International Journal of Advanced Research in Computer Science and Software Engineering

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1 Volume 2, Issue 9, September 2012 ISSN: X International Journal of Advanced Researc in Computer Science and Software Engineering Researc Paper Available online at: Performance Enancement of RSA Cryptograpy Algoritm by Membrane Computing Sala Zaer, Amr Badr & Ibraim Farag Faculty of computer & information Cairo University, Egypt Tarek Abd Elmageed Governmental Security Consultant Egypt Abstract Te problem of using RSA algoritm in cryptograpy is te long time it takes for te encryption process. To overcome te problem, RSA algoritm uses sort messages. In tis paper we make analysis for te RSA public key protocol in te framework of membrane computing to develop a membrane model of RSA algoritm wit performance improvement. Te approac as been validated by developing a simulation program in C++ of RSA algoritm using te property of parallel computing of membrane environment to enance encryption/ decryption time of te RSA algoritm. Comparing te encryption time results using normal RSA algoritm and te developed model, it was proved tat te developed algoritm is faster tan te normal one by 30% approximately. Key words: Membrane Computing, RSA, public key, encryption, decryption. 1. INTRODUCTION Membrane computing is a branc of natural computing, te broad area of researc concerned wit computation taking place in nature and wit uman-designed computing inspired by nature. Besides systems biology tat tries to understand biological organisms as networks of interactions, and syntetic biology tat seeks to engineer and build artificial biological systems, anoter approac to understanding nature as computation is te researc on computation in living cells [1], [2]. Membrane computing abstracts computing models from te arcitecture and te functioning of living cells, as well as from te organization of cells in tissues, organs (brain included) or oter iger order structures suc as colonies of cells (e.g., bacteria).te initial goal was to learn from cell biology someting possibly useful to computer science, and te area quickly developed in tis direction. Several classes of computing models were defined in tis context, inspired from biological facts or motivated from matematical or computer science points of view. A number of applications were reported in te last few years in several areas biology, Bio-medicine, linguistics, computer grapics, economics approximate optimization, cryptograpy, etc. Te models investigated in membrane computing area are called P systems. Te main components of a P system are (i) te membrane structure, (ii) te mustiest of objects placed in te compartments of te membrane structure, and (iii) te rules for processing te objects and, te membranes. Tus, membrane computing can be defined as a framework for devising cell-like or tissue-like computing models wic process multisets in compartments defined by means of membranes [3].Te membrane structure consisting of several membranes arranged in a ierarcal structure [4].A membrane structure is represented by a Venn diagram (or a rooted tree) and is identified by a string of correctly matcing parenteses, wit a unique external pair of parenteses corresponding to te external membrane, called te skin. A membrane witout any oter membrane inside is said to be elementary. Te following Example from [5] illustrates te situation: te membrane structure in Fig.1 is identified by te string. μ = [1 [ 2 [ 5 ] 5 [6 ] 6 ]2 [ 3 ]3 [4 [7 [8 ]8 ] 7 ]4 ] 1 A P system wit active membranes is a construct[6] Fig.1.A membrane structure and its associated tree. 2012, IJARCSSE All Rigts Reserved Page 10

