A Block Cipher Using Linear Congruences

Transcription

1 Joural of Computer Sciece 3 (7): , 2007 ISSN Sciece Publicatios A Block Cipher Usig Liear Cogrueces 1 V.U.K. Sastry ad 2 V. Jaaki 1 Academic Affairs, Sreeidhi Istitute of Sciece & Techology, Ghatkesar, Hyderabad 2 Kakatiya Istitute of Techology & Sciece, Waragal, Idia Abstract: We have developed a block cipher by usig modular arithmetic iverse ad liear cogrueces. The cipher cotais a key matrix called the outer key. It also icludes aother key, which cotais a set of costats ivolved i the liear cogrueces. This key is called as ier key. The cryptaalysis carried out i this paper idicates that the cipher caot be broke by ay cryptaalytic attack. This cipher is exteded to the case of a larger block wherei iterlacig ad iteratio also play a vital role. Keywords: modular arithmetic iverse, ier key, outer key, liear cogrueces. INTRODUCTION I the literature of cryptography Hill cipher [1] has bee a promiet block cipher. Cosiderig oly 26 alphabetic ters a to z, Hill developed a block cipher whose ecryptio ca be described by the equatio C=KP mod 26, (1.1) where K is a key matrix of size x, P is a plaitext vector, ad C is the ciphertext vector both havig compoets. The decryptio of the cipher is carried out by usig the relatio P=K -1 C mod 26 (1.2) where K -1 is the modular arithmetic iverse [2] of the key matrix K. From the cryptaalysis of the cipher it is see that it caot be broke by bruteforce attack whe the size of the matrix is large. However, i the case of the kow plaitext attack, it is clearly established that the cipher ca be broke by takig appropriately colum vectors of plaitext ad ciphertext. I the preset paper our objective is to modify the Hill cipher by itroducig a additioal key. To this ed, we use liear cogrueces, i the colum vector of the plaitext, wherei the cogrueces cotai the umbers correspodig to the plaitext ters. Here, we cosider a plaitext, which icludes ters that ca be represeted by ASCII code. Thus, we use mod 128 istead of mod 26 used i the Hill cipher. From the cryptaalysis developed i this paper, we fid that the cipher caot be broke by ay cryptaalytic attack. I sectio 2 of this paper, we have discussed the developmet of the cipher. I sectio 3, we have described the algorithms for ecryptio, decryptio ad preseted a procedure for the modular arithmetic iverse of a matrix. I sectio 4, we have illustrated the cipher by cosiderig a example. The cryptaalysis for this cipher has bee carried out i sectio 5.I sectio 6, we have exteded the aalysis to a larger block by iterlacig ad iteratio. The, i sectio 7, we have examied the avalache effect. Fially i sectio 8, we have preseted the computatios ad coclusios. DEVELOPMENT OF THE CIPHER Cosider a plaitext vector, which ca be represeted i the form p = (p 1, p 2, p 3, p ) T. (2.1) Let us suppose that we choose a key matrix K give by K=[K ij ], i=1 to, j=1 to, (2.2) where the matrix K is o sigular, ad its determiat is relatively prime to 128. Correspodig Author: V. Jaaki, Kakatiya Istitute of Techology ad Sciece, Waragal, Idia 556

2 The aforemetioed coditios are to be satisfied for the existece of the modular arithmetic iverse of K with respect to mod 128. Let C = (C 1,C 2,C 3.C ) T be the ciphertext vector. (2.3) Let us ow itroduce liear cogrueces give by P = K -1 C mod 128 (2.9) where K -1 is the modular arithmetic iverse of K. O usig (2.9) ad (2.8) we get p i, the compoets of the plaitext as we have idicated i the aforemetioed discussio. The problem of the ecryptio, give by the equatio (2.5) ca be writte i the form P i = (a i p i + b i ) mod 128, i=1 to, (2.4) where a i ad b i are costats, chose appropriately, as metioed below. I the process of ecryptio, the ciphertext C ca be writte as C=KP mod 128, (2.5) where P=(P 1,P 2,P 3 P ) T. C 1 C 2 C 3... C k 11 k 12. k 1 k 21 k 22...k 2 k 31 k 32 k 3 = P 3 mod k 1 k 2 k P 1 P 2.. P (2.10) Here the compoets of the plaitext vector p i ( i =1 to ) are obtaied from the cosecutive ters of the give plaitext. Here i (2.4), we choose each oe of the a i s as a odd iteger, which lies betwee 0 ad 127, ad each oe of the b i s as ay iteger lyig betwee 0 ad 127.The reaso for this choice of the values of a i ad b i, will be clear very soo. Whe a i, b i ad p i are kow to us we ca readily calculate P i by usig (2.4). O the other had, whe the P i is kow to us, p i ca be determied by solvig (2.4). As we have assumed that b i is a iteger which lies betwee 0 ad 127, we ca write (2.4) i the form P i - b i = a i p i mod 128. (2.6) Here firstly we have to obtai P i where P i =(a i p i +b i ) mod 128.For this we require the values of a i, ad b i (i=1 to ).To this ed we itroduce a key comprisig the umbers a 1,a 2. a ad b 1,b 2...b ad call this as the ier key. Subsequetly we have to apply the key matrix K for obtaiig C. Thus we cosider this key as the outer key. Here it is to be oted that i the process of decryptio firstly the outer key is to be applied ad the the ier key is to be used. Thus both the keys are to be supplied by the seder i a secret maer to the receiver. I what follows we desig the algorithms for the ecryptio, the decryptio, ad the modular arithmetic iverse of the key matrix. As a i is a odd iteger, which lies betwee 0 ad 127, it is relatively prime to 128. Thus we obtai d i, the multiplicative iverse if a i, such that a i d i mod 128=1 (2.7) where d i is the multiplicative iverse of a i. From (2.6) ad (2.7), we get p i = (P i - b i ) d i mod 128. (2.8) Now let us cosider the process of decryptio. From the equatio (2.5) we get 557 ALGORITHMS 3.1 Algorithm for ecryptio 1. for i=1 to 2. read p i, a i, b i. 3. P i =( p i a i + b i ) mod read the key matrix K. 5. C=KP mod write C.

