption keys field. The proposed algorithm could be favored by or simply roundoff errors in and out of the instructors who teach or numerical transformation domain. In many applications, such as analysis at time they want to come up with their own medical, military operations, and satellite image special matrices for their examinations or projects. The processing [24], the quality of the images transmitted to results of this paper can be appealing to the the receiver station is crucial during the decision cryptography community because k-potent matrices are making process. Therefore, perfect reconstruction of useful in digital signal encryption, which will also be the original image from the encrypted data is imperative explored in this paper. when selecting various encryption methods, in addition In the past decade or so, image encryption techniques to robustness to various attacks. were developed to keep up with the pace of the growth Images are stored in two-dimensional arrays, which of internet and multimedia communications. There are make matrices the natural candidates for the kernels of hard encryption and soft encryption approaches. Most encrypting operators. Moreover, digital images are scrambled with soft encryption, is analogous to convolution/deconvolution between which is also the choice of encryption as a component filters and signals. The matrix leaves signatures of the proposed UAS. Most image encryption methods onto the image pixels and grey levels strictly over the can be classified as the DCT-based techniques, DWT- integer field. There will be no roundoff errors in the based (Discrete Wavelet Transform) techniques, decrypted images; hence, perfect reconstruction of the transformations, and chaotic maps. Both DCT and original image is achieved. The matrix considered in DWT-based techniques are known as compression this paper is called k-potent integral matrix. It is a oriented schemes. The well-received MPEG encryption generalization of nilpotent, idempotent, and involutary was first proposed by Tang [12] and is called “zig-zag matrices. permutation algorithm”. The idea is to substitute the fixed zig-zag quantized DCT coefficient scan pattern by 2 Eigen-structure of functional k-potent a random permutation list. A number of improvements on MPEG encryption were developed thereafter [13- matrix and integral form 14]. The DWT-based method, [15-16], takes advantage As discussed in the previous section, we are looking of the efficient image compression capability of for integral matrices that satisfy (1). Some of these wavelet networks through multi-resolution analysis matrices can be adopted in image encryption as the integrated with block cipher data encryption. Some encryption keys. One of the requirements for a robust public key cryptographic systems uses Jacobian group cryptosystem is that the key space is infinite of Cab curves, which is defined by a multi-variable dimensional. Well, how many integral matrices are polynomial function to perform the encoder and space there that satisfy (1)? The answer is infinitely many. time operations [17]. The chaos-based encryption of The following study will reveal a systematic approach images employs the principle of applying chaotic maps for constructing such matrices, which turns out be

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closely related to the eigen-structure of the k-potent k 1 if j= i + k matrix. We will go through the case studies of some Jij = i, j = 1,2,..., m , k = 2,3,... (4) 0 otherwise well-known matrices, and, more importantly, extend the  results to higher k-values as in (1). We first investigate the spectral decomposition of Proof: Use mathematical induction. For the case k = 2 , nilpotent matrices. A A is such that from the matrix product formula, k A = 0 , the zero matrix, for some positive integer k m m 2 1 1 known as the index number of Nilpotency if the integer JJJJJJJij=∑ issj = ∑ issj = ii+1 ii + 1 + 2 =1 s=1s = 1 is the smallest positive integer so that Ak−1 ≠ 0 . Nilpotent matrices are useful in the design of digital FIR 2 1 if j= i + 2 which implies that Jij =  , satisfying (4). filter banks with unequal filter length. The eigen- 0 otherwise structure of a nilpotent matrix is revealed in what k+1 Now, assume (4), we obtain J , follows. Note that most of the proofs are omitted due to ij limited space. m m k+1 k 1 k k JJJJJJJij =∑is sj = ∑ is sj = i i+ k i+ k i+ k +1 =1 Proposition 2.1 The eigenvalues of a nilpotent matrix s=1 s=1 are all zeros.

k+1 1 if j= i + k + 1 Proof: Let λ be an eigenvalue and v the corresponding which establishes Jij =  . Therefore, 0 otherwise eigenvector of a nilpotent matrix A satisfying Ak =0, formula (4) for an element of the matrix J k is valid. with k ≥ 2 .Then, Av= λ v ,which yields 0 =Ak v = λ k v . m