2 Sala et al., International Journal of Advanced Researc in Computer Science and Software Engineering 2 (9), Π = (V, H, μ, w1, wm, R), Were: (i) M >= 1; (ii) V is an alpabet; (iii) H is a finite set of labels for membranes; (iv) μ is a membrane structure, consisting of membranes, labeled (not necessarily in a one-to-one manner) wit elements of H; all membranes in μ are supposed to be neutral; (V) w1... wm are strings over V, describing te multi sets of objects placed in te regions of μ; (vi) R is a finite set of developmental rules (a) a [ a v], for H, {,,0}, a V, v V (object evolution rules, associated wit membranes and depending on te label and te carge of te membranes, but not directly involving te membranes, in te sense tat te membranes are neiter taking part to te application of tese rules nor are tey modified by tem); (b) 2 a[ ] for 1 [ b] H, {,,0}, a, bv, 1 2 (Communication rules; an object is introduced into te membrane, may be modified during tis process; also, te polarization of te membrane can be modified, but not its label); (c) (d) 1 2 [ a] [ ] b, for, 1, 2 H {,,0}, a, bv (Communication rules; an object is sent out of te membrane, maybe modified during tis process; also, te polarization of te membrane can be modified, but not its label); [ a] b, for H, {,,0}, a, bv (Dissolving rules; in reaction wit an object, a membrane can be dissolved, leaving its entire object in te surrounding region, wile te object specified in te rule can be modified); 2 3 (e) 1 [ ] [ ] [ ] a b c, for, 1, 2, 3 H {,,0}, a, b, c V (Division rules for elementary membranes; in reaction wit an object, te membrane is divided into two membranes wit te same label, may be of different polarizations; te object specified in te rule is replaced in te two new membranes by possibly new objects; all te oter objects are copied into bot resulting membranes); (f) [ [ 0 ]...[ ] [ ] 1 1 k k k 1 k 1...[ for n k 1, i H,0 i n, n ] ] n 2 3 [ [ ] k 2,..., 6 {, and,0} [ ] ] [ [ ]...[ ] ], (Division of non-elementary membranes; tis is possible only if a membrane contains two immediately lower membranes of opposite polarization, + and ; te membranes of opposite polarizations are separated in te two new membranes, but teir polarization can cange; all membranes of opposite polarizations are always separated by applying tis rule) ; k 0 0 k 1 k1 n n 0 If te membrane labeled 0 contains oter membranes tan 1,..., n specified above, ten tey must ave neutral carges in order to make tis rule applicable; tese membranes are duplicated and ten become part of te content of bot copies of membrane 0; P automata, wic are symport/antiport P systems wic accept strings: te sequence of objects (because we work wit strings and symbol objects, we use intercangeably te terms object" and symbol") imported by te system from te environment during a alting computation is te string accepted by tat computation (if several objects are brougt in te system at te same time, ten any permutation of tem is considered as a substring of te accepted string; a variant, is to associate a symbol to eac multiset and to build a string by suc marks" attaced to te imported mustiest) [7]. 2012, IJARCSSE All Rigts Reserved Page 11

3 Sala et al., International Journal of Advanced Researc in Computer Science and Software Engineering 2 (9), A sequence of transitions constitutes a computation. A computation is successful if it alts; reaces a configuration were no rule can be applied. If we ave an output region specified, ten we count te objects present in te output regions in te alting configuration and tis number is te result of computation [8]. 2. RSA ALGORITHM RSA sceme is a block ciper in wic te plain text and ciper text are integers between 0 and n- 1 for some n [9]. 2.1 Description of te Algoritm RSA algoritm is specified as in [10], [11], [12], [13]. Key setup To setup te key material, user Alice performs te following steps: 1- Eac user generates a public/private key pair by selecting two large primes at random p, q. 2- Compute N=p.q 3- Compute ( N) ( p 1) ( q 1) 4- Selecting at random te encryption key e Were 1 e ( N),gcd( e, ( N)) 1 5- Solve te following equation to find decryption Key d were e d 1mod ( N) and 0 d N 6- Publis teir public encryption key KU { e, N} 7- keep secret private decryption key KR { d, q, p} Encryption To send a message m < N to Alice, te sender Bob creates te ciper text c as follows: e c m (mod N) Decryption d To decrypt te ciper text c. Alice computes c m (mod N). 2.2 Modular Exponentiation RSA uses fast exponential algoritm called "repeated square and multiply" te algoritm repeats te following process: dividing te exponent into 2, performing square and performing an extra multiplication if exponent is odd. Te algoritm is specified as in [14],[15]. Modular Exponentiation algoritm INPUT x, y, n: integers wit x 0, y 0, n 1; OUTPUT x y (modn). Mod_ exp (x, y, n) 1. If y = 0 return (1) ; 2. If y (mod 2) = 0 return (mod_exp(x 2 (mod n), y 2, n); 3. Return (x.mod_exp ((x 2 (mod n), y 2, n) ; 3. COMPACT ENCODING Te natural encoding is easy to understand and work wit; owever it as te disadvantage tat for very large numbers te membrane systems sould contain a very large number of objects, undesirable for practical reasons. We discuss compact encodings were eac object of a membrane system is represented in a more compact way, similarly to numbers in base 2 or iger. Tese compact encodings require notions and results from combinatorics over multisets. To develop encoding and decoding algoritms, we start considering te number n in base b represented by using m objects. To develop encoding and decoding algoritms, we start considering te number n in base b represented by using m objects. As a first step we must determine m, te encoding lengt, ten we look for te first (lowest) number 2012, IJARCSSE All Rigts Reserved Page 12

