PERSPECTIVES OF THE USE OF GENETIC ALGORITHMS IN CRYPTANALYSIS

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1 PERSPECTIVES OF THE USE OF GENETIC ALGORITHMS IN CRYPTANALYSIS Lal Besela Sokhum State Uversty, Poltkovskaa str., Tbls, Georga Abstract Moder cryptosystems aalyss s a complex task, the soluto of whch s estmated to exceed the volume of cryptaalyst real opportutes. Evolutoary algorthms, amely geetc algorthms, have a opportuty to dramatcally reduce the umber of assumptos. Merkle - Hellma cryptosystem s a publc key cpher, whch s based o the famous kapsak task. Splma was frst,who used a geetc algorthm to break the cpher, the a scetst has expaded ts work ths drecto. I these works carred out kow-platext attack. Yet there s the Shamr s polyomal-tme algorthm, whch the successfully breakable Merkle - Hellma system, fds the secret key va the publc key. Despte the fact that the Shamr s algorthm s polyomal-tme, for breakg a real system eed to perform a large amout of computato. The paper ams to demostrate the advatages of the use of geetc algorthms for cryptoaalyss compared wth other methods. Drectly the publc key cryptalgorthms cases, where we ca fds the secret key va the publc key. I ths sese, we have created two separate ad dfferet from the other algorthms algorthm (the geetc lbrares) have mplemeted ths algorthm attack Merkle - Hellma cryptosystem. The obtaed results allow us to coclude that the use of geetc algorthms for cryptaalyss s effectve ad what we have descrbed a algorthm for fdg the secret key s much faster tha the Shamr s algorthm. IdexTerms: geetc algorthms, cryptosystem, the system Merkley-Hellma cryptaalys. I. INTRODUCTION Merkley - Hellma Cryptosystem s a publc-key cpher based o the well-kowproblem kapsack []. Splma frst used a geetc algorthm to crack ths code [2]. Severalother researchers [3,4,5] have expaded the work ths drecto cocetratg maly o thetal parameters of geetc algorthms. At the same tme, all these papers, thsproblem s solved by fdg the decrypto of the cphertext wthout the secret key.we put the problem of computg the secret key from the publc key usg the attacks based o the kow platext usg a geetc algorthm. The results of such a attack ca be compared wth the results of the attack algorthm A. Shamr [6] verfes whether such approach provdes a real advatage. II. DENTIFIER MERKLEY - HELMAN A. The problem of the kapsack. Kapsack problem ca be formally stated as follows. Da backpack volume ad a set of objects, each wth a volume of Our goal s to fd a subset of the set for whch the equalty V = x a 56

2 where x {0, }, for all I geeral, the presece of such a subset requres exhaustve search, whose complexty s O(2 ), however, there s a specal case whe the problem s elemetary solved. B. The calculato of the publc key from the prvate kay. The algorthm Merkley - Hellma as a secret key takes codto j sequece, where each > a a j j= For ths sequece, there s a algorthm that solves the problem of the kapsack ad complexty, whch s a polyomal of (or rather lear). The publc key of the prvato, prepared by the followg trasformato. Choose a umber m, whch satsfes the codto m > a ad the umber, where, ad elemets of the publc key s calculated by the formula: b ta (mod m). C. Ecrypto ad decrypto. Message s coverted to a bt sequece ad dvded by bts. Calculate the sum of. s, s2,... sl formula = s j = x b, j blocks. Each block cotas exactly where aj s the - the elemet of the block of the ope test. s, s2,... sl ad there s a amout - cphertext. To decrypt each sum s multpled by modulo ad the problem s solved by usg a kapsack creasg over sequece for each. III SHAMIR ALGORITHM Shamr broke Merkl-Hellma algorthm whch showed that t s ot ecessary to search for t couple ad the exact over creasg sequece, whch has bee ecrypted. It s eough to fd a ad a par of over creasg sequece, whch s derved from the sequece, va followg formula a = b t (mod m). Shamr attack us kapsack system algorthm whch cossts of two parts. To a whole umber, for whch the codto s satsfed, the value for some s located these fuctos mmum terval. After accumulatg a certa umber of pots, algorthm s lookg for par va Dophate approxmato method, whch may ope the key to to calculate the secret key. IV. GENETIC ALGORITHMS Geetc algorthm based o the mechasm of atural selecto. It was frstly proposed by Hollad 975. The algorthm starts wth a radom selecto of caddates solutos (aalogues 57