Therefore, λ = 0. Proposition 2.3 reveals that the 1s’ on the off- of

a nilpotent Jordan block J are pushed over to the Proposition 2.2 Suppose the square matrix A is a m nonzero nilpotent matrix, then A is not diagonalizable. upper right corner as the power k increases. k Consequently, it is easy to verify that Jm becomes a Proof: Assume A is diagonalizable, then the spectral zero matrix if k is greater than or equal to the size of the decomposition of the matrix is given by APP= Λ −1 . k matrix Jm , i.e. Jm = 0(zero matrix) if k≥ m . According to proposition 2.1.1, A = 0 , a zeros matrix, Introduce the Jordan canonical form Λ with nilpotent contradicting that A is a nonzero matrix. Jordan blocks along its main diagonal, and the off- diagonal blocks are zero matrices, i.e. Proposition 2.2 implies that the eigen-space associated with the zero eigenvalues is degenerate. Such a matrix has a generalized spectral decomposition with Jordan J  canonical forms. In other words, there are decoupled  m1  Jordan blocks in Λ . In order to further explore the  J Ο   m2  eigen-structure of a nilpotent matrix, we introduce a    ⋱  special nilpotent matrix called nilpotent Jordan block. Λ =   (5)  Ο Jm   p  Definition 2.1 The elements of a nilpotent Jordan block J    ⋱  satisfy Jij =1, if j= i +1, and Jij = 0 , if j≠ i +1.    J   mn  The following proposition will be used to link the index where the dimension of the Jordan block J is number of nilpotency for a nilpotent matrix to the size of mi the largest nilpotent Jordan block associated with the mi by m i , i= 1,2,..., n . We have the following result. matrix. Proposition 2.4 Let Λ be the Jordan Canonical form (5) Proposition 2.3 Let Jm be an m by m nilpotent Jordan with nilpotent Jordan blocks along its main diagonal. If

k k mp−1 block, and let Jij be an element of the matrix Jm , mp = max{ m1 , m 2 ,..., mn }, then Λ ≠ 0 and k = 2,3,..., then, m Λp = 0.

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Proof: It is easy to verify that periodic matrices. A square matrix A such that AAk = for k to be a positive integer is called a periodic matrix.  q  If k is the least such integer, then the matrix is said to Jm   1  have period k-1. The well-known q  J  2  m2  i.e. AA= , is obviously a special case of the periodic    ⋱  matrix to be studied here. Periodic matrices are useful in Λq =   (6)  q  digital signal encryption such as image coding. We  Jm  begin with exploring the spectral properties of a periodic  p   ⋱  matrix.    q   J  Proposition 2.6 Let A be a periodic matrix with index  mn  number k and letλ be an eigenvalue of A, then Therefore, according to Proposition 2.3, the λ ∈{0} ∪{ei2 mπ /( k− 1) , m= 0,1,..., k − 2} . m −1 m −1 J p in Λ p is a nonzero matrix because one mp Proof: Let λ and v be the eigenvalue and corresponding mp −1 mp−1 element in J is nonzero, i.e. J =1. On eigenvector of A respectively. From Avλ v , one has mp ( mp ) = 1,mp Ak v=λ A k−1 v = λ k v . Hence, λv= λk v , which implies the other hand, let q= mp , from Proposition 2.3, each λk − λ = 0 . The solution set of this equation is mp block matrix on the main diagonal of Λ is a zero k−2 {}0,1 ∪ ei2mπ /( k− 1) . matrix because the power mp either exceeds or equals the { }m=1

size of any block matrix Jm because i Proposition 2.6 tells us that the eigenvalues of a periodic mp mp = max{ m1 , m 2 ,..., mn }. Hence, Λ = 0. matrix are distributed around the unit circle or possibly at the origin. The next proposition addresses the Proposition 2.4 implies that the Λ is a diagonalizability of periodic matrices. nilpotent matrix, and the index number of the Nilpotency for Λ equals the dimension of the largest nilpotent Proposition 2.7 Periodic matrices are diagonalizable. Jordan block in Λ . The following result links the index number of a Proof: Let A be a periodic matrix. Assume A is not nilpotent matrix to the size of the largest nilpotent diagonalizable. Then, the matrix has a Jordan Jordan block associated with the matrix, which also decomposition APP= Λ −1 , where Λ is similar to (5). plays an important role in the symbolic construction of Since A is not diagonalizable, there exists a Jordan block nilpotent matrices to be discussed later. in (5) such that it has the following form:

Proposition 2.5 The index number of a nilpotent matrix λs 1 0 ... 0  equals the size of the largest nilpotent Jordan block    λ 1 ... 0  associated with the matrix.  s    J =   (7) ms  ⋱ ⋱  Proof: Let A be a nilpotent matrix. Then, there exists a   nonsingular matrix P such that a Jordan decomposition    λs 1  of the matrix exists, i.e. APP= Λ −1 , where Λ is given    λs  by (5). From APPPPk ()−1 k k −1 , one can see = Λ = Λ k k that A = 0 if and only if Λ = 0. According to As discussed earlier, APPk= Λ k −1 , where Λk has a Proposition 2.4, the smallest positive integer k satisfying k similar expression to (6). We will show that JJm≠ m , Λk = 0 equals the size of the largest nilpotent Jordan s s which leads to the contradiction to A being periodic. To block in Λ. Therefore, the index number of a nilpotent this end, we rewrite JN= Λ + , where Λ is a ms s s s matrix, I<0,0 > , equals the size of the largest nilpotent with λ on the main diagonal, and N is Jordan block associated with the matrix. s s a nilpotent Jordan block (see definition 2.1). Use Our next group of matrices is in the category of binomial expansion to obtain the following

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k min{k , ms} k Proposition 2.8 further reveals the connection between a JNCNCNk = Λ + =n Λ k− n n =n Λ k− n n ms () s s ∑ k s s ∑ k s s unipotent matrix and a periodic matrix from the circular n=0n= 0 (8) distribution of their eigenvalues.

 k1 k− 1 2 k − 2 ms−1 k− ms +1  λsC k λ sC k λ s ... Ck λs  Proposition 2.9 Unipotent matrices are diagonalizable  m 2   λkCCC1 λ k− 1 2 λ k − 2... s− λk− ms +2  s k s k s k s  The proof of this proposition is almost identical to the =  ⋱    proof of Proposition 2.7 due to the comparable structure  ⋱ 2k− 2   Ckλ s  between these two matrices. Similar to (9), the eigenvalue    λk  of a unipotent matrix with index number k must satisfy  s  the following equation:

k In deriving (8), we used Proposition 2.3. It is easy to see λ −1 = 0 . (10) that JJk ≠ from (8). Hence, a periodic matrix must ms m s Since the treatment for the skewed k-potent matrix is be diagonalizable. exactly the same as that for the previously discussed k- potent matrices, we summarize the results as follows on Unlike nilpotent matrices, the eigen-space of a periodic the skewed k-potent matrix. matrix is non-degenerate. A periodic matrix is similar to A skew-periodic matrix A satisfies the constraint with a diagonal matrix via a similarity transformation. This index k ≥ 2 , AAk = − . We have the following results result is useful for numerical formation of periodic for the spectral properties of skew-periodic matrices. matrices. The index number of a periodic matrix obviously Proposition 2.10 Let A be a skew-periodic matrix with relates to the periodicity of the matrix as seen from the index number k and letλ be an eigenvalue of A, then definition of a periodic matrix. We would like to point it i(2 m+ 1)π /( k − 1) out that the eigenvalue (except zero) of a periodic matrix λ ∈{0} ∪{e ,m = 0,1,...,k − 2} . with period ν must satisfy the following equation: The eigenvalues (except zero) of a skew-periodic matrix λν −1 = 0 . (9) are solutions of the following equation

Condition (9) gives another criterion for identifying a λk−1 +1 = 0 . (11) periodic matrix with certain periodicity. In what follows we look into the case of unipotent Proposition 2.11 Skew-periodic matrices are matrices. A unipotent matrix extends the involutary diagonalizable. matrix to a higher-order power matrix. To be exact, a unipotent matrix satisfies Ak = I, k ≥ 2. It is easily seen A skew-unipotent matrix A satisfies the from the definitions that a unipotent matrix must also be a constraint AIk =− . We have the following results for periodic matrix, but not the other way around unless the the spectral properties of skew-unipotent matrices. periodic matrix is also invertible. Again, we are interested in exploring the spectral properties of unipotent Proposition 2.12 Let A be a skew-unipotent matrix with matrices. index number k and let λ be an eigenvalue of A, then k−1 i(2 m+ 1)π / k Proposition 2.8 Let A be a unipotent matrix with index λ ∈ e . { }m=0 number k and let λ be an eigenvalue of A, then λ ∈{ei2 mπ / k , m = 0,1,2,..., k − 1} The eigenvalues of a skew-unipotent matrix with index number k satisfy the following equation:

