Security Analysis of the Efficient Chaos Pseudo-random Number Generator Applied to Video Encryption

Transcription

1 Journal of Electronic Teting (2018) 34: Security Analyi of the Efficient Chao Peudo-random Number Generator Applied to Video Encryption Dragan Lambić 1 Alekandar Janković 1 Muheer Ahmad 2 Received: 8 July 2018 / Accepted: 4 November 2018 / Publihed online: 8 November 2018 Springer Science+Buine Media, LLC, part of Springer Nature 2018 Abtract Recently, chao baed peudo-random number generator (PRNG) for video encryption wa propoed. Security analyi preented in thi paper reveal eriou problem. Chaotic map ued in analyzed PRNG do not enhance it ecurity due to a coniderable number of initial value that lead to fixed point. Alo, baed on 6 known iteration, an attacker can recontruct a ecret key ued in the working tage of the analyzed PRNG baed on attack whoe complexity i much maller than the etimated key pace. Therefore, ecurity of the analyzed PRNG i much lower than expected and it hould be ued with caution. Some potential improvement of the analyzed PRNG are propoed which could eliminate perceived hortcoming of the original verion. Keyword Chao Video encryption Peudo-random number generator Cryptanalyi Cryptography 1 Introduction A recently recognized, multimedia data i frequently being tranmitted through variou unafe channel. For thi reaon, ecurity of multimedia data uch a video file i attracting more and more attention. Some feature of video encryption uch a huge amount of data and higher requirement of real-time capacity are not completely upported by conventional cipher primarily intended for encryption of textual data [18]. Hence video encryption Reponible Editor: S. Bhunia Dragan Lambić dragan.lambic@pef.un.ac.r Alekandar Janković alekjankovic@gmail.com Muheer Ahmad muheer.ce@gmail.com 1 Faculty of Education, Univerity of Novi Sad, Podgorička 4, Sombor, Serbia 2 Department of Computer Engineering, Jamia Millia Ilamia, New Delhi, , India algorithm have been developed in recent year which are intended to protect the video content from unauthorized acce [7, 8, 18]. Security of cryptographic ytem developed for encryption of image and video file depend largely on feature of random number generator ued in thee ytem [9, 17]. There are two type of random number generator, truerandom number generator (TRNG) and peudo-random number generator (PRNG). TRNG are baed on nondeterminitic phyical procee and ome of them can achieve high peed but they are conidered unpredictable and uceptible to failure due to external fluctuation [12, 15]. On the other hand, PRNG are baed on digital algorithm which can provide random-like behaviour. Main diadvantage of PRNG i that they are necearily periodic [2], but on the other ide they are flexible, practical and can be ued at low cot [12]. Chao ha been widely ued in PRNG deign [4, 6, 12] due to it high enitivity to initial value, mixing, random-like behavior and ergodic behavior which are conidered a deirable propertie in cryptography. In recent year, variou chaotic map were ued a a ource of randomne in PRNG. For example, logitic map and it variation uch a quantum logitic map and piecewie logitic map have been ued in a ignificant number of PRNG [2, 5]. Chaotic map uch a piecewie-linear one-

2 710 J Electron Tet (2018) 34: dimenional map, multi-modal map, tinkerbell map, etc. arealowidelyuedinprng[6, 15, 16]. Many of the propoed cryptographic ytem and PRNG baed on chao uffer from eriou ecurity problem caued by inadequate election and application of chaotic map [1, 19, 20]. For example, initial value of multi-modal map are ued a a ecret key of peudo-random bit generator but ome feature of the ued multi-modal map caued exitence of certain number of weak key [11]. Alo, 99 percent of the key pace of PRNG baed on quantum chaotic map i compoed of weak key due to inadequate election of interval for control parameter of chaotic map [10]. Another caue of low ecurity, which i not directly connected to the feature of ued chaotic map, i inadequate deign of PRNG. For example, inadequate deign of PRNG baed on peudorandomly enhanced logitic map enabled ituation in which approximately 2 50 different ecret key produce the ame initial condition and control parameter of enhanced logitic map. Such deign error ignificantly reduce key pace, making thi PRNG non-reitant to brute-force attack [9]. Such ecurity problem can enable attacker to fully break analyzed cryptoytem and obtain ecret data and ecret key or can lead to partial reduction of ecurity which attacker cannot exploit to full extent with technological reource available today. In any cae, identification of uch ecurity iue i very important becaue level of ecurity of uch cryptoytem i far lower than expected. For that reaon, even cryptoytem with partially reduced ecurity are conidered broken [14]. Due to contant development of technological reource and cryptanalyi technique, uer mut be aware of reduced ecurity level which can lead to total break in the future. In [18] a novel peudo-random number generator i propoed, which i ued in a video encryption cheme. Thi PRNG i partially baed on two one-dimenional chaotic map. Due to it unuually large key, it can function properly and provide ufficient level of ecurity without thee two chaotic map. Unfortunately, ecurity analyi preented in thi paper indicate that chaotic map ued in analyzed PRNG often exhibit non-chaotic behavior which can lead to lower level of ecurity. Alo, baed on everal known iteration of thi PRNG, attacker can calculate all ubequent output value with complexity which i le than the etimated ecurity level. The ret of thi paper i organized a follow. In Section 2, the analyzed PRNG i decribed. Analyiof two chaotic map and cryptanalyi of the analyzed PRNG are preented in Section 3. In Section 4 the potential ecurityenhancement cheme i preented. Concluion are drawn in Section 5. 2 Decription of the Analyzed PRNG 2.1 Chaotic Map Analyzed PRNG [18] ue two chaotic map: x n+1 = 2xn 2 1 (1) y n+1 = 4yn 3 3y n (2) where x,y [ 1, 1]. In paper [18] x and y are decribed a real number from interval [ 1, 1] and key pace for each of them i etimated at = It i not clear how thee number hould be implemented becaue none of the two mot ued computer number format for toring real number have thi preciion. Single-preciion floatingpoint format ha 32 bit but 9 bit are ued for toring ign and exponent, therefore key pace of only 2 24 i available in thi format. Double-preciion floating-point format ha 64 bit and preciion of 2 52 which i higher than etimated key pace for x and y. 2.2 The Structure of the Analyzed PRNG Analyzed PRNG ha two tage, initialization tage and working tage. Initialization tage i ued only to etup initial tate of the analyzed PRNG K baed on ecret key K. BearinginmindthatK and K have ame number of bit and repreent ame parameter of the analyzed PRNG, attacker can ue K intead of K to obtain complete equence produced by the analyzed PRNG. For thi reaon initialization tage i not of interet to the attacker and therefore decription of analyzed PRNG will focu only on working tage. In the decription of the analyzed PRNG following notation i ued: the bitwie excluive OR operation, the modulo 2 32 addition, a L the leftmot 16 bit of integer a, a R the rightmot 16 bit of integer a, a b the concatenation of tring a and b. The ecret key of the analyzed PRNG conit of following parameter: 16 integer ( 0,..., 15 ), each repreented by 31 bit, two 32-bit integer R 1 and R 2 and two real number x 0 and y 0 from interval [ 1, 1] which are repreented by 33 bit each in paper [18]. In total, ecret key conit of 626 bit which i a very large number of bit compared to current required level of afety of 128 bit [3]. Secret key of uch length i more complicated for generation, ue and afe keeping but on the other hand it offer very high level of ecurity even if PRNG i partially broken. Secret key K