4 Sala et al., International Journal of Advanced Researc in Computer Science and Software Engineering 2 (9), m represented using m objects. i=0 Tis number is 1 N (b, i) te count of all numbers represented using less tan m objects, and it is lower or equal to n. Tus we determine m from te following equation: i0 m 1 i0 N ( b, i) n 0 TABLE1. MOST COMPACT ENCODINGS (MCE) IN BASES 1,2,3. Decimal MCE1 Natural encoding MCE 2 MCE Successor in MCE Time complexity: O (1). Te successor of a number ii tis encoding is computed in te following manner: eiter we ave an object 0 aid te rule 0s 1, transforms tis 0 in to an 1, or we ave a number encoded using only objects 1 ten aid te rule 1 0 s transforms all 1s into 0s; moreover te rule s 0 produces an additional Predecessor in MCE Time complexity: O (1). Te predecessor of a number is computed by turning an 1 in to a 0 by te rule 1s 0 wenever we ave objects 1; oterwise we consume one 0 by te rule 0s u, aid transform all te oter objects 0 into 1. By rule 0 1 u. 3.3 Multiplication MCE Time complexity: O (n1 n2) = O (n2) if n1 = n2 =n. We implement multiplication in a similar manner to addition, coupling a predecessor wit an adder. Te idea is to provide te first number to a predecessor, and perform te addition iteratively until te predecessor reaces 0. Te evolution is started by te predecessor working over te first number. Te predecessor activates te adder by passing a communication token. Te adder is modified to use an extra backup membrane wic always contains te second number. Wen te adder is triggered by te predecessor, it signals te backup membrane wic supplies a fres copy of te second number to te adder, and a new addition iteration is performed. At te end of te iteration. At te end te adder sends out a token to te predecessor. Te procedure is repeated until te predecessor reaces. 2012, IJARCSSE All Rigts Reserved Page 13

5 Sala et al., International Journal of Advanced Researc in Computer Science and Software Engineering 2 (9), 4. RSA ENCRYPTION IN MEMBRANE COMPUTING Te MCE multiplication can be used in calculating te encryption equation of RSA: C m e (mod N) Using Modular Exponentiation as dividing te exponent into 2, performing square and performing an extra multiplication wit membrane computing system. Te evolution process of te P system for RSA encryption is sown in Fig. 2. Te P system model of RSA encryption can be described as follows. Π = (V, H, μ, w1, wm, R), M= 4 H= a b c u=[1[2]2[3]3[4]4]1 V1= {[x,1,1]} V2= {[x,2,1],[x,2,2]} V3= {[x,3,1]} V4= {[x,2,2],[x,1,1],[x,2,1]} R1= {[x,1,1]} R2= {([x,2,1]^2) % [x,2,2]} R3= {BitSiftRigt([x,3,1])} R4= {([x,1,1]*[x,2,1]) % [x,2,2]} A modular power function wit tree parallel treads as described in Fig. 3 is used in simulating te encryption equation using te property of parallelism of membrane computing. Te pseudo-code of te function is described as follows: Function modular_pow(base, exponent, modulus) result := 1 wile exponent > 0 if (exponent mod 2 == 1): result := (result * base) mod modulus exponent := exponent >> 17 base = (base * base) mod modulus return result Fig.2. Te evolution process of te P system for RSA encryption. 2012, IJARCSSE All Rigts Reserved Page 14