3 3.2 Algorithm for decryptio. 1. for i=1 to read a i, b i. 2. read K,C. 3. Fid K P= K -1 C mod for i=1 to Fid d i such that a i d i mod 128=1. 6. p i =( P i - b i ) d i mod write p i 3.3 Algorithm for modular arithmetic iverse // A is a x matrix. N is a positive iteger with which //modular arithmetic iverse is carried out. Here N= Fid the determiat of A. Let it be deoted by, where Fid the iverse of A. The iverse is give by [A ji ]/. 3. for i = 1 to N if ( (i) mod N = 1 ) d = i; // is relatively prime to N. break; 4. B=(d[A ji ] ) mod N // B is the modular arithmetic iverse of A ILLUSTRATION OF THE CIPHER Let us cosider the plaitext Bur the forest as soo as the thieves eter ito it. Let the key matrix K be take i the form K = p = ( ) T. (4.2) Let us choose the ier key i the form (a 1 a 2 a 3 a 4 a 5 a 6 b 1 b 2 b 3 b 4 b 5 b 6 ) = ( ) (4.3) O usig (4.2) ad (4.3), ad the ecryptio algorithm 3.1, we get ( P 1 P 2 P 3 P 4 P 5 P 6 ) = ( ) (4.4) ad C = ( ) T. (4.5) The o usig the algorithm 3.3, the modular arithmetic iverse of the matrix K, deoted by K -1,is obtaied as K -1 = It ca be readily verified that K -1 K mod 128 = K K -1 mod 128 = I. The o usig the algorithm 3.2, we get P= ( ) T (4.7) ad (p 1 p 2 p 3 p 4 p 5 p 6 ) = ( ). (4.8) The values of p 1.p 6 give by (4.2) ad (4.8) are the same. Thus we get back the plaitext. CRYPTANALYSIS (4.6) I the process of ecryptio we have Here =6. (4.1) C=KP mod 128, (5.1) Now let us focus our attetio o the first six ters of the plaitext uder cosideratio. The first six where P i = (a i p i + b i ) mod 128, i=1 to. (5.2) ters are Bur t. Here whe the ciphertext C is kow to us, the Cosiderig the correspodig ASCII codes, the plaitext vector p ca be obtaied as plaitext p ca be foud if the matrix K, ad the a i ad 558

4 the b i are kow to us, i.e if the outer key ad the ier key are kow to us. The key matrix is of size. The a i s are i umber ad the b i s are also i umber. The umber of combiatios of the secret key, icludig the outer key K ad the ier key comprisig a i ad b i, ca be determied as follows. The outer key K cotais 2 umbers wherei each oe ca be represeted i terms of 7 biary. Thus the key space correspodig to this key is give by 2 2. Now let us cosider the key space of the ier key. As the a i s are purely odd umbers, lyig betwee 0 ad 127, they ca be represeted by 7 biary wherei the least sigificat bit all the while remai as 1. As the umber of a i s is, the possible umber of combiatios of the a i s is 2 6. As b i is ay umber that lies betwee 0 ad 127, the possible umber of combiatios of b i s is 2. Thus the total search space for the secret key icludig the outer key K, ad the ier key a 1, a 2. a, b 1,b 2...b ca be obtaied as = For every possible key of the key space, the attacker has to fid the plaitext till he gets a meaigful oe. This process is to be carried out at least with half of the possible keys. If a plaitext correspodig to a secret key, ca be obtaied, by the decryptio process i 10-7 sec., the the time T required for computig with half of the possible keys, is give by T= ceturies, where =10-7 / = Let us ow cosider the kow plaitext attack. I this case we kow as may pairs of plaitext ad ciphertext as we require. For each plaitext vector p we are to fid P s for all possible combiatios of the values of a i ad b i i.e, for each possible ier key. Thus the umber of P s correspodig to each p is (2 6 ) (2 7 ) =2 13 as a i is a odd iteger lyig betwee 0 ad 127, ad b i is ay iteger lyig betwee 0 ad 127. If we take plaitext vectors ad form a matrix with these vectors, the the umber of matrices cotaiig the correspodig Ps is I view of the example give i sectio 4,we have =6.I this case the umber of matrices cotaiig P s are 2 13x36. Now, let us suppose that fidig the modular arithmetic iverse of each matrix takes 10-7 sec., the fidig all possible modular arithmetic iverses will take 2 13x sec ceturies. Thus we coclude that the stregth of the cipher icreases eormously, ad it caot be broke by the kow plaitext attack. A LARGER BLOCK CIPHER USING INTERLACING AND ITERATION I sectio 2, we have developed the cipher for a block of ters. Let us ow exted the aalysis for a block of 2 ters by itroducig the cocepts iterlacig ad iteratio. I this sectio, the procedures used for ecryptio ad decryptio are preseted i Figures 1 ad 2. I the procedure of ecryptio, the block cosistig of 2 ters is divided ito two blocks left half ad right half each cotaiig ters. O these two halves the same procedure, discussed i sectio 2, is applied. The we obtai ters as output o both the sides. The ters o each side are coverted ito biary. 559