λ Proof: Let and v be the eigenvalue and corresponding k eigenvector of A respectively. From Av= λ v , one has λ +1 = 0 . (12) k k k k A v= λ v . Hence, v= λ v , which implies λ =1.The Proposition 2.13 Skew-unipotent matrices are k−1 solution set of this equation is {}1 ∪ ei2 mπ / k . diagonalizable. { }m=1

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Finally, we examine the spectral properties of the In summary, we categorize four groups of k-potent functional k-potent matrix as follows, matrices: (i) nilpotent matrices, (ii) periodic and unipotent matrices, (iii) skew-periodic and skew- Ak =α I + β Ar , αβ ≠ 0, k > r ≥ 1 (13) unipotent matrices, and (iv) the functional k-potent matrices. The classification is based on the

characteristics of the eigenvalue/eigen-space of the It can be verified that the eigenvalues of a functional k- matrices. The results presented above will be used to potent matrix (13) are solutions of the following manufacture such matrices symbolically, i.e. all k-potent equation matrices are constructed over the integer field. k r λ− βλ − α = 0 (14) Our objective in this work is to develop an algorithm for constructing integral k-potent matrices. In particular, Since α ≠ 0 , all the eigenvalues are nonzero. Therefore, (skew-) periodic and (skew-) unipotent matrices are the matrix is nonsingular. useful in digital signal encryption. Instructors who teach Linear Algebra and Numerical Analysis may find the Proposition 2.14 Functional k-potent matrices (13) are proposed algorithm useful as they may want to come up diagonalizable. with a number of such k-potent matrices for students to practice with the related concepts in matrix theory. Proof: Assume the matrix is not diagonalizable. With The idea is simple. A power-induced matrix can be the spectral decomposition of A, APP= Λ −1 , where easily constructed via the spectral decomposition Λ is given by (5). The Jordan block in Λ is given by (7). formula, i.e. Therefore, this Jordan block must satisfy (13). Given APP= Λ −1 (15) that λs ≠ 0 because the eigenvalues are nonzero. One has, according to (8), where P is an and Λ is either a diagonal matrix or a block diagonal matrix in Jordan k min{k , m } form. It is easy to see that, as long as Λ is k-potent, the k s JNCNk = Λ + =n Λ k− n n =CNn Λ k− n n matrix A is k-potent of the same type. In what follows, ms () s s ∑ k s s ∑ k s s n=0n= 0 we introduce different ways for constructing the Λ -

 k1 k− 1 2 k − 2 ms−1 k− ms +1  matrix so that it is a power-induced matrix satisfying a λsCC k λ s k λ s ... Ck λs   m 2  predetermined index number.  λk CCC1 λk− 1 2 λk − 2... s− λk− ms +2   s k s k s k s  =  ⋱    Case (i): Nilpotent matrices  ⋱ C2λk− 2   k s  According to Proposition 2.4, the -matrix in  k  Λ  λs  (15) is guaranteed nilpotent with certain index number if Λ consists of nilpotent Jordan blocks, and and the size of the largest nilpotent Jordan block equals the index number. The following matrix, r r r n r− n n for example, is a nilpotent matrix with index 4, αIAINICN+ β = α + β () Λs + s =α + β ∑ r Λ s s 4 n=0 i.e. Λ = 0 .

min{}r , ms =αICN + β n Λ r− n n 0 1 0 0 0 0 ∑ r s s   n=0 0 0 0 0 0 0  r1 r− 1 2r − 2 ms−1 r − m s + 1    α+ βλsCCC r βλ s r βλ s ... rβλ s     0 0 0 1 0 0 r1 r− 1 2r − 2 ms−2 r − m s + 2 Λ = . (16)  α+ βλsCCC r βλ s r βλ s... r βλ s  0 0 0 0 1 0     = ⋱     0 0 0 0 0 1  ⋱ C2βλr− 2     r s  0 0 0 0 0 0  k   α+ βλs  Case (ii): Periodic matrices Comparing the two matrices, the off-diagonal entries Proposition 2.7 and equation (9) are the keys are mismatched because k> r . Therefore, the functional for constructing periodic -matrix. For the sake k-potent matrices (13) are diagonalizable. Λ of argument, let ν be the period of and let Λ

