Polynomial Encryption Using The Subset Problem Based On Elgamal. Raipur, Chhattisgarh , India. Raipur, Chhattisgarh , India.

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1 Polyomal Ecrypto Usg The Subset Problem Based O Elgamal Khushboo Thakur 1, B. P. Trpath 2 1 School of Studes Mathematcs Pt. Ravshakar Shukla Uversty Rapur, Chhattsgarh 92001, Ida. 2 Departmet of Mathematcs, Govt. N.P.G. College of Scece Rapur, Chhattsgarh 92001, Ida. Abstract I ths artcle we have demostrated a polyomal message whch have bee ecrypted through the Merkle-Hellma ecrypto scheme ad set by use of Elgamal. So that oly the proposed recever s effcet to decode the message. Our result also llustrated wth the help of a mathematcal eample ad algorthm.. Keywords: Elgamal, Kapsack problem, super creasg vector, subset sum problem. 1. Itroducto Lattces were frst studed by Mathematcas Joseph Lous Lagrage ad Carl Fredrch Gauss. I 199, Mkls Ajta showed the use of lattces as a cryptography prmtve semal result. There are two fudametal computatoal problems assocated wth the lattce are shortest vector problem ad closest vector problem. I geeral, CVP s to kow be NP-hard ad SVP s to kow be NP-hard uder a certa radomed reducto hypothess []. The Merkle Hellma system s based o the subset sum problem [1]. The subset sum problem s a specal case of the kapsack problem. The subset sum problem s hard, ts decso problem was show to be NP-complete by Karp []. The cocept of the super creasg subset problem was coed by Merkle-Hellma Ralph Merkle ad Mart Hellma used the subset problem to create a cryptosystem for ecrypt data [1]. I ths system super-creasg kapsack vector s s created ad the super-creasg property s hdde by creatg a secod vector M by modular multplcato ad Permutato. Here the vector M s the publc key of the cryptosystem ad s s used to decrypt the message. The subset sum problem s to fd a subset of a gve set of postve tegers..., such that the elemets 1 the subset sum up to some gve teger s. The subset sum problem s a NP complete problem [] combatoral optmato. The kapsack problem selects the most useful tems from a umber of tems gve that the kapsack has a certa capacty. Kapsack problems are wdely used to model solutos, dustral problems such as publc key cryptography. Ralph Merkle ad Mart Hellma used the subset problem to create a cryptosystem to ecrypt data. A super-creasg kapsack vector s s created ad the super-creasg property s hdde by creatg a secod vector M by modular multplcato ad permutato. The vector M s the publc key of the cryptosystem ad s s used to decrypt the message [2]. ENCRYPTING MESSAGES: Ths cryptosystem performs ecrypto two steps. Frst the polyomals are coverted to ther bary equvalet. These polyomals are ecrypted through the Merkle-Hellma ecrypto scheme whose ma dea s to create a subset problem whch ca be solved easly ad the to hde the super- creasg ature by modular multplcato ad permutato. Secodly, these ecrypted polyomal are further ecrypted though the use of EL-GAMAL cocepts. Publshed by: PIONEER RESEARCH & DEVELOPMENT GROUP ( 38