5 Plaitext: 2 Roud 1 Plaitext: 2 chars 392 Roud 16 Iterlace Iterlace Roud 2 P Roud 16 P Decompose Decom pose Decompose Decom pose P bit Roud 2 Roud 1 Iterlace Ciphertext:2 ters Decompose Decom pose Ciphertext:2 ters bit Fig.1: Procedure for Ecryptio i the case of larger block cipher cotaiig 2 ters Fig.2: Procedure for Decryptio i the case of larger block cipher cotaiig 2 ters 560

6 The process of iterlacig ca be described as follows. The first bit of left side is placed as the first bit of a array. The first bit of the right side is placed as a secod bit i the array. The the secod bit of the left is placed i the third place of the array. This process is cotiued till we exhaust all the o both the sides. The procedure described above costitutes oe roud. This process is repeated for 16 rouds ad the ultimately we get the ciphertext. The process of decryptio is completely a reverse process of ecryptio. The decompose, used i decryptio, is a fuctio i which the 14 of the 2 ters are separated ito o the left side ad o the right side. By usig the process of ecryptio we are able to ecrypt 2 ters ad obtai the correspodig ciphertext. Further, by performig decryptio we are able to obtai the plaitext of legth 2 ters from the ciphertext. I order to illustrate the above procedure, let us cosider the plaitext Bur the for (6.1) which is cosistig of 12 ters. Followig the ecryptio procedure give i Fig.1, the ciphertext correspodig to the plaitext (6.1), is obtaied i terms of biary as follows (6.2) O applyig the procedure give i Fig.2, the ciphertext (6.2) ca be coverted ito the plaitext give by (6.1). AVALANCHE EFFECT Let us cosider the plaitext give by (6.1). O chagig the first ter B i the plaitext to C, the plaitext assumes the form Cur the for. (6.3) I biary bit represetatio (6.1) ad (6.3) differ oly oe i biary bit as B ad C differ i oe bit. Now o applyig the ecryptio procedure give i Fig.1, o the modified plaitext (6.3), we get the ciphertext i the form (6.4) Here we otice that (6.2) ad (6.4) differ i 30. This idicates that the avalache effect is ot at all less sigificat. Cosider the effect o accout of a chage i the ier key, which plays a promiet role o the cipher. Let us ow we take the ier key as (6.5) istead of (4.3). O usig this key ad the ecryptio procedure give i Fig.1, we get the ciphertext i the form (6.6) O comparig (6.2) ad (6.6) we fid that they differ i 31. This agai shows that the avalache effect is sigificat. COMPUTATIONS AND CONCLUSIONS I this paper we have developed a block cipher by usig liear cogrueces ad modular arithmetic iverse. Firstly the plaitext vector is modified by usig the liear cogrueces. These cogrueces cotai a set of costats, which form a key called the ier key. The modified plaitext is operated by a key matrix called as outer key The ier ad outer keys are used i the process of ecryptio ad i the process of decryptio, ad they form the secret key. This cipher is exteded to the case of larger block (block size is doubled) by itroducig iterlacig ad iteratio. Here all the programs for ecryptio ad decryptio are writte i C laguage. O performig computatios we have obtaied the ciphertext for a give plaitext ad vice-versa. The ciphertext obtaied i the case of the plaitext Bur the forest as soo as the thieves eter ito it. is give by From the cryptaalysis it is clealy see that the cipher caot be broke by ay cryptaalytic attack. The iterlacig ad the iteratio stregthe the cipher remarkably. From the aalysis preseted i this paper we fid that the outer key ad the ier key both play a vital role i stregtheig the cipher. The stregth is further ehaced i the case of the larger block. REFERENCES 1. Cryptography ad Network Security, William Stalligs, 3 rd Editio, Pearso Educatio 2. O the Modular Arithmetic Iverse i the Cryptology of Hill cipher, V.U.K.Sastry, V.Jaaki, Proceedigs of North America Techology ad Busiess Coferece, Caada 561