Faster Image Encryption: A Semi-Tensor Product Approach

Transcription

1 207 Internatonal Conference on Computer Scence and Applcaton Engneerng (CSAE 207) ISBN: Faster Image Encrypton: A Sem-Tensor Product Approach Shpng Ye, Jnmng Wang *, Zhenyu Xu and Chaoxang Chen Zhejang Shuren Unversty, 3005 Hangzhou, Chna ABSTRACT Due to the large-scale of the mages, they suffer from a long tme to fnsh the process of encrypton and decrypton. In ths paper, we propose a novel mage-encrypton algorthm to mprove the performance of processng tme based on the sem-tensor product (STP). The STP s a novel matrx product that works by extendng the conventonal matrx product n cases of unequal dmensons. In paper, we construct a small reversble key matrx by usng the Kronecker product. Then, we use ths key matrx to change the values of pxels n the orgnal mage by applyng the sem-tensor product. As a result, the dmensons of the orgnal mage are much larger than the dmensons of the key matrx, and the amount of data that s calculated durng the encrypton and decrypton process s effectvely reduced. Meanwhle, the proposed algorthm can be used to encrypt and decrypt mages of dfferent szes. Experments were carred out usng mages of varous szes. The expermental results were compared wth prevous methods and the proposed method outperformed the others n terms of securty and processng tme. INTRODUCTION Wth the development of multmeda communcaton technology, dgtal data transmsson through wred and wreless networks has ncreased sgnfcantly. Such data nclude audo, vdeo, mages, and fles, and they have prompted a surge n traffc volume and network nfrastructure n recent years. Thus, securty problems durng transmsson and storage have also ncreased, and due to the large-scale of the mages, they suffer from a long tme to fnsh the process of encrypton and decrypton. Consequently, t needs a tradeoff between encrypton and computaton durng ts transmsson and storage. Many researchers have focused on the securty of mage-data transmsson, and several mage-encrypton algorthms have been proposed based on matrx transformatons [, 2], chaos theory [4-8], and transformaton domans [9-2]. Meanwhle, the Kronecker product has been used to mprove the transformaton performance [3-6]. The Kronecker product s a product of one element to a whole matrx, whch can quckly construct a large-dmenson matrx by some small-dmenson matrces. Such as, a transform matrx was generated usng the Kronecker product of two 6 6 matrces, whch s helpful for fast computaton and effcent constructon. The sem-tensor product (STP) s the product of one element and a block from the other matrx, and t s a novel matrx product that works by extendng the conventonal matrx product n cases of unequal dmensons. In [7], the 72

2 STP was used to provde a Boolean network representaton n a lnear system for nonlnear feedback-shft regsters (NFSR), and ths research demonstrates that the STP approach s helpful n desgnng NFSR for stream cphers n cryptography. Thus, nspred by the successful applcaton of the Kronecker product and the STP, and to mprove the effcency of mage encrypton, we propose a new algorthm for mage encrypton. The proposed algorthm s less computatonally expensve durng the encrypton and decrypton process. Furthermore, t offers lossless decrypton and robustness, avods expandng the transmtted data, and needs less tme to process. The proposed algorthm uses a small encrypton matrx to encrypt the mage wth the large-scale mage matrx. Our proposal s advantageous for several reasons. Frst, the algorthm effectvely reduces data computaton durng the encrypton and decrypton process. Consequently, the encrypton and decrypton operaton s more effcent, owng to the use of a small encrypton matrx to encrypt or decrypt a large-scale mage matrx. Second, the proposed algorthm can be appled for the encrypton of mages of dfferent szes for secure transmsson and storage. Fnally, our proposal ntroduces several key parameters, such as the dmensons of the encrypton matrx, the dfferent elements of the encrypton matrx, and the number of encrypton teratons. These parameters mprove the robustness of the encrypted mage. To verfy the accuracy and effectveness of the proposed mage-encrypton algorthm, we tested t usng DICOM medcal mages, and gray-scale mages. After usng the STP algorthm to encrypt the mages wth an 8 8 encrypton matrx, we compared the hstogram of the orgnal wth the encrypted mage, and we studed the correlaton of adjacent pxels, nformaton entropy, tme consumed, and ts susceptblty to dfferental attacks. The results of these tests demonstrate that the algorthm proposed here offers secure nformaton protecton and satsfes the processng tme requred by standard applcatons. Moreover, several encrypton keys are generated durng the mageencrypton process, and ths greatly enhances the space of the keys and mproves the robustness of the encrypted mage. The remander of ths paper s organzed as follows. In Secton II, the prelmnares of the Kronecker product and the STP are ntroduced. In Secton III, we descrbe the proposed algorthm for mage encrypton and decrypton. In Secton IV, we present the expermental results. Secton V offers a dscusson of these results. Fnally, Secton VI concludes the paper and dscusses future research. PRELIMINARIES In ths secton, we dscuss some necessary prelmnares to the Kronecker product and the STP, whch are adopted n the subsequent sectons. The Kronecker product s a product of one element to a whole matrx, whch can quckly construct a large-dmenson matrx by some small-dmenson matrces. The STP of matrces was ntroduced by Cheng [8]. It s a generalzaton of the conventonal matrx product. 73

3 Kronecker Product Defnton : For the Kronecker product of matrces (also called the tensor product ), f A ( aj ) M m n, B ( bj ) M p q, then the Kronecker product s defned as follows [9]: a B a B a B 2 n A B, () a B a B a B m m2 mn where s the Kronecker product for matrces, ab j ( 2,,,m, j 2,,,n) s tself a p q matrx, and A B s the set of mp nq matrces. Hence, the followng theorem can be proved: Theorem : Suppose C ( Ckl ) M m n,,2, K. Accordng to the defnton of the Kronecker product, we have K K C C, (2) K f m m, K K n n, then C Mm n. 2 2 Proof: By Def., when K =2, C C C holds, where C s a matrx of 2 2 m n. 2 k k k k Let C C C2 C k, 2 k K, where C s a m n matrx. k By Def., there exsts C k = (C Ck) Ck, where C s a k k K m n matrx. Then, C Mm n holds (when k K). The theorem s thus proved. Corollary If all C are nvertble, such that, 2, K, M m n, then Sem-tensor Product C K K ( C ) C M. (3) mn In [8], the STP s presented as an extenson of the conventonal matrx product. For a conventonal matrx product, f the column number of A (Col(A)) s equal to the lne number of B (Row(B)), then matrces A and B are multplcatve. The STP of matrces, on the other hand, extends the conventonal matrx product n cases of unequal dmensons. That s, Col(A) Row(B). Frst, we consder the conventonal matrx product. Let U and V be m- and n-dmensonal vector spaces, respectvely. Assume F L(U V, ). That s, F s a blnear mappng from U V to. Denote by { u,,u m } and { v,,v n } the bases of U and V, respectvely. We call S = ( s j ) the structure matrx of F, where sj F( u,v j ), (4) and where 2,,,m, j 2,,,n. m If we let X x uu, otherwse wrtten as X ( x,, x ) T m U, and f n Y yvv, otherwse wrtten as Y ( y,, y ) T m V, then 74

4 F( X,Y) T X T SY (5) Followng ths, and denotng the rows of S by S, m, and S, we can alternatvely calculate F n two steps: 2 m Step : Calculate xs, xs 2,, xms and take ther sum. m Step 2: Multply xs by Y (whch s a standard nner product). It s easy to check whether ths algorthm produces the same result. In the frst step, t seems that we have ( S n S ) X. Ths calculaton motvates the STP, defned as follows. Defnton 2: Let T be an np -dmensonal row vector and let X be a p - dmensonal column vector. Splt T nto p equal blocks, named T, p, T, whch are n matrces. Defne a left sem-tensor product, denoted by, as follows: T X p T x n. (6) Defnton 3: Let X ( x,, x s ) be a row vector, and let Y be a column vector Y ( y,, y ) T t. Case : If t s a factor of s,.e., f s t n, then the n - dmensonal row vector defned as X Y t k X yk n (7) k s called the left sem-tensor nner product of X and Y, where ( t n X X,, X ), X,,,t. Case 2: If s s a factor of t,.e., f t s n, then the n - dmensonal column vector defned by X t k k n Y : x Y (8) k s called the left sem-tensor nner product of X and Y, where Y ((Y ) T,,( Y t ) T ) T n, X,,,t. mn pq Defnton 4: Let A and B. If ether n s a factor of p.e., f nt p (denoted as A t B) or f p s a factor of n.e., n pt (denoted as A t B) then the (left) STP of A and B can be denoted by C { C j } A B, as follows: C conssts of m q blocks, and each block s defned as j C A B,,, m, j,, q, j where A s the -th row of A, and B j s the j-th column of B. In comparng the conventonal matrx product wth the Kronecker product and the STP of matrces, t s easy to see that there are sgnfcant dfferences between them. For the conventonal matrx product, the product s element-toelement. The Kronecker product s a product of one element to a whole matrx, and the STP s the product of one element and a block from the other matrx. The STP of matrces s a generalzaton of the conventonal matrx product. That mn pq s, f A, B, and n p, then A B AB. 75

5 Consequently, when the conventonal matrx product s extended to the STP, almost all of ts propertes are nevertheless mantaned. Ths s a sgnfcant advantage to usng the STP [8]. We provde the followng theorems and corollares. mn Defnton 5: Suppose A( a j ). If A s a square matrx ( m n), and A exsts, then by Def. 4 and notcng that have mn mn mn mtp ( A B) A B (see [8]), we A A I. (9) p p p p Defnton 6: Suppose A( a j ), B( b j ), and F ( f j ). The STP satsfes the assocatve law: ( A B) F A ( B F). (0) The STP has a wde range of applcatons: n nonlnear systems control for structural analyss and control of Boolean networks, n systems bology as a soluton to Morgan s Problem, etc. [8]. Furthermore, there are also applcatons for the STP n the feld of data encrypton and decrypton. To change the elements n A, whch are defned n Def. 4, we can use the conventonal matrx product or the STP. By comparng the two dfferent matrx products, we can show that less computaton s needed for the STP than the conventonal matrx product. Ths encourages us to adopt the STP for mage encrypton and decrypton on systems wth constraned resources. ALGORITHM DESCRIPTION In ths secton, we frst ntroduce the encrypton algorthm, whch uses a small key matrx to encrypt an mage wth a large-scale mage matrx. We then explan the decrypton algorthm by the nverse key matrx. Encrypton Algorthm Consder a dgtal mage PM Nwth M pxels n the row and N pxels n the column, and let P, j( {,2,, M}, j {,2,, N} ) be any pxel from the mage. If there exsts an nverse rectangular matrx Sn n, then take N nt and use Def. 4 to derve the STP. Thus, the followng equaton holds: QM N PM N Sn n, () where QM Ns the encrypted mage matrx after changng the value from each pxel n PM N. Multplcaton then expands Eq. (), such that the followng holds:,,2, n 2, 2,2 2, n QM N PM N Sn n PM nt Sn n, (2) M, M,2 M, n n k, where P s sk j S k, n n, and P can be determned as,,, j k k. j k, P { p, p,, p } P, (3),( k ) t,( k ) t2, kt 76

6 k, where P s the k-th block of the -th lne, and each block has t pxels, M, k n. Gven the characterstcs of the mage, each pxel value can be denoted wth a sngle byte for both gray-scale and color mages, whch means that the value of pxel ranges from 0 to 255. If the mage s a medcal mage whether magnetc resonance (MR), computed radology (CR), or those classfed by DICOM as other (OT) the pxel value can be expressed n 6 bts, 2 bts, and 0 bts, respectvely. In ths case, we modfy Eq. (2) to obtan the fnal encrypton results wth the followng matrx: mod(,, Mmax B ) mod(, n, Mmax B ) QM N, (4) mod( M,, Mmax B ) mod( M, n, Mmax B ) where M max B denotes the maxmum value per pxel, derved as follows: 256, f bt-depth 8 024, f bt-depth 0 M max B. 4096, f bt-depth , f bt-depth 6 By Theorem, we can assume that nvertble matrces for Sl lexst such that the nverse rectangular matrx Sn n can be quckly constructed as follows: k Snn Sl l (5) k where l n. Ths s helpful for an effcent constructon and for speedng up the encrypton process. To mprove and contrast the encrypton securty of the encrypted mage, we encrypt the orgnal mage K rounds, as follows: Q P S S S P S. (6) M N M N nn nn nn M N nn Above all, ths algorthm encrypts dgtal mages usng the encrypton matrx Sn n at much smaller dmensons than the mage matrx PM N. Sgnfcantly less data s needed, and the computatonal complexty of the encrypton algorthm s reduced. Decrypton Algorthm To decrypt the encrypted mage QM N, the frst step nvolves ensurng that S n n exsts. Notng that ( A B) A B (see [8]), we can use Eq. () such that PM N QM N Sn n, and from PM N ( PM N Sn n) S nn, by Defnton 5, we can obtan PM N PM N ( Sn n S nn ). Usng Defnton 5, we have K 77

7 PM N PM N In n PM N. (7) Thus, we obtan the decrypted mage. By Theorem, Corollary, and Eq. (5), then, the nverse matrx S n n of Sn n can be obtaned quckly wth the followng equaton: k ( ) Snn Sl l, (8) k where l n, and ( S l ) l s the nverse matrx of S l l, 2,,,k. From the above descrpton, t s clear that once an nverse matrx S n n wth the approprate dmensons s constructed, then the dgtal mage can be encrypted and decrypted. After generatng the nverse matrx S n n by Eq. (8), we can use the prncple n Eq. () and the matrx calculaton n Eq. (2). Next, the characterstcs of the mage pxel values can be combned. Referrng to Eq. (4) durng the encrypton process, we can obtan the decrypted mage PM N as follows: mod(,, Mmax B ) mod(, n, Mmax B ) PM N, (9) mod( M,, Mmax B ) mod( M, n, Mmax B ) where, s, n k,, j Q s k k, j S k j n n, k, k, Q s defned as P, and M, k n. Thus, the process of encryptng and decryptng the orgnal mage are completed wth an encrypton matrx of smaller dmensons than the orgnal mage. There are two mportant and unque features to the proposed algorthm. Frst, the amount of data that s calculated durng the encrypton and decrypton process s effectvely reduced by usng data from a smaller matrx to encrypt and decrypt a large matrx mage. Second, the proposed algorthm can be used to encrypt and decrypt mages of dfferent szes. As explaned above, f we are gven an n-dmensonal matrx Sn n, then n s sutable for the number of columns n the orgnal mage PM N, provded that N n t, where t s a postve nteger. Thus, f the number of columns and rows PM' N' changes, such that N ' n t', where t' s also a postve nteger, the algorthm proposed here proceeds n the same manner, wthout any need for modfcatons. Constructng the Key Matrx Gven ther senstvty to the ntal condtons and control parameters, chaotc maps had been wdely used n data encrypton [4-8]. However, there are dsadvantages to chaotc maps, ncludng lmted precson, perod loops, and dffculty measurng the cycle. The decmal representatons of rratonal numbers are nether perodc nor termnatng, and that the next uncalculated decmal unt mght be any number between 0 and 9. In such cases, t s sad to be sutably chaotc [20]. Thus, we adopt rratonal numbers when constructng the key matrx. For = , let I ( d, L) [ ( d, L ), ( d2, L2 ), ( d, L ), ] be a set of ntegers from the rratonal number I, where L s the length of a segment chosen from I, whle 78

8 the ntal dgt of the segment s d dgts away from the decmal pont. Usng the defnton of I ( dl, ), we can obtan the followng: ( dl, ) [ (3,4), (5,3), (8,2), ], (20) where (3,4) =592, (5,3) =926, and (8,2) =53, etc. The parameters L and d can be determned by the sender and the recever together, where d, L, and [, ). From the above descrpton, let { u, v} ( d, L). The encrypton matrx S2 2 can be derved as follows: u S2 2 v uv, where S2 2s an nverse matrx. That s, det( S) 2 2. Usng Eq. (6), we can derve S4 4, S88, and, as follows: S S S u u u v ( uv ) uv u( uv ) v uv ( uv ) u( uv ), 2 2 v v( uv ) v( uv ) ( uv ) 44 and then we can obtan S88 by S2 2 S4 4 Therefore, we know that f we use dfferent { u, v} ( d, L), the elements S2 2, S4 4, and S88 wll change accordngly. Wth Eq. (9), we can quckly obtan S as follows: 88 uv u S2 2 v, S S S ( uv ) u( uv ) u( uv ) u v( uv ) ( uv ) uv u v( uv ) uv ( uv ) u, 2 v v v 44 and then we can obtan S 88 by S2 2 S4 4 Thus, we construct the encrypton matrx Sn n usng the above procedure. Indeed, there are other ways to construct Sn n, and the reader s nvted to generate the matrx usng any such procedure. Durng the data transmsson, there s no need to transmt the encrypton matrx or any parameters. Rather, the recever can merely use the rule of common agreement between the sender and recever to construct the decrypton matrx. EXPERIMENTAL RESULTS To verfy the proposed mage-encrypton method, gray-scale mages of dfferent dmensons and bt-depths were used, ncludng a CR chest mage 79

9 (hereafter CR-Chest ) (dmensons: ) obtaned from [2], Lena (dmensons: 52 52), Peppers (dmensons: ), Mandrll (dmensons: ), and Boats (dmensons: ). The CR-Chest has a bt-depth of 0, whereas the others have a bt-depth of 8. In ths paper, we conducted an expermental analyss of the proposed encrypton algorthm usng the followng gray-scale mages: CR-Chest, Lena, Peppers, Mandrll, and Boats. All of these mages were encrypted wth the matrx S88 and decrypted by. From Eq. (7), let K=4. Then, S 88 Q P S S S S P S. M N M N M N 88 Once the mages are encrypted, they can be decrypted as follows: PM N QM N S8 8 S8 8 S8 8 S8 8 QM N S8 8. The experments were conducted as follows: Encrypton procedure: Input: the n-dmenson of the key matrx; the source mage PM N; the number of encrypton rounds K; and the varables L and d for the key matrx. Output: Returns encrypted mage QM N. Step. The elements u and v from the set ( dl, ) are obtaned wth Eq. (20), and the matrx S2 2 s constructed. Step 2. The n n encrypton matrx Sn n s obtaned by Eq. (5). Step 3 The varables of t for the block of the source mage PM N are calculated by t N n, whch s defned n Def. 4. Step 4: The source mage PM Ns encrypted nto the cphered mage QM Nn Eqs. (2) and (4). If the current round s not the fnal round of encrypton, Step 4 s repeated. Otherwse, the encrypton process s complete. Decrypton procedure: Input: The varables L and d for the key matrx; the n-dmenson of the key matrx; the encrypted mage QM N; and the number of encrypton rounds K. Output: Returns decrypted mage PM N. Step. The n n encrypton matrx S n n s obtaned by Eqs. (20) and (8). Step 2. The varables of t for the block of the encrypted mage QM Nare calculated by t N n, whch s defned n Def. 4. Step 3: The encrypted mage QM Ns decrypted nto the decrypton QM Nn Eq. (7) and (9). If the current round s not the fnal round of encrypton, repeat Step 3 agan. Otherwse, the encrypton process s complete. The expermental results are provded n Fgs. 3, where the elements of the encrypton and decrypton matrces were obtaned as follows: u (6,2) =26, v (0,2) =58, where d =6, L =2, d 2 =0, and L 2 =2. In Fgures. -3, Frame (a) shows the orgnal mage. Frame (c) shows the encrypted mage from the orgnal n Frame (a) usng the key S88. Frame (e) shows the decrypted mage. Frames (b), (d), and (f) are the hstograms for (a), (c), and (e), respectvely. 4 80

10 Fgure. Comparson of (a) the Orgnal CR-Chest Image wth (c) the Encrypted Image, and (e) the Decrypted Image, and ther Correspondng Hstograms (Dmensons: , Bt-depth: 0). Fgure 2. Comparson of (a) the Orgnal Lena Image wth (c) the Encrypted Image, and (e) the Decrypted Image, and ther Correspondng Hstograms (Dmensons: 52 52, Bt-depth: 8). Fgure 3. Comparson of (a) the Orgnal Peppers Image wth (c) the Encrypted Image, and (e) the Decrypted Image, and ther Correspondng Hstograms (Dmensons: , Bt-depth: 8). 8

11 EXPERIMENTAL ANALYSIS Statstcal Analyss HISTOGRAM FOR ENCRYPTED IMAGE An mage hstogram llustrates how the pxels n an mage are dstrbuted, and ths s done by graphng the number of pxels n terms of ther ntensty. We calculated and analyzed the hstograms for the encrypted and orgnal mages to reveal how ther respectve content dffers. The hstograms are shown n Fgures. 3. In Frames (b) and (d) for these fgures, t s clear that the hstograms of the encrypted mage are farly unform, and that they dffer sgnfcantly from the hstograms of the orgnal mage. Meanwhle, we employ varances of hstograms to evaluate the unformty of the encrypted mages. The varance of hstograms s shown as follows: n n 2 Var( Z) ( ) 2 z z j, 2 n j where n denotes the gray values n the hstogram, and z and z j are the number of pxels wth gray values equal to and j, respectvely. As shown n TABLE I, lower varance to the encrypted mages ndcates hgher unformty, and the decrypted mages have the same varance as the orgnal, ndcatng that there s no dstorton between the orgnal and the decrypted mage. TABLE I. VARIANCES OF THE HISTOGRAMS. mage (Dmenson, Bt-depth) orgnal encrypted decrypted CR-Chest ( , 0) 5.406e e e+7 Lena (52 52, 8) 7.475e+5.823e e+5 Peppers ( , 8).3500e e e+6 Mandrll ( , 8) e e e+6 Boats ( , 8) e e e+6 All the orgnal mages of dfferng szes were encrypted wth the same encrypton matrx S88. Consequently, the algorthm proposed here s compatble wth dfferent-szed mages nsofar as the dmensons of Sn n are sutable for the number of columns n the orgnal mage. RELEVANCE OF ADJACENT ELEMENTS To test the correlaton of pxels (vz., vertcal, horzontal, and dagonal), we randomly selected 2,000 pars of adjacent pxels from both the orgnal and the encrypted mage, and calculated the correlaton coeffcents of pxels. TABLE II lsts the correlaton coeffcents calculated from the orgnal mages and ther correspondng mages cphered wth the proposed algorthm. From TABLE II, we can see that the correlaton coeffcents are always hgh. Whereas they are very close to n the plan-text mages, they are sgnfcantly reduced n the cphered mages. It s clear that our algorthm effectvely reduces the correlaton between adjacent pxels. 82

12 TABLE II. CORRELATION COEFFICIENTS OF CR-CHEST, LENA, PEPPERS, MANDRILL, BOATS AND THEIR ENCRYPTED IMAGES IN THE PROPOSED ALGORITHM. Orgnal CR-Chest Orgnal Lena Orgnal Peppers Orgnal Mandrll Orgnal Boats Vertcal Horzontal Dagonal Encrypted CR-Chest Encrypted Lena Encrypted Peppers Encrypted Mandrll Encrypted Boats Vertcal Horzontal Dagonal The correlaton comparson tests wth other encrypton algorthms are lsted n TABLE III. From TABLE III, we can see that the correlaton coeffcents from the proposed algorthm are lower than other schemes, except HC-DNA. These results confrm that our proposed encrypton s resstant to statstcal attacks amed at dscoverng the correlatons between adjacent pxels. TABLE III. COMPARING THE CORRELATION COEFFICIENTS OF CR-CHEST, LENA, PEPPERS, MANDRILL, AND BOATS. Image Ref. [4] Ref. [5] Ref. [8] AES HC- [22] DNA[24] Proposed CR-Chest Vertcal Horzontal Dagonal Lena Vertcal Horzontal Dagonal Peppers Vertcal Horzontal Dagonal Mandrll Vertcal Horzontal Dagonal Boats Vertcal Horzontal Dagonal

13 Informaton Entropy Analyss Informaton entropy s thought to be one of the most mportant features n randomness [23]. The nformaton entropy Hm ( ) s calculated wth the followng formula: 2 n H( m) p( m ) log p( m ) (2) 0 2 TABLE IV. THE INFORMATION ENTROPY OF ORIGINAL AND ENCRYPTED IMAGE. CR-Chest Lena Peppers Mandrll Boats Orgnal Image Encrypted Image where m s the message, p( m )= M max B ( M max B s derved from Eq. (5)) represents the probablty of symbol m, and the entropy s expressed n bts. Suppose that a message s such that each symbol s encoded wth eght bts. For randomness, the entropy value should be eght, deally. However, the entropy value of the message s often less than eght, though t should nonetheless come close to ths deal. For messages where each symbol s encoded wth ten bts, we can reach the deal Hm= ( ) 0. Ths deal entropy was acheved wth the CR- Chest mage, for nstance. We used Eq. (2) to calculate the nformaton entropy for the encrypted mages. TABLE IV shows the entropy for the gray-scale values, whch all come close to the deal value. Ths ndcates that the proposed scheme has successfully hdden the nformaton randomly, such that the probablty of accdental nformaton leakage s very low. TABLE V provdes a comparson of the proposed method wth other algorthms n terms of the nformaton entropy. The values obtaned from the proposed scheme are much closer to the deal and other algorthms, and thus the nformaton hdden wth the proposed scheme s more random than t s wth other schemes. TABLE V. COMPARING THE INFORMATION ENTROPY OF CR-CHEST, LENA, PEPPERS, MANDRILL, AND BOATS. Image Bts/pxel Ref. [4] Ref. [5] Ref. [8] AES [22] HC- DNA[24] Proposed CR-Chest Lena Peppers Mandrll Boats Dfferental Attack As a general requrement for all mage-encrypton schemes, the encrypted mage should dffer consderably from ts orgnal form. Such dfferences can be measured by means of two crtera namely, the number of pxel change rate (NPCR) and the unfed average changng ntensty (UACI) [8]. 84

14 Let C (, j) and C 2 (, j) be the gray level of the pxels at the -th row and j-th column of the encrypted mages, respectvely before and after a pxel from the orgnal mage changes (sze: M N pxels). The NPCR for these two mages s defned as M N NPCR [ D(, j)] 00%, (22) M N j where D(, j) s defned as 0, f C(, j) C2(, j) D(, j)., f C(, j) C2(, j) The UACI s calculated as follows: M N C(, j) C2(, j) UACI [ ] 00%, (23) M N M j max B where M max B s derved from Eq. (4). We measured the NPCR and the UACI for the orgnal and encrypted mages. The results are shown n TABLE VI. The results show that our algorthm s robust aganst dfferental attacks. TABLE VI. NPCR AND UACI OF THE ENCRYPTED IMAGES. Image Ref. [4] Ref. [5] Ref. [8] AES [22] Proposed NPCR(%) CR-Chest Lena Peppers Mandrll Boats UACI(%) CR-Chest Lena Peppers Mandrll Boats Key Space and Senstvty Analyss KEY SPACE The key space should be suffcently large to render brute-force attacks unfeasble. In the proposed algorthm, the keys consst of the followng: () the elements of the key matrx, (2) the n-dmenson of the key matrx, and (3) the K 64 number of encrypton rounds. Gven a precson of 2 and takng an M N - szed mage as an example f the same key matrx s used n dfferent encrypton rounds, the key space can be calculated as follows: 64 (2 ) n S n t, where n n denotes the elements of the key matrx, and t s the varable for the block of the source mage PM N, defned as t N n. If n =2, S = t 2. However, f we construct a larger key matrx, such as an 8 8 key matrx, then the key space S s t 2. If dfferent key matrces are used n dfferent encrypton rounds, the key space would be: 64 nn K S [(2 ) t], 85

15 where K s the number of encrypton rounds, meanng that the key space S s much larger than t was formerly. Thus, the proposed algorthm wth such a large key space s suffcent for relable and practcal use. KEY SENSITIVITY To test the key senstvty of the proposed algorthm, we agan used the NPCR. The key matrx S2 2 was constructed followng the detals provded n the secton Constructon of Key Matrx. Then, we constructed S88. The expermental results are provded n Fg. 4, where the parameters u and v range from 0 to 00, and the parameter dfference was u [0,] and v [0,]. Further, the encrypton rounds were, 2, and 4 tmes, and the same key matrx was used durng each encrypton round. Seen from the Fgure 4, the senstvty remans above 99% regardless of the parameters or encrypton rounds. Ths shows that our algorthm has a hgh level of senstvty to the ntal key. Ths guarantees the securty of the proposed algorthm aganst brute-force attacks to some extent. Fgure 4. Key-senstvty Test for the Proposed Algorthm: Boats mage (Dmensons: , Bt-depth: 8). Speed and Complexty Analyses To evaluate the runnng speed, we performed tests on the encrypton speed of the proposed algorthm, comparng t wth [5], [7], and the AES. The proposed algorthm was mplemented n Vsual C on an Intel laptop, runnng Wndows 8. The CPU tmes requred for the encrypton and decrypton processes are shown n TABLE VII. In all cases, the processng tme was merely a few seconds, but the processng tme nevertheless ncreased as the dmensons of the mage grew. These results show that the proposed algorthm runs faster than the AES algorthm, [5], and [7]. To analyze the executon tme of the encrypton process, we studed the tme consumed durng the multplcatons and addtons. Let the orgnal mage be PM N, and let the key matrx be Sn n, where n N. 86

16 2 For K encrypton rounds, the proposed algorthm requres O( Mn K ) teratons for multplcaton and O( Mn( n ) K) teratons for addton, whereas for the 2 conventonal matrx product, the teratons are O( MN K ) and O( MN( N ) K), respectvely. Therefore, the proposed algorthm s relatvely fast. TABLE VII. TIME-CONSUMING COMPARISONS. Image AES [22] (ms) Ref. [5] (ms) Ref. [7] (ms) Proposed (ms) CR-Chest ( ) Lena (52 52) Peppers ( ) Mandrll( ) Boats ( ) Test on another Images To further test the effectveness and performance of the proposed algorthm, we used another 33 mages: medcal mages obtaned from [2] and other mages obtaned from [25]. As shown n TABLE, the column Corr. s the average correlaton of the encrypted mage n three drectons; the column Ds. s the dstorton rato between the orgnal and the encrypted mage; and all of the expermental results were obtaned after four encrypton rounds wth S88. From TABLE VIII, we can see that the proposed algorthm performs well and that t s effectve for mage encrypton. TABLE VIII. MORE TEST ON THE PROPOSED ALGORITHM. Orgnal mage NPCR UACI Tme Corr. Entropy (Dmenson, Bt-depth) (%) (%) (ms) Ds. CT-abdo (52 52, 8) OT-a7 (52 52, 8) OT-colon (52 52, 8) OT-hp (52 52, 8) MR-an2 ( , 2) MR-an (52 52, 2) CT-bran (52 52, 6) CT-ort (52 52, 6) MR-head ( , 6) MR-knee ( , 6) nemacb ( , 8) nemacl ( , 8) nemacl2 ( , 8) CONCLUSIONS In ths paper, we proposed a small encrypton matrx to encrypt and decrypt mages, where the dmensons of the orgnal mage are larger than the key matrx. We carred out statstcal analyses, and we measured the processng tme, nformaton entropy, NPCR, and UACI of the proposed algorthm. The results from these experments demonstrate the securty and hgh-performance of the proposed mage-encrypton procedure. The proposed algorthm s unque n a number of ways. Frst, a small matrx s used to encrypt and decrypt mages from a relatvely large matrx. Thus, the 87

17 computatons of the data are effectvely reduced, and the operatonal effcency of encrypton process s enhanced. Second, ths algorthm s compatble wth mages of dfferent dmensons usng the same encrypton matrx. Thrd, the ntroducton of the key s smple: by varyng the elements of the key matrx Sn n, the dmenson n, and K encrypton rounds, a suffcently large key space can be generated, whch s useful for ncreasng the securty of the algorthm. Fnally, an nverse encrypton matrx Sn n and ts nverse matrx S n n can be constructed quckly usng the tensor product, whch s helpful for decreasng the tme consumed by the proposed algorthm. Thus, the algorthm s smple, effcent, and easly mplemented. In future work, we plan to analyze the securty of the proposed algorthm theoretcally, execute the securty experments of brute-force attack, knownplan-text attack, and chosen-cpher-text attack, etc. We also plan to study the STP n conjuncton wth chaotc dynamc symbols to facltate the encrypton of color mages. ACKNOWLEDGMENT The authors wsh to thank the anonymous revewers for ther valuable comments and for ther help n fndng errors. We apprecate ther assstance n mprovng the qualty and the clarty of the manuscrpt. Ths work was supported by the Scence and technology project of Zhejang Provnce, Chna (Grant No. 204C33058, 205C33074, 205C33083). REFERENCES. L. L. Zhao, Z. L. Fang, and Z. C. Gu Novel algorthm of dgtal mage scramblng and encrypton based on magc cube transformaton, Journal of Optoelectroncs Laser, 9(): J. George, S. Varma, and M. Chatterjee, 204. Color mage watermarkng usng DWT-SVD and Arnold transform, Inda Conference (INDICON), 204 Annual IEEE, Pune, Inda, December -3, D. M. Chen A feasble chaotc encrypton scheme for mage, Proc of Internatonal Workshop on Chaos-Fractals Theores and Applcaton, Shenyang, Chna, November 6-8, Abdo, S. Lan, I. A. Ismal, and M. Amn, etc A cryptosystem based on elementary cellular automata, Communcatons n Nonlnear Scence and Numercal Smulaton, 8(): Kanso, and M. A. Ghebleh A novel mage encrypton algorthm based on a 3D chaotc map, Communcatons n Nonlnear Scence and Numercal Smulaton, 7(7): H. J. L, and X. Y. Wang. 20. Color mage encrypton usng spatal bt-level permutaton and hgh-dmenson chaotc system, Optcs Communcatons, 284: Y. Q. Zhang, and X. Y. Wang A symmetrc mage encrypton algorthm based on mxed lnear nonlnear coupled map lattce, Informaton Scences, 273: X. Wang, and D. Luan A novel mage encrypton algorthm usng chaos and reversble cellular automata, Communcatons n Nonlnear Scence and Numercal Smulaton, 8(): K. Yadav, S. Vashsth, H. Sngh, and K. Sngh A phase-mage watermarkng scheme n gyrator doman usng devl's vortex Fresnel lens as a phase mask, Optcs Communcatons, 344():

18 0. Mehra, and N. K. Nshchal Optcal asymmetrc watermarkng usng modfed wavelet fuson and dffractve magng, Optcs and Lasers n Engneerng, 68: T. Banch, A. Pva, and M. Barn On the mplementaton of the dscrete Fourer transform n the encrypted doman, IEEE Trans. Inf. Forenscs Securty, 4(): K. Kab, B. J. Saha, and C. Pradhan Enhanced dgtal watermarkng scheme usng fractal mages n wavelets, Computng, Communcaton and Networkng Technologes (ICCCNT), 204 Internatonal Conference on, IEEE, Hefe, Chna, July -3, M. H. Lee, B. S. Rajan, and J. Y. Park A generalzed reverse jacket transform, Crcuts and Systems II: Analog and Dgtal Sgnal Processng, IEEE Transactons on, 48(7): M. H. Lee, X. D. Zhang, and X. Jang Fast parametrc recprocal-orthogonal jacket transforms, EURASIP Journal on Advances n Sgnal Processng, : H. B. Kekre, T. Sarode, and S. Natu Performance analyss of watermarkng usng Kronecker product of orthogonal transforms and wavelet transforms, Internatonal Journal of Scentfc & Engneerng Research, 5(2): S. Bouguezel A recprocal-orthogonal parametrc transform and ts fast algorthm, Sgnal Processng Letters, IEEE, 9(): Zhong and D. Ln On maxmum length nonlnear feedback shft regsters usng a Boolean network approach, Control Conference (CCC), rd Chnese. IEEE, Nanjng, Chna, July 28-30, D. Z. Chen, H. S. Q, and Z. Q. L. 20. Analyss and Control of Boolean Networks: A Semtensor Product Approach. Sprnger, London, pp N. Vartak On an applcaton of Kronecker product of matrces to statstcal desgns, The Annals of Mathematcal Statstcs, 26(3): Prasad, and K. L. Sudha. 20. Chaos mage encrypton usng pxel shufflng, Computer Scence & Informaton Technology, 4(): S. Barr Medcal mage samples. [Onlne]. Avalable: Natonal Insttute of Standards and Technology FIPS 97: Advanced Encrypton Standard [Onlne]. Avalable: csrc.nst.gov/publcatons/fps/fps97/fps-97.pdf. 23. Awad, and D. Awad Effcent mage chaotc encrypton algorthm wth no propagaton error, ETRI journal, 32(5): Zhan K, We D, and Sh J, et al Cross-utlzng hyperchaotc and DNA sequences for mage encrypton, Journal of Electronc Imagng, 26(): Test Images. [Onlne]. Avalable: 89