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Γ ={0} ∪ei2 mπ / ν , m = 0,1,...,ν − 1 be the set must be satisfied for the corresponding matrices. ν { } For mathematics instructors, it is preferred to work of eigenvalues of the periodic matrix. It is with integral matrices, i.e. the elements of a matrix are sufficient that all integers, mainly because the arithmetic is symbolic as far as additions and multiplications are concerned, which Λ = diag(λ1 , λ 2 ,..., λs ) , (17) also implies that there are no roundoff errors. We are able to achieve this when constructing the Λ -matrix, see (21), for instance. where λi ∈ Γν ,i = 1,2,..., s , which guarantees Instead of constructing the -matrix, one can take that the Λ -matrix (12) is a periodic matrix with Λ advantage of the for the characteristic period ν . The Λ -matrix can also be written as a block diagonal matrix as follows polynomial [3]. In general, the companion matrix of an nth degree characteristic polynomial

Λ = diag(BBB , ,..., ) (18) 1 2 m n n−1 P()λ= λ + an−1 λ +... +a1 λ + a 0 as long as the eigenvalues of each block is given by B, i= 1,2,..., m , belong to Γν . This setting i gives us some flexibility for constructing 0 0 ... 0 −a  periodic matrices. One can also mix the  0  eigenvalues of the Λ -matrix in (17) or (18) to 1 0 ... 0 −a   1  construct periodic matrices with a higher index Λ = 0 1 ... 0 −a  (22) number. To this end, let the eigenvalues of p  2  Λ  ⋮  be chosen from the following set    ⋯  0 0 1 −an−1 Γ = Γ ∪ Γ ∪... ∪ Γ , (19) ν1 ν 2 νt The characteristic polynomials of various kinds of k- and let potent matrices are given by equations (9)-(12) and (14). Formula (15) can be used if one wants to construct a ∗ dense integral k-potent matrix, where both P and P−1 ν= LCM( ν1 , ν 2 ,..., νt ) , (20) in (15) have to be integral matrices. In what follows, let

n× n where LCM stands for least common multiple, Z represent the set of n by n integral matrices. The then, it can be verified that the period of Λ is following proposition gives the necessary and sufficient 1 n n n n ν∗ . For example, the following matrix is a condition for AZ−∈ × if AZ∈ × . periodic matrix with period 12, Proposition 2.15 Suppose AZ∈ n× n and A is a  1 3 0 0   nonsingular matrix, then AZ−1 ∈ n × n if and only if −1 − 2 0 0 A = ±1)det( .   (21) Λ =    0 0 0 1   Proposition 2.15 gives us a guideline for constructing  0 0− 1 0 such an integral P-matrix. We can simply use the

following formula for P, because the eigenvalues of the first block are

1 3 − ±i , which are solutions of (9) with P= UL , (23) 2 2 ν = 3 , and the eigenvalues of the second block where U is an upper triangular integral matrix with 1’s in (21) are solutions of (9) with ν = 4. on the main diagonal and L is a lower triangular integral matrix with 1’s on the main diagonal. It is obvious The treatment for constructing the other Λ -matrices, that P =1, according to Proposition 2.15, P−1 is an i.e. unipotent, skew-periodic, and skew-unipotent matrices, is essentially the same as that for periodic integral matrix. With an integral P-matrix from (23), we matrices because the eigen-structures among those obtain a dense integral nilpotent matrix calculate from matrices are similar. When constructing such matrices, (16), one should realize that the equations (9), (10), and (11)

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 22 44 114 183 2− 317 pixel so that the original image is no longer   recognizable. This masking process is in essence a −13 − 26 − 68 − 110 − 4 188    filtering process because each row (column) in the    20 40 104 167 2− 289 encryption key matrix is treated as a digital filter with A =    9 18 48 77 3− 133 finite impulse response. Due to the randomness and   magnitude of the filter coefficients, the original image is −14 − 28 − 74 − 118 − 2 206    transformed into a rather different image by way of a  12 24 63 101 2− 175 filter banks. . We adopt the previously studied k-potent matrices for It is verified in MatLab that A4 = 0, which has the same the encryption key matrix, particularly the unipotent or index of nilpotency as that of (11). A more sophisticated periodic matrices. The nilpotent matrix can also be used algorithm for constructing integral similarity for image encryption with some special treatment such transformation matrices can be found in [25]. as diagonal perturbation, but we will not elaborate here. We also constructed an integral periodic matrix from The cryptosystem proposed in this paper consists of (21) as follows associate keys and primary keys. The function of the associate key T1 is to divide the original image into sub-  −8 − 1 2 4  images, not necessarily the same sizes, followed by   another associate key T to permute the pixels of the  22 4− 5 − 12 2 A =   . sub-image for pre-scrambling. The permutation key is −42 − 8 9 21    nothing but a product of elementary matrices. The    11 3− 2 − 6  mathematical setting is given as following for the pre- encryption stage: It is hard to tell that the matrix above is a periodic matrix with period 12 unless one literally calculates the power T: ZM×N → Zmi× n i , m < M , n < N , i = 1,2,..., k 1i i i (24) matrix A13 without peeking at the eigenvalues of A. ∑mi = M,. ∑ ni = N For more general k-potent matrices, the eigenvalues