2 We choose a prme umber p ad a prmtve root g mod p ad also choose a radom epoet a 0... p 2 ad calculate `A' such that A = g a mod p We choose aother radom teger b 0... p 2 usg these detals we fd out the value of `B' such that B = g b mod p Here (p,g, A) s a publc key ad prvate key s a. Thus the formula to ecrypt the message usg Elgamal cryptosystem s c = A b m mod p 3. Mathematcal Eplaato: Step 1: The frst step s to choose key of legth 7 bts. These are used to perform the frst ecrypto process. Step 2: The secod step s to covert the polyomal of the message to bary. The bary sequece s represeted by the varable y. Step 3: Choose a super creasg sequece of umber of postve tegers. A super creasg sequece s oe where every umber s greater tha the sum of all precedg umbers. S = ( s,,,... ) 1 s2 s3 s Step : The forth step s to choose a radom teger () such that f s = 0 ad a radom teger r, such that gcd (, r) = 1 where r ad are co prme. The sequece s ad the umbers ad r be the prvate key of the cryptosystem. All the elemets ( s,,,... ) 1 s2 s3 s of the sequece s are multpled wth the umber r ad the modulus of the multple s take by dvdg wth the umber. Now calculate the sequece k = ( k,,,... ) 1 k 2 k 3 k Where k = r s mod The publc key s k ad prvate key s (r,, s). Step : The message s ecrypted by multplyg all the elemets of sequece k wth the correspodg elemets of sequece y where y s the - th bt of the message ad y 0,1. The umbers are the added to create the ecrypted message M forms the cpher tet of the cryptosystem. Therefore,. Eample: Publshed by: PIONEER RESEARCH & DEVELOPMENT GROUP ( 39 M Ecryptg the message = = 0 Step 1: The frst step s to covert the Compao matr bary equvalet- A= Step 2: The secod step s to choose a super-creasg sequece s s created s = (1, 3, 11, 23, 72, 11, 28) Ths problem s easy because s s a super-creasg sequece. Step 3: The bary sequece s y = (,,,... ), choose a umber that s y y y y greater tha the sum of all super creasg sequece the = 7 ad choose a umber r that s the rage [1; ) where r = 1. The prvate key cossts of, s ad r. To calculate a publc key, geerate the sequece k by multplyg each elemet s by r mod k k = (1,, 1, 3, 1, 339, 21) because k1 = 1 1 mod 7 = 1 k2 = 3 1 mod 7 = k3 = 11 1 mod 7 = 1 k = 23 1 mod 7 =3 k = 72 1 mod 7 = 1 k = 11 1 mod 7 = 339 k7 = 28 1 mod 7 = 21 where the sequece k s a publc key. Step : The message s ecrypted by multplyg all the elemets of sequece k wth the correspodg elemets of sequece y ad addg the resultg sum. Therefore, the ecrypted message s k M= y =0 Ecryptg the message. Its bary equvalet s , Where k = (1,, 1, 3, 1, 339, 21) ad y = ( ) y

3 The, M = 1 1 = 22 Step : Ths s ecrypted usg elgamal cocepts. Here we choose a prme umber p=283, g=179 ad choose radom umber b=113. The value of `a' s choose s 10. Therefore, A = mod 283 A = 7 Thus the message s ecrypted accordg to the formula C = A b M mod p Thus the ecrypted code for the frst character becomes C = mod 283 = 81 These codes are set to the recever. Thus the message to be trasmtted to the recever s DECRYPTING MESSAGES Durg the decrypto process, the blocks of ecrypted code are separated. O these blocks decrypto s performed usg the ELGAMAL cocepts. The formula used s p 1 a M = B C mod p The output of t s decrypted usg Merkle-Hellma Kapsack cryptosystem decrypto process. To decrypt the message M, the recpet of the message would have to fd the bt stream whch satsfes the Equato [1] k M= y =0 The frst step s to calculate the modular multplcatve verse of `r r mod [3]. Ths s calculated usg the Eteded 1 Eucldea algorthm. Ths s deoted by r. The secod step s to multply each elemet of the ecrypted 1 message (M) wth r mod. Sce r was chose such that gcd (r, ) =1. The largest umber the set whch s smaller tha the resultg umber s subtracted from the umber. Ths cotues utll the umber s reduced to ero [].. Eample. Step 1: Decryptg the message: C= 0081 Step 2: Decryptg ecrypted code. Perform the decrypto process o C=0081 usg the cocepts of ELGAMAL. Here we chooses g=179 ad radom umber b=113. Therefore, `B = mod 283 `B = 237 Thus the message s ecrypted accordg to the formula, M= p 1 a C mod p B Thus M = mod 283 =22 Step 3: The modular verse of 1 1 mod 7 s calculated usg the eteded Eucldea algorthms ad was foud out to be 1. Step : The ecrypted message M s 22 ad s = (1, 3, 11, 23, 72, 11, 28). Aga, 22 1 mod 7 = 1 Now decompose 1 by selectg the largest elemet s whch s less tha or equal to 1. The selectg the et largest elemet less tha or equal to the dfferece, utl the dfferece s = -3 = 1 1-1= 0 Thus, the bary sequece becomes The polyomal equvalet to ths bary sequece s Publshed by: PIONEER RESEARCH & DEVELOPMENT GROUP ( Ecrypto Process Algorthms I ths secto, we propose algorthms for above ecrypto ad decrypto mathematcal eample. A.. Algorthm for teratve multplcato of bary umber ad super creasg umber: Iput : Bary umber a(m) ad super-creasg array b[m]. Iput : Se of array. Itale : mul = 0, = 0 whle( <= ) mul = mul a[]*b[]; retur (mul) B. Algorthm: Itale : r = 1 ad = 7 Iput : Se of array elemet: =0 whle(i <= se) put : array elemet of k

4 =0 whle( <= se) c=r*s[] k()=c mod retur (k()) C. A algorthms for ecrypted message: fucto myfu (,y, ) A = 1 j = 1 whle(j <= y) A=(A * ) mod retur (A) Iput : g = 179 Iput : b = 113 Iput : p = 283 Now we create a fucto pt = elgamal mod (g, b,) Output : Pla tet Thus the algorthms for the scheme C = a b m mod p s : fucto ecrypt(, y, m, ) k=1 j=1 whle(j <= y) k=(k*) C=(k*m) mod retur (C).2 Decrypto Process Algorthms: A. Algorthm: Itale : g= 179, b = 113 Iput : g Iput : b put : p =1 whle( <= b) B = 1 B = (B g) modp retur (B) B. Algorthm for Iverse modulo : Fucto v (a; b) =1 whle ( < b) um=(a ) mod f(um=1) retur() C. Algorthm: Iput : Se of array Iput : Eter check value (say 1) Iput : Eter array (s) elemets =1 whle( <= se) Iput : array elemets whle( <= se) j=1 whle(j <= se) f(s()> s(j)) temp=a() a() = a(j) a(j) = temp =1 whle(check value! = 0) f(check value > = s()) check value = check value - s() retur (check value) Publshed by: PIONEER RESEARCH & DEVELOPMENT GROUP ( 1

5 . Coclusos Ths paper epla how to ecrypt ad decrypt data by the workg of subset sum problem through the use of Elgamal cocepts. The whole cryptosystem was demostrated by ecrypted a polyomal t. ad the decryptg Refereces [1] M. Hellma ad R. Merkle, Hdg formato ad sgatures trapdoor kapsacks, IEEE Tras. Iform. Theory, Vol. 2, 1978, [2] A. Meees, P.vaOorschot ad S.Vastoe, Hadbook of Appled Cryptography, CRC Press 199. [3] W. Dffe ad M. Hellma, New drectos cryptography, IEEE Tras. Iform. Theory, Vol. 22, 197, -. [] J. Hoffste, J. Ppher ad J. H. Slverma, A Itroducto to Mathematcal Cryptography, Udergraduate Tets Mathematcs, Sprger [] Ashsh Agarwal, Ecryptg Message usg the Merkle Hellma Kapsack Cryptosystem, Iteratoal Joural of Computer Scece ad Network Securty Vol ,12-1. [] Rchard M. Karp, Reducblty amog combatoral problems, Complety of Computer Computatos, Raymod E. Mller ad James W. Thatcher (eds.) Pleum Press, NY, Publshed by: PIONEER RESEARCH & DEVELOPMENT GROUP ( 2