Image Encryption Using Chaotic Signal and Max Heap Tree

Transcription

1 Image Encrypton Usng Chaotc Sgnal and Max Heap Tree Farborz Mahmoud 1, Rasul Enayatfar 2, and Mohsen Mrzashaer 1 1 Electrcal and Computer Engneerng Department, Islamc Azad Unversty, Qazvn Branch, Iran {Mahmoud,Mrzashaer}@QazvnIAU.ac.r 2 Department of Electrcal and Computer Engneerng, Islamc Azad Unversty, Froozkooh Branch, Iran r.enayatfar@aufb.ac.r Abstract. In ths paper, a new method s proposed for mage encrypton usng chaotc sgnals and Max-Heap tree. In ths method, Max-Heap tree s utlzed for further complexty of the encrypton algorthm, hgher securty and changng the amount of gray scale of each pxel of the orgnal mage. Studyng the obtaned results of the performed experments, hgh resstance of the proposed method aganst brute-force and statstcal nvasons s obvously llustrated. Also, the obtaned entropy of the method whch s about s very close to the deal amount of 8. Keywords: Image Encrypton, Max-Heap Tree, Chaotc Sgnal. 1 Introducton Together wth the rapd rate of multmeda products and vast dstrbuton of dgtal products on nternet, protecton of dgtal nformaton from beng coped, llegal dstrbuton s of great mportance each day. To reach ths goal, varous algorthms have been proposed for mage encrypton [1-4]. Recently, due to the wdespread use of chaotc sgnals n dfferent areas, a consderable number of researchers have focused on these sgnals for mage encrypton [5-9]. One of the most mportant advantages of chaotc sgnals s ther senstvty to the ntal condtons and also ther nose-lke behavor whle beng certan. In [5], the method of movng pxels s proposed for mage encrypton. In [6], an algorthm s proposed whch s based on a key for the encrypton of the mage (CKBA 2 ). In ths method a chaotc sgnal s utlzed to determne the amount of gray scale of the pxels. Later researches have shown that the aforesad method s not secure enough [7]. In ths paper, a new method s proposed for mage encrypton usng chaotc sgnals and Max-Heap tree to make the encrypton algorthm more complex and secure, n whch the mplementaton of Max-Heap tree has caused that, even when the ntal value of the chaotc functon s revealed, the real amount of gray scale of each pxel cannot be accessed. In the followng secton, Max-Heap trees are prmarly ntroduced n bref, and then the proposed method s analyzed. In the expermental results M. Sorell (Ed.): e-forenscs 2009, LNICST 8, pp , ICST Insttute for Computer Scences, Socal Informatcs and Telecommuncatons Engneerng 2009

2 20 F. Mahmoud, R. Enayatfar, and M. Mrzashaer secton, the functonalty of ths method s studed through some experments. The reversblty of the method s studed n the next secton and fnally the conclusons are drawn. 2 Max-Heap Tree One of the specal trees whch s wdely appled n computer scences s Max-Heap tree. Ths tree s a complete bnary tree n whch the ntal value of each node s larger than or equal to the keys of ts chldren. Inserton of nformaton n ths tree s done as: the tree s always flled from left to rght n the last lne and then the next lne. Whle nsertng a new node, t s nserted n the far left empty space of the last lne (not flled yet), so that the tree s always complete. Then the heapfcaton s done, n whch a node n the lowest pont mght be replaced by ts parent as many tmes as t takes to be heapfed. For nstance, n case the nserton of 9 dgts s as 5, 8, 2, 3, 4, 7, 9, 20, 14, the resultng Max-Heap tree s as follows: Fg. 1. Max-Heap Tree 3 Chaotc Sgnal Chaos s a phenomenon that occurs n defnable nonlnear systems whch are hghly senstve to ntal values, and trend to show random-lke behavor. If such systems satsfy the condtons of Lapanov exponental equaton, wll contnue to be n the chaotc mode. The man reason why these sgnals are utlzed n mage encrypton s the defnablty of the system whle beng random-lke; ths caused the output of the system seem random to the nvaders. Snce t s defnable by the encrypter, t s decodable. The advantages of these functons are studed n two parts: a) Senstvty to the ntal value Ths means that mnor varaton of the ntal values can cause consderable dfferences n the next value of the functon, that s when the ntal sgnals vares a lttle, the resultng sgnal wll dffer sgnfcantly.

3 Image Encrypton Usng Chaotc Sgnal and Max Heap Tree 21 b) Random-lke behavor In comparson wth the generators of ordnary random numbers, n whch the seres of generated random numbers are capable of regeneraton, the random-numbergeneraton methods utlzed n chaotc functon algorthms are able to regenerate the same random numbers, havng the ntal value and the transform functon. Eq. (1) s one of the most well-known sgnals to have random-lke behavor and s known as Logstc Map Sgnal. ( ) X n+ 1 = rx n 1 X n (1) The Logstc Map sgnal wll have a chaotc behavor n case the ntal value s X 0 ( 0,1) and r = In fg. 2, the sgnal behavor wth ntal value s X 0 = 0. 5 and r = can be seen. The nput of the system s the scanned mage of several dfferent persons handwrtng of 26 Englsh captal letters. Fgure 2 represents sample dfferent handwrtten letters. Fg. 2. The chaotc behavor of sgnal (1) n ts 500 teratons 4 The Proposed Method In ths method, a bnary Max-Heap tree s made by non-repettve random numbers from 0 to 255, wth random order, generated by the chaotc functon of Logstc Map. Ths functon needs an ntal value to start out. To ncrease the level securty, an 80- bt key s used to generate the ntal value of the sgnal (Eq. (1)). Ths key can be defned as an ASCII character of the form: K0, K1,..., K9( ASCII ) (2) In ths key, K determnes an 8-bt block of the key. The bnary form of the mentoned key s as follows:

4 22 F. Mahmoud, R. Enayatfar, and M. Mrzashaer K01, K02, K03, K04, K05, K06, K07 K = K08,..., K91, K92, K 93 K, K, K, K, K,( Bnary) (3) The ntal value s resulted by Eq. (4). X K K K K = / 2... K 2 + K n7 n8 80 (4) On the other hand and as seen n fg. 2, the varaton range of the sgnal s [0,1].Ths range s dvde nto P parts whose sze s determnes by: ε = 1/ P (5) Based on ths segmentaton, the range of the th part s determned by: (( 1 ) ε, ε) (6) In ths method, P s 256 (the number of gray scales). In the followng part, the range n whch X 1 that s generated by Eq. (1) and the ntal value of X 0, wll be determned. The number of ths range s chosen as the frst order, provded that ths amount was not prevously located n the range; ths wll contnue as long as the sgnal magntude s located n all P parts. Fnally, non-repettve random order wll be generated n the range of (0,255) as: ( ) Iteraton = t1, t2,..., tr (7) Now, the frst value of the teraton wll be put nto the root and the second one (based on the Max-Heap tree structure) n the tree; ths wll contnue as long as all the numbers have flled the tree. Fnally, a bnary Max-Heap tree of 256 nodes wll be generated n each node of whch there s a unque number from 0 to 255. Ths tree s used to change the gray scale of the mage pxels. In the nest stage, 50 percent of the pxels of the frst row of the mage are selected by the use of Eq. (1), Eq. (2) (p=the mage wdth) and the ntal value of X r (the last number generated by the chaotc sgnal n the last stage). The root of the tree generated n the prevous stage replaces the frst pxel of the next lne. Knowng the tree structure, the chldren of the each node of the tree are put n a separate pxel of the mage. Then, the value of each node s xored wth the value of the pxel t s n. Ths wll contnue up to the last lne. In ths stage, three ponts are of great mportance: a) The poston of the chldren of a node on the pxel s ths way: f the node s n the poston (x,y) of the mage, the left-hand-sde chld s at (x+1,y-1) and the rghthand-sde chld s at (x+1,y+1).

5 Image Encrypton Usng Chaotc Sgnal and Max Heap Tree 23 b) In a pxel whch contans more than one node, the value of all nodes and the value of the pxel are xored wth each other (together) (nodes 15 and 10 n fgs. 3a and 3b). c) The mage s assumed to be a node. Fgs. 3a and 3b are examples of the proposed method, n whch a 4 4 mage and a Max-Heap tree of 6 nodes are consdered. In fg. 3b, by nsertng the root on pxel 2 and assumng the mage to be sphercal, node 3 wll be placed on pxel 12. Fg. 3. (a) The root s located n pxel 7. (b) The root s located n pxel 2. 5 Expermental Result A proper encrypton method must be resstance and secure to varous types of nvason, such as cryptanalytc nvasons, statstcal nvasons and brute-force nvasons. In ths secton, besdes the effcency of the proposed method, t s studed n terms of statstcal and senstvty analyses, n case of key changes. The results show that the method stands a hgh securty level aganst varous types of nvasons. 5.1 Hstogram Analyss Hstogram shows the numbers of pxels n each gray scale of an mage. In fg. 4, the orgnal mage s seen n frame (a) and the hstogram of the mage n red, green and blue scales are seen n frames (b), (c) and (d), respectvely. Also, n frame (e), the encrypted mage (usng key ABCDEF ABCD n a 16-scale) can be seen. In frames (f), (g) and (h), the hstogram of the encrypted mage n red, green and blue scales can be seen, respectvely. As seen n fg. 4, the hstogram of the encrypted mage s totally dfferent from that of the orgnal one, whch restrcts the possblty of statstcal nvasons.