are usually non-integers. However, it is always feasible T: Zmi× n i → Z mi × n i , i = 1,2,..., k . to make use of the companion matrix (22) for the 2i (25) characteristic polynomials of the k-potent matrices, such TEEE2i= i 1 i 2 ⋅⋅⋅ is as (9)-(12) and (14), combined with the integral similarity P in (23). The following where Eij is an that exchange the matrix satisfies matrix equation AIA4 =3 − 2 , rows of a matrix if left-multiplied or columns of the

matrix if right-multiplied. −9 − 16 − 7 24    The primary key can be formulated via a product of  7 12 6− 17 unipotent and/or skew-unipotent matrices as follows A   . =   −2 − 3 − 1 5    TAAA=k1 k 2 ⋅⋅⋅ kt (26)  1 2 2− 2  M 1 2 t

Let Xi be a sub-image from (24) to be scrambled, with 3 Applications to image encryption matching dimensions to assure multiplicability between An image is formed from MN samples arranged in a TXM and i , the encrypted image is obtained as two-dimensional array of M rows and N columns such as YTXi= M i . The decryption key is given by a photo, an image formed of the temperature of a integrated circuit, x-ray emission from a distant galaxy, a n k TAAA−1 =( − 1) p nt− k t t−1− t − 1 ⋅⋅⋅ n1− k 1 (27) satellite map from Google Earth. M t t−1 1 In imaging terminology, each sample of the image is ni called a pixel. Each pixel is attributed a value called where ni is such that Ai = ± I, i = 1,2,..., t and p grayscale ranging from 0 to 255, where 0 is black, 255 is represents the number of skew-unipotent matrices white, and the intermediate values are shades of gray. applied in (26). With (27), the original image is For the purpose of image encryption, we apply a series −1 of encryption key matrices to mask an image via matrix recovered from Yi via XTYi= M i . It is also ready to multiplications. This will alter the gray level of each be seen that the decryption process only involves matrix

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multiplication with additions and multiplications between integers. Therefore, lossless image encryption/decryption is guaranteed, see Fig. 1 for an example. The encryption key consists of three 5 by 5 unipotent matrices. Interestingly enough, functional k-potent matrix can add more complexity to the encryption scheme, proposed as follows. Consider the functional k-potent matrix, which is also an extension of (13),

m k i AIA= + ∑βi (28) i=1

where 0

Step 1. Pre-multiply Y by Ai−1 , respectively, to get i 0 ZAXi = , i=1,2,..., m , with AI= .

k− m k Step 2. Pre-multiply Zm by A to get WAX= .

m Step 3. XWZ= −∑βi i due to (28). i=1 Another image encryption with the above proposed method is shown in Fig.2.

In this paper, we studied the eigen-spaces of various (b) k-potent matrices, including Nilpotent, periodic, Figure 1. (a) original picture of mathematicians; involutary, and skew-periodic matrices. Extensions are (b) encrypted image of selected faces (courtesy of made to more general functional k-potent matrices. An MatLab image processing toolbox) immediate application of the results is seen in digital image encryption. The methodology proposed in this paper can also be extended to other functional matrices satisfying special constraints, similar to the ones for the k-potent matrices, and such constraints have imprints on the eigen-space structures of the matrices.