6 Sala et al., International Journal of Advanced Researc in Computer Science and Software Engineering 2 (9), Fig. 3. Parallel treads description. 5. EXPREMINTAL RESULTS Te performance of te proposed algoritm of RSA using membrane computing is simulated using software written in C++(using visual c++) We compared te performance to te performance of normal RSA on te same platform. We used 3.2 giga ertz Intel processor wit one giga byte ram. Te results is sown in table 2. Te encryption time in seconds was recorded as a function of number of decimal digits of te two primary numbers products (N) in decimal digits. Te results is graped as in Fig. 4. TABLE 2 COMPARISON BETWEEN ENCRYPTION TIME OF NORMAL RSA AND MEMBRANE RSA Number of Decimal Digits N(digits) Average Normal Time T n (ms) Average Membrane Time T m (ms) , IJARCSSE All Rigts Reserved Page 15

7 Sala et al., International Journal of Advanced Researc in Computer Science and Software Engineering 2 (9), Fig.4 Comparison between encryption times of normal RSA and membrane RSA 6. CONCLOUSION AND FUTURE WORK In tis paper, we propose an analysis of RSA using te membrane computing environment. We make a comparison wit normal RSA and our proposed system can improve te performance of te encryption time of te algoritm. te proposed algoritm also enance te security of RSA by using large prime numbers and large exponents in addition to proving te ability of membrane computing in performing massive calculations wit ig performance, in relatively sort time. REFERENCES [1] D. Endy. Foundations for engineering biology. Nature, pp 438: , (2005). [2] J. Dassow, G. P_AUN "Journal of Universal Computer Science, vol. 5, no. 2, pp 33-49", (1999). [3] O. H. Ibarra, G. Paun Membrane Computing: General View Te European Academy of Sciences, (2007). [4] G. Paun, Membrane Computing: An introduction Springer Verlag, Berlin, ISBN: , (2002) [5] P. Sosik, Alfonso Rodriguez-Patton Membrane computing and complexity teory: A caracterization of PSPACE Journal of Computer and System Sciences73, pp , (2007). [6] H. Adorna, G. Paun, M. J. PEREZ- JIMENEZ " On Communication Complexity in Evolution- Communication P Systems "Romanian Journal Of Informartion " Volume 13, Number 2, pp , (2010). [7] G. P aun, M. J. Perez Jimenez "Solving Problems in a distributed Way in Membrane Computing: DP systems" Int. J. of Computers, Communications & Control, ISSN , E-ISSN Vol. V, No. 2, pp , (2010). [8] G.Paun Application of Membrane Computing Springer- Verlag, Berlin, ISBN: (2002). [9] W. Stallings" Cryptograpy and Network Security Principles and Practices", fift Edition, ISBN: (2011). [10] C. Paar.jan pelzl "understanding cryptograpy "Spring Verlag,Berlin, ISBN : , (2010). 2012, IJARCSSE All Rigts Reserved Page 16

8 Sala et al., International Journal of Advanced Researc in Computer Science and Software Engineering 2 (9), [11] B. Esslinger "Te cryptool script cryptograpy, matematics and more", available at [ttp: // (2010). [12] B. Scneier "Applied cryptograpy Protocol, Algoritms and Code in C" ISBN: (1996),. [13] J. Katz and y. lindel "introduction to modern cryptograpy", ISBN: , (2008). [14] W. mao "modern cryptograpy, teory and practice ", ISBN , (2004). [15] R. Oppliger "Contemporary Cryptograpy" Artec House, ISBN , (2011). 2012, IJARCSSE All Rigts Reserved Page 17