6 24 F. Mahmoud, R. Enayatfar, and M. Mrzashaer Fg. 4. (a) the man mage, and (b), (c) and (d) respectvely show the hstogram of the lena mage of sze n red, green and blue scales, and (e) shows the encrypted mage usng the key, ABCDEF ABCD n a 16-scale. (f), (g) and (h) show the hstogram of the encrypted mage n red, green and blue scales. 5.2 Correlaton Coeffcent Analyss Statstcal analyss has been performed on the proposed mage encrypton algorthm. Ths s shown by a test of the correlaton between two adjacent pxels n plan mage and cphered mage. We randomly select 1000 pars of two-adjacent pxels (n vertcal, horzontal, and dagonal drecton) from plan mages and cphered mages, and calculate the correlaton coeffcents, respectvely by usng the followng two formulas (see table1 and Fg. 5(a) and (b)): where, cov 1 N N =1 ( x, y) = ( x E( x ))( y E( y )) r xy ( xy) ( ) D( y) cov, D x = (8) E 1 N N = 1 1 N N = 1 ( x) = x, D( x) = ( x E( x )). 2

7 Image Encrypton Usng Chaotc Sgnal and Max Heap Tree 25 Here, E(x) s the estmaton of mathematcal expectatons of x, D(x) s the estmaton of varance of x, and cov(x, y) s the estmaton of covarance between x and y, where x and y are grey-scale values of two adjacent pxels n the mage. Fg. 5. (a) Correlaton analyss of plan mage; (b) Correlaton analyss of cphered mage Table 1. Correlaton coeffcent of two adjacent pxels n two mages Plan Cphered Horzontal Vertcal Dagonal Plan Cphered Informaton Entropy Analyss The entropy s the most outstandng feature of the randomness [13]. Informaton theory s a mathematcal theory of data communcaton and storage founded by Claude E. Shannon n 1949 [14]. There s a well-known formula for calculatng ths entropy: 2N 1 1 = log = 0 Ps ( ) P( s ) H S (9) where P (s) represents the probablty of symbol s and the entropy s expressed n bts. Actually, gven that a real nformaton source seldom transmts random messages, n general, the entropy value of the source s smaller than the deal one. However, when these messages are encrypted, ther deal entropy should be 8. If the output of such a cpher emts symbols wth an entropy of less than 8, then, there would be a possblty of predctablty whch threatens ts securty. The value obtaned s very close to the theoretcal value 8. Ths means that nformaton leakage n the encrypton process s neglgble and the encrypton system s secure aganst the entropy attack. Usng the above-mentoned formula, we have got the entropy H(S) = , for the source s= 256. ( )

8 26 F. Mahmoud, R. Enayatfar, and M. Mrzashaer 5.4 Key Space Analyss In a proper method, key should have enough space so that the method s resstant aganst brute-force nvasons. In the proposed method, there can be 2 80 ( ) dfferent combnatons of keys. Scentfc results have shown that ths number of key combnatons s suffcent for a proper resstance aganst brute-force nvasons. 5.5 Key Senstvty Analyss In fg. 6b, the encrypton of the mage for fg. 6a, usng the encrypton key of ABCDEF ABCD s seen. The encrypton of the same mage s also done usng the keys BBCDEF ABCD and ABCDEF ABCE, respectvely seen n fgs. 6c and 6d. Fg. 6. The result of mage encrypton for an mage (fg. 6a): usng the encrypton key of ABCDEF ABCD n 7b, the encrypton keys BBCDEF ABCD and ABCDEF ABCE, respectvely seen n 7c and 7d In order to compare the obtaned results, the average of correlaton coeffcent (horzontal, vertcal and dagonal) of some specfc ponts s calculated for each par of encrypted mages (table 2). The obtaned results show that ths method s senstve to even small changes of the key. For nstance, the effect of the change n a pxel of the orgnal mage on the encrypted mage was measured usng two standards of NPCR and UACI [10,11]; NPCR s defned as the varance rate of pxels n the encrypted mage caused by the change of a sngle pxel n the orgnal mage. UACI s also defned as the average of these changes. These two standards are as follows: D (, j) j NPCR = 100% W H 1 C1(, j) C2(, j) UACI = 100% W H, j 255 where H and W are respectvely the length and wdth of the mages, and C 1 and C 2 are two encrypted mages of two mages whch are dfferent n one pxel. D s defned as: (10)