3 J Electron Tet (2018) 34: i ued in initialization tage a initial value of parameter which are after 32 iteration of thi tage tranformed into K which i ued a initial value of parameter in working tage. Working tage of the analyzed PRNG ue bit reorganiation function and function F(X 0,X 1,X 2 ). Bit reorganiation() {X 0 = 15L 14R ; X 1 = 11R 9L ; X 2 = 7L 5R ; X 3 = 2R 0L } F(X 0,X 1,X 2 ) {D = (X 0 R 1 ) R 2 ; D 1 = R 1 X 1 ; D 2 = R 2 X 2 ; R 1 = D 1 INTx i ; R 2 = D 2 INTy i ;} where INTx i and INTy i are 32-bit nonnegative binary integer repreentation of x i and y i, i i index of iteration and D i 32-bit output of function F(X 0,X 1,X 2 ). 2.3 The Execution Procedure of the Analyzed PRNG In the initialization tage of the analyzed PRNG, whoe detailed decription can be found in paper [18], ecret key K i tranformed to the input of the working tage K which can be alo conidered a a ecret key. Baed on K, attacker can obtain complete equence produced by analyzed PRNG. Let K i = ( 0,i,..., 15,i, R 1,i,R 2,i,x i,y i ) be the input of each iteration i of the working tage and 32-bit word Z i output of each iteration of the working tage, where K 0 = K. In each iteration of working tage following algorithm i ued: { Bit reorganiation(); Z = F(X 0,X 1,X 2 ) X 3 ; 16,i = ,i ,i ,i ,i + ( ) 0,i mod (231 1); If 16,i = 0thenet 16,i = 231 1; For each 0 j<16 et j,i+1 = j+1,i ; } Firt 100 output Z i are dicarded and ubequent value of Z i for i 100 are ued. 3 Security Analyi 3.1 Chaotic Map Analyi Analyzed PRNG ue two chaotic map. Firt map (Eq. 1) ha two fixed point x = 1andx = 1 2 and great number of value of x which after certain number of iteration lead to one of thee two fixed point. In paper [13] itihownthat 89 percent of initial point of thi chaotic map, implemented in digital computer, lead to fixed point. For thi reaon there i a high probability that thi chaotic map will produce ame value after certain number of iteration. Second map (Eq. 2) ha three fixed point y = 1, y = 1 and y = 0 and great number of value of y which after certain number of iteration lead to one of thee fixed point. Becaue econd chaotic map i cubic function, each of thee three fixed point can be obtained on the bai of three value, each of thee three value can be obtained on the bai of another three value and o on. For thi reaon, if we conider i iteration of thi chaotic map, there i 3 i value of y 0 which lead to ome of thee three fixed point. However, becaue y i repreented with 32 bit plu one bit for ign in paper [18], the vat majority of thee value will be found in the ame interval of length 1 and therefore a 2 32 greater number of real value will be repreented with the ame dicretized value. Becaue none of the two mot ued computer number format for toring real number have preciion of 32 bit, experiment with ingle-preciion (24 bit preciion) and double-preciion (52 bit preciion) floating-point format are performed in order to prove exitence of initial value which lead to fixed point. When ingle-preciion floatingpoint format i ued for implementation of the analyzed PRNG, about 0.86 percent of initial point lead to one of the fixed point in le than 100 iteration of the analyzed PRNG. Even 7.86 percent of initial point lead to one of the fixed point in le than 1000 iteration of the analyzed PRNG, which repreent very large part of key pace for uch a mall number of iteration. We mut have in mind that in ingle-preciion floating-point format only 24 bit are ued, therefore key pace of only 2 24 i available in thi format which i le than 2 32 mentioned in paper [18]. For bigger key pace we aume that greater number of iteration of chaotic map will be required to reach fixed point, o we can ue data baed on ingle-preciion floatingpoint format a maximal etimation of part of the key pace which lead to fixed point in mentioned number of iteration. When double-preciion floating-point format i ued for implementation of the analyzed PRNG, about percent of initial point lead to one of the fixed point in

4 712 J Electron Tet (2018) 34: le than 100 iteration. It i obviou that greater key pace influenced uch a ignificant reduction in percent of initial point which lead to fixed point, o greater number of iteration hould be teted. More than 1 percent of initial point lead to one of the fixed point in le than 10 million iteration of the analyzed PRNG. Becaue analyzed PRNG i intended for video encryption, very large number of iteration of PRNG (and chaotic map) i required. For thi reaon analyzed chaotic map could reach fixed point during encryption proce and exhibit non-chaotic behavior even when double-preciion floating-point format i ued. We mut have in mind