(a)

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MPEG video data efficiently, Proc. ACM Multimedia, 219-229, 1996. [13] S. U. Shin, K.S. Kim, and K. H. Rhee, A secrete scheme for MPEG video data using the joint of compression and encryption, Proc. Inform. Security Workshop (ISW ’99), v. 1729 (lecture notes on computer science), 191-201, 1999. [14] Wenjun Zeng and Shawmin Lei, Efficient frequency domain selective scrambling of digital video, IEEE Trans. Multimedia, v. 5, no.1, 118-129, 2003. [15] P. P. Dang and P. M. Chau, Image encryption for secure internet multimedia applications, IEEE Trans. Consum. Electron., v. 46, no.3, 395-403, 2000. [16] R. P. Elias, O. V. Villegas, M. L. Sanchez, and V. G. C. Sanchez, Automatic Inspection Using Edge (b) Preserving Wavelet Lossy Image Coding by Means of Modified SPIHT, WSEAS Trans. Signal Process., v. 2, Figure 2. (a) original fingerprint image; (b) scrambled no. 11, 1515-1522, 2006. image of selected areas of the fingerprint (courtesy of [17] B. Ontiveros, I. Soto, and R. Carrasco, A New MatLab image processing toolbox) Cryptography Algorithm Using Cab Curves and LDPC for Wireless Communication Systems, WSEAS Trans. Math., v. 6, no.2, 422-424, 2007. References: [18] J. Scharinger, Fast encryption of image data using [1] S. H. Friedberg, A. J. Insel, L.E. Spence, Linear chaotic Kolmogorov flows, Journal of Electron. Algebra, 4th Ed., Prentice Hall, Upper Saddle River, NJ, Imaging, v. 7, no. 2, 318-325, 1998. 2002. [19] J.C. Yen and J. I. Guo, A new chaotic key-based [2] G. Strang, Introduction to Linear Algebra, 3rd Ed., design for image encryption and decryption, Proc. IEEE Wellesley-Cambridge Press, MA, 2003. Int. Conference Circuits and Systems, v.4, 49-52, 2000. [3] G. H. Golub and C. F. Van Loan, Matrix [20] H. Zhang, X. F. Wang, Z. H. Li, D. H. Liu, and Y. Computations, 2nd Ed., The Johns Hopkins University C. Lin, A new image encryption algorithm based on Press, Baltimore, Maryland, 1989. chaos system, Proc. IEEE Intern. Conference Robotics, [4] G. W. Stewart, Matrix Algorithms, Volume II: IS and Signal Process, 778-782, 2003. Eigensystems, SIAM, Philadelphia, PA, 2001. [21] J. Fridrich, Symmetric ciphers based on two- [5] J. H. Wilkinson, The Algebraic Eigenvalue Problem, dimensional chaotic maps, Int. Journal Bifurcation & Oxford Science Publications, NY, 1965. Chaos, v. 8, no.6, 1259-1284, 1998. [6] J. Rimas, Computing of Positive Integer Powers for a [22] G. Jakimoski and L. Kocarev, Analysis of some Special Kind of Matrices, WSEAS Trans. Math., v. 6, recently proposed chaos-based encryption algorithms, no.2, 289-295, 2007. Physics Letters A, v. 291, no. 6, 381-384, 2001. [7] S. L. Steinberg, J. P. Zingano, and P. R. Zingano, On [23] S. Li, X. Mou, Y. Cai, Z. Ji, and J. Zhang, On the nilpotent singular systems, J. Comput. Appl. Math. 137, security of a chaotic encryption scheme: problems with 97-107, 2001. computerized chaos in finite computing , [8] J. K. Baksalary and O. M. Baksalary, Idempotency of Computer Physics Communications, v. 153, no. 1, 52- linear combinations of two idempotent matrices, Linear 58, 2003. Algebra Appl. 321, 3-7, 2000. [24] A. Kulkarni and S. Mccaslin, Knowledge Discovery [9] J. K. Baksalary, O. M. Baksalary, and G. P. H. Styan, from Satellite Images, WSEAS Trans. Signal Process., v. Idempotency of linear combinations of an idempotent 2, no. 11, 1523-1530, 2006. matrix and a tripotent matrix, Linear Algebra Appl. 354, [25] Y. Wu and A. C. Vosler, An Anti-Symmetric Key 21-34, 2002. Algorithm for Signal Encryption, Proceedings of the [10] O. M. Baksalary, Idempotency of linear WSEAS Inter. Conf. on Signal Process., pp 140-145, combinations of three idempotent matrices, two of which 2007. are disjoint, Linear Algebra Appl. 388, 67-78, 2004. [11] J. K. Baksalary, O. M. Baksalary, and H. Özdemir, A note on linear combinations of commuting tripotent matrices, Linear Algebra Appl. 388, 45-51, 2004. [12] L. Tang, Methods for encrypting and decrypting

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