9 Image Encrypton Usng Chaotc Sgnal and Max Heap Tree 27 1 f C1 = D(, j) = 0 otherwse (, j) C (, j) The obtaned values of an mage wth the sze are as follows: NPCR=0.9952%, UACI=0.339% The value obtaned n table 2 clearly shows that ths method s resstant to dfferental attacks. 2 Table 2. The average of correlaton coeffcent (horzontal, vertcal and dagonal) of some specfc ponts for each par of the mages encrypted mage-fg 6b and 6c 6c and 6d 6b and 6d correlaton coeffcent Decodng an Encrypted Image One of the vtaltes of an mage encrypton method s ts reversblty of the encrypted mage to the orgnal one. Usng the proposed method, decodng an encrypted mage takes place as follows: As mentoned n secton 3, one of the sgnfcant propertes of chaotc functons s that, havng the ntal value and the transform functon, the seres of the numbers generated by the functon can be regenerated. In ths paper, havng the key, the ntal value of the chaotc functon can be generated; therefore, the requred seres of numbers for the generaton of Max-Heap tree s provded. Then, the last generated value by the chaotc functon of the prevous stage s used to choose the frst pxel of the frst lne. Unlke the encrypton method, the tree s not transformed on the chosen pxel; nstead, the poston of the pxel s saved n the PosPxel seres. Then the poston of the second pxel of the frst lne s determned by the chaotc functon. Thereby, the poston of half of the pxels of the frst lne s saved n PosPxel. Ths wll contnue through the last lne, where the followng seres s produced: n ( 1,1), ( 1,2 ),..., 1, 2 n ( 2,1), ( 2,2),..., 2, PosPxel = n ( n,1), ( n,2),..., n, 2 n n 2 In PosPxel, n each element of j, I determnes the row number and j the normalzed form of the frst generated number by the Logstc Map n the range of [0,n]. In the next step, the pxel n the poston of the last value of PosPxel, (n, n/2), s used as the frst pxel to be transformed and gone under Xor operaton (as explaned

10 28 F. Mahmoud, R. Enayatfar, and M. Mrzashaer n secton 4). Ths wll contnue up to the frst value of the PosPxel seres, (1, 1). And fnally, the decoded mage s regenerated. 7 Concluson In ths paper, a new method of mage encrypton has been proposed, whch utlzes chaotc sgnals and the Max-Heap tree for hgher complexty. As seen n the expermental results, ths method shows a very proper stablty aganst dfferent types of nvasons such as decodng nvasons, statstcal nvasons and brute-force ones. The hgh entropy of the method (7.9931) shows the capabltes of the proposed method. References 1. Mtra, A., Subba Rao, Y.V., Prasanna, S.R.M.: A New Image Encrypton Approach usng Combnatonal Permutaton Technques. Internatonal Journal of Computer Scence, (2006) 2. Chang, C.-C., Yu, T.-X.: Cryptanalyss of an encrypton scheme for bnary mages. Pattern Recognton Letters, (2002) 3. Josh, M., Chandrashaker, Sngh, K.: Color mage encrypton and decrypton usng fractonal Fourer transform. Optcs Communcatons, (2007) 4. Roterman, Y., Porat, M.: Color mage codng usng regonal correlaton of prmary colors. Image and Vson Computng, (2007) 5. Alsultanny, Y.A.: Random-bt sequence generaton from mage data. Image and Vson Computng, (2007) 6. Yen, J.-C., Guo, J.-I.: A New Chaotc Key-Based Desgn for Image Encrypton and Decrypton. In: Proceedngs IEEE Internatonal Conference on Crcuts and Systems, vol. 4, pp (2000) 7. L, S., Zheng, X.: Cryptanalyss of a Chaotc Image Encrypton Method. In: Proceedngs IEEE Internatonal Symposum on Crcuts and Systems, Scottsdale, AZ, USA, vol. 2, pp (2002) 8. Kwok, H.S., Tang, W.K.S.: A fast mage encrypton system based on chaotc maps wth fnte precson representaton. In: Chaos, Soltons and Fractals, pp (2007) 9. Behna, S., Akhshan, A., Ahadpour, S., Mahmod, H., Akhavan, A.: A fast chaotc encrypton scheme based on pecewse nonlnear chaotc maps. Physcs Letters A, (2007) 10. Chen, G., Mao, Y.B., Chu, C.K.: A symmetrc mage encrypton scheme based on 3D chaotc cat maps. In: Chaos, Soltons & Fractals, pp (2004) 11. Mao, Y.B., Chen, G., Lan, S.G.: A novel fast mage encrypton scheme based on the 3D chaotc baker map. In: Int. Bfurcat Chaos, pp (2004) 12. Pareek, N.K., Patdar, V., Sud, K.K.: Image encrypton usng chaotc logstc map. Image and Vson Computng, (2006) 13. Young, L.-S.: In: Branner, B., Hjorth, P. (eds.): NATO ASI Seres, p Kluwer Academc Publshers, Dordrecht (1995) 14. Shannon, C.E.: Bell Syst. Tech. J. 28, 656 (1949)