that double-preciion floating-point format enable key pace of 2 52 that i 2 20 time bigger than key pace of 2 32 mentioned in paper [18]. Therefore maller number of iteration will be required to reach fixed point. We can ue data baed on double-preciion floatingpoint format only a a minimal etimation of part of the key pace which lead to fixed point in mentioned number of iteration. Baed on previou data we can conclude that ignificant number of initial point of both chaotic map ued in the analyzed PRNG lead to fixed point after certain number of iteration. Thee initial point are not grouped in maller number of interval and therefore they cannot be completely avoided when the ecret key i elected in a random manner. Bearing in mind that the analyzed PRNG i intended for video encryption and that amount of data in video file i today meaured by hundred of million or even billion of bit, it i expected that the analyzed PRNG mut be capable to function properly in more than a everal hundred million iteration becaue only 32 output bit are generated in one iteration. When we compare expected number of required iteration (everal hundred million iteration) with experimentally obtained number of iteration for which part of key pace leading to fixed point i greater than one percent (10 million iteration for double-preciion format or 1000 iteration for ingle-preciion format), it i quite poible that chaotic map will diplay non-chaotic behavior. Therefore we can aume that chaotic map do not contribute to the afety of the analyzed PRNG after certain number of iteration. All value of ecret key which lead to reduced ecurity of cryptoytem are called weak key. In thi cae, all value of chaotic map which lead to fixed point can be conidered a weak key of the analyzed PRNG. Bearing in mind that exitence of only one weak key i unacceptable in PRNG ued for cryptographic purpoe we can conclude that analyzed chaotic map, whoe number of weak key i expreed in percentage of total key pace, hould not be ued in PRNG intended for ecure application uch a video encryption. 3.2 Cryptanalyi One of the baic propertie of a PRNG, which influence it ecurity, i unpredictability. If the PRNG i unpredictable, attacker can not gue value of next output bit with a probability greater than 50 percent, regardle of the number of known conecutive output bit of ome PRNG. In the analyzed PRNG, value of input parameter in each iteration of working tage are ued to calculate all ubequent output value. Therefore we can conider input parameter of ome iteration a a ecret key of analyzed PRNG. In thi ection it will be hown that baed on everal known iteration of thi PRNG, attacker can obtain all value of input parameter of ome iteration of working tage with complexity which i le than etimated ecurity. Therefore, attacker can calculate all ubequent output value of analyzed PRNG by attack which complexity i leer than etimated ecurity of guee. Let K i = ( 0,i,..., 15,i, R 1,i,R 2,i,x i,y i ) be the input of each iteration i of the working tage and 32-bit word Z i output of each iteration of the working tage, where K 0 = K. For calculation of Z i attacker need to know only value of X 0,i,X 1,i,X 2,i,X 3,i,R 1,i and R 2,i that i 15L,i, 14R,i, 11R,i, 9L,i, 7L,i, 5R,i, 2R,i, 0L,i,R 1,i and R 2,i. However, due to need to update value of parameter for next iteration, attacker alo need to know value of 15R,i, 13,i, 10,i, 6,i, 0R,i,x i and y i. Aume that the attacker know the output value of the firt 6 iteration of the analyzed PRNG Z 0,...Z 5.Goalof the attacker i to obtain input of the working tage K which can alo be conidered a a ecret key. Entire proce of the propoed attack i preented in Fig. 1. Each row in Fig. 1 repreent data of one iteration of the analyzed PRNG, while each column repreent a manner in which value of integer (left and right part) are obtained. In Fig. 1, gueed value are denoted by letter G, calculated value by letter C and value equal to ome value from other iteration i are denoted by = j,i. The propoed attack i performed in the following tep: Step 1: In working tage of the analyzed PRNG output Z i calculated baed on equation Z = F(X 0,X 1,X 2 ) X 3. If attacker guee value of input parameter X 0,0,X 1,0, X 2,0,R 1,0,R 2,0 of function F, he can calculate value of X 3,0 = F(X 0,0,X 1,0,X 2,0 ) Z 0 baed on known Z 0. Therefore, value of 2R,0, 0L,0 (which are part of X 3,0 ) are calculated baed on gueed 15L,0, 14R,0, 11R,0, 9L,0, 7L,0, 5R,0,R 1,0,R 2,0 and known Z 0. Attacker alo need to gue value of 15R,0, 13,0, 10,0, 6,0, 0R,0,x 0 and y 0 in order to calculate 16,0 which will be ued in next iteration. Total