3 of chromosomes), represeted bary strgs. By applyg geetc operators of selecto, crossover ad mutato to each geerato of caddates solutos to get a ew geerato of caddates solutos. Thus, these rows are a ew geerato average better tha the prevous (depedg o the choce of the target, or as ofte referred to, a ftess fucto).the most mportat advatage s the ablty of geetc algorthms parallel to the realzatos of the process of fdg a soluto that sgfcatly reduces the attack. IV. ALGORITHM Our method of attack s qute dfferet from the methods used the above metoed works. Besdes, we created a geetc algorthm qute dfferet from other geetc algorthms (dfferet the selecto crtero ad crossover process). Our geetc algorthm s descrbed fle geetc2.h created by us. I geetc class of the fle four fuctos are descrbed: the ftess fucto (bool ftess(vector<populatca>&v)), the crossover fucto (vod crossover (vector<populatca>&v)), the mutato fucto (vod ftess(vector<populatca>&v)) ad the selecto fucto (vod selektca(vector<populatca>&v)). a) The ftess fucto determes the extreme crease each member (soluto-caddate) of the populato trasmtted to t. The ftess value of the soluto-caddate the sequece s equal to the quatty of the extremely creasg members. b) The selecto fucto chooses the selecto-caddates, whch the most fulfll the ftess fucto,.e. ther ftess values are hgher tha those of others. I case the populato sze s Lwe choose oly L/5 soluto-caddates. Exactly these soluto-caddates form ew geeratos. c) The crossover fucto receves the populato of the soluto-caddates. From ths populato we choose soluto-caddates wth t ad t2 umbers pars by meas of a radom geerator takg to accout that t ad t2 do ot cocde wth each other ad the used par s ot repeated. Each soluto-caddate s dvded two parts (at the mdpot). d) The mutato fucto chages oe byte of each soluto-caddate. We choose the dex by the radom geerator ad chage the relevat bt,.e. f the bt value s zero t s chaged, ad o the cotrary, f t s, the t s chaged zero. Our goal s, usg the above descrbed algorthm, to fd such (u,m) par, by whch we wll be able to fd the extremely creasg sequece by the followg formula: b = au(mod m), whereu=t - (3) The algorthm s realzed o C++ laguage base. It cossts of the preparato ad ma parts. I the preparato part the formato-to-be-trasmtted s cphered by the Merkle-Hellma algorthm. We took { b, b2,... b} extremely creasg sequece, m module root ad selected t multpler, by meas of whch we calculated ope key a = b t(mod m) ad cphered the formato-to-be-trasmtted by the formula (3). The workg chart of the ma algorthms s as follows:. The tal populato s represeted by m root, whch s talzed by radom geerator (t s represeted bary system). The sze of each member (soluto-caddate) of the populato s d*, where s equal to the legth of the ope key ad d=2. The soluto-caddates are trasformed to bary system. 2. Lke the Shamr algorthm we take the frst four members of the ope key ad calculate the verse of t multpler by m root u = p * m / a, where u = t, <= p <= a, 0 <= < 4. Thus, we receve the populato all probable multplers. We set lmts for selectg u multpler. Besdes (u,m)=ad u<m, u multpler multpled by the thrd member of the 58

4 ope key must exceed m root. By addg ths lmt we reduce the soluto-caddates of u multpler,.e. make the algorthm more purposeful. We fd the relevat closed key by (3) formula for all probable caddates of (u,m)par. 3. We determe the crtero for selecto by the ftess fucto. I ths case the crtero for selecto s the extreme crease the closed key obtaed as a result of the fourth phase. I case the sequece s extremely creasg,.e. the value of the ftess fucto s less tha, we pass to the followg phase. 4. By meas of the crossover fucto we carry out the crossover operato for the chose soluto-caddates. 5. For the receved soluto-caddates the secod, thrd ad fourth phases are repeated. I case the ftess fucto of ay soluto-caddate s equal to, t meas the desred result s obtaed ad the program stops fuctog. Otherwse, we pass to the followg phase. 6. The selecto fucto chooses the L/5 (L s the sze of the tal populato)umber soluto-caddates, the ftess fuctos of whch are hgher. 7. 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 table shows the results of the expermets. Kkey legth Populato sze The umber of expermets carred out Repeatg the average umber of geetc operators (crossover operato) Average executo tme st st st st Accordg to the expermet results t s obvous that by the use of geetc algorthms the Merkle-Hellma cryptosystem s cracked qute quckly. Cosequetly, we may coclude that use of geetc algorthms wll be useful for cryptaalyss of other asymmetrc cryptosystems as well. REFERENCES [] S Merkle R.C., Hellma M.E. Hddg formato ad sgatures trapdoor Kapsak, IEEE Tras. Iform. Theory, IT-24 (978), pp [2] Spllma R. Cryptaalyss of Kapsack cphers usg geetc algorthms. Cryptologra, October, 993. [3] Yasee, Sahasrabuddhe. A Geetc Algorthm for cryptaalyss of Chor Rvest Kapsack Publc key cryptosystem, Thrd teratoal coferece o computatoal tellgece ad multmeda applcatos, 999. [4] Garg P., Shastr A. A Improved Cryptaalytc Attack o Kapsack Cpher usg Geetc Algorthm. Iteratoal Joural of Iformato Techology 3:

5 [5] Muthureguatha R., Vekatarama D., Rajasekara P. Cryptaalyss of Kapsack Cher Usg Parrallel Evolutoary Computg. Iteratoal Joural of Recet Treds Egeerg, Vol., No, May [6] Shamr A. A Polomal-Tme Algorthm for Breakg the Basc Merkle-Hellma Cryptosystem. IEEE Trasactos o Iformato Theory Vol., IT-30, No5, september 984, pp Artcle receved: