GESJ: Computer Scece ad Telecommucatos 206 No.2(48) ISSN 52-232 UDC 004.02 USING GENETIC ALGORITHM FOR CRYPTANALYSIS CRYPTOALGORITHM MERKLE-HELLMAN Lal Besela Sokhum State Uversty,Poltkovskaa str., Tbls, Georga Abstract The artcle cosders possblty of usg geetc algorthms cryptaalyss, amely for crackg Merkle Hellma cryptosystem. The expermetal results show the effcecy usg of geetc algorthms for cryptaalyss moder cryptosystems. Idex Terms: geetc algorthms, cryptosystem, the system Merkley-Hellma cryptaalyss. I. Itroducto Merkle-Hellma s famous artcle was publshed 978 [], whch descrbes the publc key (asymmetrc) crypto system, based backpack task [2] Oe of the partcular case, whch ca be summed up as follows: V the capacty kapsack ad B = { b, b2,... b} obects set, whch have a certa capactes. Our task s to fd a B subset of the abudace B B of the elemets the equato to be executed. V = = 49 b x where, x {0, }, =,2,...,. If x = t meas that the subect should be put kapsack, ad f x = 0 the subect does ot vest kapsack. As s well kow [2], kapsack task belogs to the group of NP complexty of the problem, but ths partcular case, f the B set s the cremetal sequece, the sequece of each b member satsfes the codto > b b, = The there's the task of the lear complexty of the algorthm [2]. Merkle ad Hellma bult the ope crypto system usg of ths system. The ecrypto key A = { a, a2,... a} s ot reasoable for the creasg sequece, where A the sequece of each member a the followg method: a = b t (mod m) () ad where m, t Z the followg codtos are fulflled: m > b = ( t, m) = Secret decrypto of the key s trple ( B, m, t). Ope Text, whch represets zeros everywhere ad roll sequece, durg the legth of the ecrypto tme wll be dvded to a umber L of blocks ad performs the role x {0, } of abudace. Ecrypted text S, S2,..., SL sums, whch are calculated by the formula:
GESJ: Computer Scece ad Telecommucatos 206 No.2(48) ISSN 52-232 S = = x a Ope Text s eeded to repar the problem solved backpack above opto lear complexty algorthm, the kow B ad the Ascedg ad m ad t optos. To do ths, multpled t - by the sum of each module m / S = S t (mod m) (3) ad becomes a backpack to the soluto of lear complexty algorthm separately for each S of the sum metoed, whe t s kow B the ascedg sequece. I order to break the adversary system, ad t wll have to fd the Ope Text soluto of NP complexty of the task, whch s practcally mpossble, whe two hudred to three hudred elemets of B the sequece of the umber are chaged. At a glace, ths cryptosystem really was protected from ay attacks ad was the fastest publc key system, the use of whch has a large capacty to ecrypt texts, but t was dscovered that he had some falg[3], wth whch the famous scetst A.Shamr used polyomal dffculty Algorthm ad broke the system [4]. From ths falg frst to ote that the secret key from the publc key recepto, ulke other ope-key crypto systems are ot oe-way fucto. Also, as t tured out, ot ecessarly to fd exactly the (t 0, m 0 ) par, wth the help of whch over creasg sequece returs the key - Not over creasg sequece. As t tured out, all of the cremetal sequece, from whch the addtoal beds A ot use ca be obtaed ascedg secret key, or a key o the attack. These vulerabltes by usg A.Shamr us cryptoalgorthm to attack the system, whch cossts of two parts. I the frst part of the algorthm to a whole umber, whch satsfed the codtos for the u / m values for some a of these fuctos s the mmum terval. Such umbers to fd the algorthm Dophate approxmato method for (u, m) pars, whch wll be possble to ope the key to the secret key to calculate. II. Geetc algorthms Geetc algorthms orgally used for solvg optmzato problems. Over tme, he foud the use of scece varous felds. Geetc algorthms based o bologcal evoluto s oe of the basc prcple: the fght to save the evromet as much as possble adaptato of the populato, whch s acheved by stregtheg ad developmet of ew geeratos of more ad better features. Geetc algorthms for modelg of ths prcple s as follows: radom soluto set-elected caddates ad the populato of the geetc operators: selecto, crossovg ad mutato usg a ew geerato of caddates accepted soluto, whch s closer to the average of the real soluto, tha those of the prevous geerato. It depeds o how we use geetc algorthms ad the qualty of the crtera we have selected, or how to use the ftess fucto. Geetc algorthms are oe of the maor advatages of search algorthms other tha the possblty of ther parallelzato, whch substatally reduce the attack. (2) III. Usg geetc algorthm Merkle-Hellma algorthm s eeded to spol Cryptographc algorthms for the aalyss of geetc algorthms use a ew drecto, whch s stll uable to settle the practcal cryptologsts. There are dozes of works whch authors try to 50
GESJ: Computer Scece ad Telecommucatos 206 No.2(48) ISSN 52-232 show the advatages of ths approach ca have a comparso wth other methods. Our goal s to demostrate the advatages of the use of geetc algorthms for the aalyss of crypto Compared wth other methods. It was at ths pot we took a crypto system has bee broke Merkl-Hellma, the breakg of whch we tred usg geetc algorthms, ad we compared the results obtaed by the results of Shamr's algorthm. There are some hard work, where merkl-hellma crypto system s explaed by meas of geetc algorthms, but all these cases the attack s made by meas of a cpher text [7]. These studes, however, we're lookg for the secret key of the publc key to the techque smlar to Shamr. Have developed a ew heurstc methods, whch resulted the use of geetc algorthms s to make more accurate ad quck. I ths artcle, the results of the study ca be used other asymmetrc crypto systems ad software crypto aalyss. IV. Problems explorato As metoed above there are works, whch dscusses the use of geetc algorthms Merkle- Hellma crypto system for cryptaalyss. But our attack method s totally dfferet from the methods used the works. Also, we use the two dfferet geetc algorthm, whch s dfferet from other geetc algorthms (dfferet selecto crtera ad the qualty of the crossover process). We carry out attacks o the cpher text of the key meas. Our task s to fd a (u, m) par, the publc key to fd the cremetal sequece of the followg formula b = au(modm) Where u=t -. (4) Asked to solve the problem, we have establshed two dfferet algorthms. Ther sale at the base of the C ++ laguage. Each algorthm cossts of the preparato ad the ma part. Becomg part of the preparato of the formato - Hellma algorthm to ecrypt Merkle: We took { b, b2,... b} a creasg sequece, m module base ad have selected the t multpler, whch calculated the publc key a = b t(mod m) ad (3) the formula to be trasferred ecrypted formato. The frst algorthm to work as follows:. The data from the soluto set-caddates from the populato, whch s to take pctures of a radom geerator to talze. The tal soluto set-caddate of the Ascedg (closed key), whch s represeted a bary form. Its sze s equal to, where s the sze of the publc key, ad each of ts members (the gee) Sze 2d * -- + - created equal, where d s the proportoalty coeffcet, ad for each bt rate dex. Each member of the bts starts at. 2. Wth the help of a radom geerator for each of the Ascedg troducg m base (bary form), whose legth s equal to d * (t must be the result of more tha Ascedg elemets). 3. The populato of each of the caddate ad the correspodg soluto set-m base resettled to the decmal system. 4. Use exstg data, () obtaed from the formula Dophate equato b t km = a, where k = 2,3,4; We fd t a multtude of factors. 5. t multplcato of the may select oly those t, for whch t <m ad (t, m) =. 6. The soluto set-secret three caddates for () the formula calculatg the ope key. 7. Ftess fucto we set the qualty of the selecto crtera. I our case, the qualty of the selecto crtera, t s the 6th step of the Publc key elemets to match the Publc key elemets. If the umber of tems matchg the key legth s equal to or, the the results obtaed ad the algorthm s complete, but f you do ot match the umber of key elemets of the algorthm ad cotue movg to the ext step. 8. The selecto fucto of the selecto fucto becomes the L / 2 (L s the tal populato sze), the umber of caddates for the soluto set-elect, whose ftess fucto s hgher. 5
GESJ: Computer Scece ad Telecommucatos 206 No.2(48) ISSN 52-232 9. The soluto set-elected caddates carry out the fucto of crossover. Crossovg Fucto are as follows: A) radom geerator, the soluto set, choosg betwee two caddates, the caddate soluto set (the selecto s made so that breedg pars are ot repeated). B) Each caddate soluto set-dvded to two parts ad the adopted two ew soluto setcaddate. Usually, two paret soluto set-up as a caddate for the four successor, but our case we get oly two successor caddate soluto set, we are terested oly the read. Ascedg. 0. After Crossover operato, we calculate the soluto set-caddates' ftess fucto, f ay soluto set-caddate's ftess fucto s ot equal to, the repeat o the 2d, 3rd, 4th, 5th, 6th, paragraph 7, 8, 9 steps 5 tmes, f we got the desred result, we cotue to work o the algorthm, otherwse movg to the ext step.. The use of mutato fucto. The repeat the 0th steps ad decde upo ay results of the algorthm to work. The algorthm of the expermets showed that most of the ftess fucto does ot exceed / 2-'s (half the legth of the key). Ths result s ot the best, but ths result may be ecrypted formato from the dea. That's why we chaged our approach ad establshed a secod algorthm, whch work as follows:. The tal populato of the m base, whch talze take pctures of radom geerator (to be submtted a bary system), whle the populato of each member (soluto set-caddates) sze s d * ; Where s equal to the legth of the publc key, ad d = 2; 2. take pctures of the soluto set - Dozes of caddates for trasfer to the system. 3. Shamr algorthm take the frst four members of the publc key ad calculate the verse of multplcato t, where the m base u = p m / a, u = t, <= p <=, 0 <= < 4 ; * To select u multpler, we mpose certa restrctos. Besdes ths (um) = ad u <m, multpler u, multpled by the thrd member of the publc key, must be greater tha the base m. Whe you add ths restrcto, we reduce the set of u caddate-solutos. (u, m) pars all of the possble caddates for a ew key to the closed (3) formula. 4. Ftess fucto we set the qualty of the selecto crtera. I ths case, the qualty of the selecto crtera, the 4th step of the prvate key of the ascedg. If the sequece of the rse of the ftess fucto value s equal to, the the results obtaed ad the algorthm s complete, but f the ftess fucto - tha movg to the ext step. 5. Crossover fucto soluto set-elected caddates are carryg out the fucto of crossover. Ths algorthm s essetally dfferet from the prevous algorthm Crossovg fucto from the fucto. A) Radom geerator, the soluto set, choosg betwee two caddates, the caddate soluto set (the selecto s made so that breedg pars are ot repeated). B) Each caddate soluto set s dvded to two parts (the mddle pot) ad ther Crossovg results a four-caddate soluto set. 6. The ewly-adopted soluto set for caddates to repeat the 2d, 3rd, 4th steps, f ay of the caddate's ftess fucto soluto set-- created equal, the the desred result s obtaed, ad the applcato s closed, otherwse movg to the ext step. 7. the selecto fucto selecto fucto becomes the L (L s the tal populato sze) of the soluto set-selecto of caddates, whose ftess fucto s hgher. 8. selected soluto set-caddates for repeat steps 5 ad 6. We have dcated that the process wll repeat 0 tmes. If ths process s repeated, 0 tmes ad we do t get the desred result, oly ths case we use a mutato, or chage the fucto of the gee, ad the repeat the 2d, 3rd ad 4th steps. Whe we get the desred results, we stop workg. But tests showed that o f the mutatos feature s ot eeded, ad the hybrdzato of a maxmum of 5 tmes usg we get the desred result. a 52
GESJ: Computer Scece ad Telecommucatos 206 No.2(48) ISSN 52-232 Key legth s 20, the populato sze of 50 Iput data Secret key m t 23 27 268 535 47 232 4578 9290 859 3787 74365 48900 297807 59562 9223 2382452 4764898 952980 905965 38928 76238574 337 Publc kay 40799 42259 888956 774595 3804599 7668904 585226 3084930 6666347 470705 7952983 36469856 7296293 69680654 63947 5002062 2378848 4758028 908452 38099385 The result (aga after Secret key m t - crossover 76565 55033 9434097 355972053 989765234870785 2829323 operato) 482396448 964792896 954429202 43579572 9643464323 3854499999 64634809024 2792294623 29633070245 5507599942 562907823 272959735878 594568234578 2456789345670 25789087654329 223456789023456 Lead tme 8,72 sec. Expermetal results show that the use of geetc algorthms Merkle -Hellma crypto system s broke very quckly. Therefore, we ca coclude that the use of geetc algorthms wll be effectve for other asymmetrc crypto systems for cryptaalyss. Referece [] Merkle R.C., Hellma M.E. Hddg formato ad sgatures trapdoor Kapsak, IEEE Tras. Iform. Theory, IT-24 (978), pp. 535-530. [2] Martello, S. ad P. Toth, Kapsack Problems: Algorthms ad Computer Implemetatos, Joh Wley &; Sos, West Sussex, Eglad, 990. [3] A. Salomaa, Publc-Key Cryptography, Sprger-Verlag, 990. [4] Garg P, Shastr A. A Improved Cryptaalytc Attack o Kapsack Cpher usg Geetc Algorthm. Iteratoal Joural of Iformato Techology, 3(3) (2006) 6. [5] Muthureguatha R., Vekatarama D., Raasekara P. Cryptaalyss of Kapsack Cher Usg Parrallel Evolutoary Computg. Iteratoal Joural of Recet Treds Egeerg, Vol., No, 2009. [6] Shamr A. A Polomal-Tme Algorthm for Breakg the Basc Merkle-Hellma Cryptosystem. IEEE Trasactos o Iformato Theory. Vol., IT-30, No5, 984, pp. 699-704. [7] R. Geetha Rama Geetc Algorthm soluto for Cryptaalyss of Kapsack Cpher wth Kapsack Sequece of Sze 6. Iteratoal Joural of Computer Applcatos (0975 8887) Volume 35, No., 20. Artcle receved: 206-06-09 53