5 J Electron Tet (2018) 34: Fig. 1 Repreentation of the propoed attack complexity of firt tep of thi attack i 2 349, becaue 349 bit of the ecret key K 0 are gueed while 32 bit are calculated. Step 2: After each iteration of working tage hift of value of i performed by j,1 = j+1,0 for each 0 j < 16. Therefore, gueed and calculated value from previou iteration are now aigned to new parameter. For example, to parameter 15,1 value of calculated parameter 16,0 i aigned. In Fig. 2, proce of hifting value i preented. We can ee for example that value of 2R,0 calculated in firt iteration (denoted with 0) i hifted to 1R,1 and 0R,2 in next two iteration. Value of R 1,1,R 2,1,x 1 and y 1 are calculated baed on two chaotic map and function F(X 0,0,X 1,0,X 2,0 ). Therefore, for thi iteration of analyzed PRNG attacker know value 15,1, 14,1, 13R,1, 12,1, 10R,1, 9,1, 8L,1, 6L,1, 5,1, 4R,1, 1R,1 of ecret key K 1 Ȧttacker need to gue value of 11R,1 and 7L,1 in order to calculate X 3,1 that i 2R,1, 0L,1. Calculation i baed on known Z 1 in the ame way a in the previou tep. Now attacker need to gue only value of 13L,1, 10L,1, 6R,1, 0R,1 in order to obtain all neceary value of parameter ued for calculation of parameter for next iteration. Total complexity of econd tep of thi attack i 2 92, becaue 92 bit of ecret key K 1 are gueed while 32 bit are calculated. It i important to mention that baed on gueed and calculated value of in thi Fig. 2 Example of hift of value of iteration we can obtain value of ome in previou iteration due to j,1 = j+1,0.infig.2 we can ee for example that baed on value of 0L,5 calculated in fifth iteration (denoted by 4) we can obtain value of 1L,4 and 2L,3 from previou two iteration. Therefore, baed on gueed and calculated value in thi tep attacker know value of 14L,0, 12R,0, 11L,0, 8L,0, 7R,0, 3R,0, 1,0 of the ecret key K 0. Step 3: Baed on calculation and data from previou iteration attacker know value of 15,2, 14,2, 13,2, 12,2, 11,2, 10R,2, 9,2, 8,2, 7L,2, 6L,2, 5,2, 4,2, 3R,2, 1R,2, 0R,2 of the ecret key K 2. In thi tep attacker need to gue 10L,2 and 6R,2 and calculate X 3,2 that i 2R,2, 0L,2, baed on known Z 2. Total complexity of third tep of thi attack i 2 29, becaue 29 bit of ecret key K 2 are gueed while 32 bit are calculated. Baed on gueed and calculated value of in thi iteration we can obtain value of ome in firt iteration due to j,2 = j+1,1 = j+2,0. Therefore, baed on gueed and calculated value in thi tep attacker know value of 12L,0, 8R,0, 4R,0, 2L,0 of the ecret key K 0. Step 4: Baed on calculation and data from previou iteration attacker know value of 15,3, 14,3, 13,3, 12,3, 11,3, 10,3, 9,3, 8,3, 7,3, 6L,3, 5,3, 4,3, 3,3, 2R,3, 1R,3, 0R,3 of the ecret key K 3. In thi tep attacker need to gue only 6R,3 and calculate 0L,3 baed on known Z 3. Total complexity of fourth tep of thi attack i 2 15, becaue 15 bit of ecret key K 3 are gueed while 15 bit are calculated. Baed on gueed and calculated value of in thi iteration we can obtain value of 9R,0, 3L,0 of the ecret key K 0. Step 5: Baed on calculation and data from previou iteration attacker know all bit of the ecret key K 4 except for 0L,4 which i calculated baed on known Z 4. Thi tep do not increae complexity of propoed attack becaue no bit were gueed and only 15 bit are calculated. Baed on calculated value 0L,4 we can obtain value of 4L,0 of the ecret key K 0. Step 6: In thi tep all bit of the ecret key K 5 are known except for 0L,5 which i calculated baed on known Z 5. Baed on calculated value 0L,5 attacker know value of 5L,0 and therefore obtain the whole ecret key K 0. Baed

6 714 J Electron Tet (2018) 34: on the ecret key K 0 attacker can generate all peudo random number produced by analyzed PRNG. Total complexity of propoed attack i = which i time maller than etimated ecurity of Thi attack can be even more efficient due to mentioned problem with chaotic map. Becaue great number of initial point x 0,y 0 lead to fixed point, there i poibility that after ome x i,y i all obtained value of chaotic map will be equal to fixed point. Therefore, attacker can kip arbitrary number of iteration and perform thi attack baed on known Z i,...,z i+5. For higher value of i there i greater probability that value of x i,y i will be equal to fixed point. For thi reaon attacker can probably reduce earch for x i on two fixed point and earch for y i on three fixed point. Therefore complexity of earch for x i,y i will be reduced from 2 66 to only 6 guee. In thi cae total complexity of propoed attack i approximately which i dratically le than etimated ecurity of the analyzed PRNG of Although the analyzed PRNG i till afe againt attack performed with technological reource which are available today (due to extremely large key pace), it ecurity level i far lower than etimated and for that reaon we can conider thi cryptoytem broken [14]. Uer of thi PRNG mut be aware of reduced ecurity level becaue future development of technology can lead to ituation in which thi attack can be feaible. Preented cryptanalyi technique can be applicable to other imilar chaotic or non-chaotic PRNG which are baed on XOR operation and do not ue whole ecret key in each iteration. Alo thi attack can be ueful for deigner of new PRNG, becaue it indicate which chaotic map and deign method lead to reduced level of ecurity. 4 The Potential Security-enhancement Scheme Due to the great number of initial value of chaotic map which lead to fixed point, level of ecurity of the analyzed PRNG doe not correpond to the length of the ecret key K. Becaue ecurity of the analyzed PRNG i good regardle of performance of ued chaotic map, thee chaotic map hould be excluded from the analyzed PRNG. Alo, all bit of hould be ued in each iteration of the analyzed PRNG in order to avoid attack decribed in previou ection. For above mentioned reaon following change hould be made to bit reorganiation function and function F : Bit reorganiation() {X 0 = 15L 14R ; X 1 = 11R 9L ; X 2 = 7L 5R ; X 3 = 2R 0L ; X 4 = 14L 11L ; X 5 = 9R 7R ; X 6 = 5L 2L } F(X 0,X 1,X 2,X 3,X 4,X 5,X 6 ) {D = (X 0 R 1 X 4 R 2 1 ) X 3 ; R 1 = (R 1 X ) X 1 ; R 2 = R 2 X 2 X ;} Recommended change hould not influence peed of the analyzed PRNG ignificantly. In changed verion there i bigger number of XOR operation than in original verion but ue of chaotic map i completely eliminated which hould keep the peed of the analyzed PRNG at approximately ame level. The ecret key of new verion of the analyzed PRNG conit of 16 integer ( 0,..., 15 ) repreented by 31 bit each and two 32-bit integer R 1 and R 2. Two real number x 0 and y 0 are no longer part of the ecret key which i now 560 bit long. Becaue all parameter of the PRNG are ued in each iteration, attacker can not calculate any part of the ecret key baed on attack whoe complexity i lower than Therefore, ecurity of the improved verion of the analyzed PRNG correpond to the length of the ecret key. 5 Concluion In thi paper ecurity analyi of the efficient chao peudorandom number generator applied to video encryption i preented. Analyi of chaotic map ued in the analyzed PRNG how that thee chaotic map do not enhance ecurity of the analyzed PRNG due to the fact that coniderable number of initial value lead to fixed point. Alo, baed on 6 known iteration of the analyzed PRNG, data ufficient for recontruction of whole peudo random equence can be obtained with complexity which i much maller than the etimated key pace. Therefore, ecurity of the analyzed PRNG i much lower than expected and it hould be ued with caution. Preented cryptanalyi technique can be applicable to other imilar PRNG baed on XOR operation which do not ue whole ecret key in each iteration. Some potential improvement of the analyzed PRNG are propoed which could eliminate perceived hortcoming of the original verion. Compliance with Ethical Standard Conflict of interet The author declare that they have no conflict of interet.

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