Novel Encryption Schemes Based on Catalan Numbers

Transcription

1 D. Sravaa Kumar, H. Sueetha, A. hadrasekhar / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: Novel Ecryptio Schemes Based o atala Numbers 1 D. Sravaa Kumar H. Sueetha 3 A. hadrasekhar 1 Reader i Physics, SVLNS Govermet ollege, Bheemuipatam, Visakhapatam Dt., Idia Assistat Professor i Egieerig Mathematics, GITAM Uiversity, Visakhapatam, Idia, 3 Professor i Egieerig Mathematics, GITAM Uiversity, Visakhapatam, Idia Abstract:- Applied Number theory has so may applicatios i cryptography. Particularly iteger sequeces play very importat i cryptography. Earlier cryptographic algorithms were desiged usig the iteger sequeces of Fiboacci umbers ad Lucas umbers. I the preset paper ovel cryptographic algorithms were proposed basig o iteger sequeces of atala umbers. Key Words:- atala umber, ecryptio, decryptio Itroductio:- Biomial oefficiets:- Biomials which are sums of two terms always occur i mathematics. There is symmetric way of expadig the positive itegral powers of biomials. Let ad r be oegative itegers. The biomial coefficiet is defied as r! where 0 r, if r the r r!( r)! r is defied as zero. Iteger Sequece: - A iteger sequece is a sequece of itegers which may be specified explicitly by givig a formula for its th term, or implicitly by givig a relatioship betwee its terms. For example, the sequece 0, 1, 1,, 3, 5, 8, 13, (the Fiboacci sequece) is formed startig with 0 ad 1 ad the addig ay two cosecutive terms to obtai the ext oe: a implicit descriptio. The sequece 0, 3, 8, 15, is formed accordig to the formula 1 for the th term: a explicit defiitio. Defiitio:- atala umbers [1] were discovered by Belgia mathematicia Eugee. atala i atala umbers are sequece of atural umbers. The atala umber is defied as! 1, 0 ( 1)!! 1 Every atala umber is a iteger. The various atala umbers are 1,1,,5,14,4,13,49,1430,486,.. Defiitio: - The secod way of defiig the atala umber is!! 1 (!) ( 1)!( 1)!!!( 1)! This defiitio cofirms that every atala umber is a iteger. Seger s Recursive Formula:- I 1761, Seger usig the additio ad multiplicatio priciples, published a secod order recurrece formula T, where if greater tha or equal to 3. The Seger s recurrece formula ca be developed usig a -go with vertices A 1,A.A. The Seger s recurrece formula T is 1 T T T T T T T... T T, 3 k k k Defiitio:- From Euler s triagulatio problem the atala umber ca also be defied [3] recursively as = 1, 1 = 1, = 1, 0.(1) I 1941 H. Urba coectured the recurrece relatio. He first computed 3, 4, 5 ad the he made the followig observatios P a g e

2 D. Sravaa Kumar, H. Sueetha, A. hadrasekhar / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: With this clear patter Urba iferred that = 1, 1...(1) Explicit formula of :- From the recursive formula the explicit formula of atala umber ca be derived as (4)(46) ( 1) (4 )(4 6)(4 10) ( 1) ( 1) A approximate value of atala umber ca be foud usig Stirlig s approximatio for factorials as ( 1) Ubiquitous Nature of atala Numbers :- Like Fiboacci umbers ad Lucas umbers, atala umbers are also ubiquitous [7]. atala sequece is the most frequetly ecoutered sequece presets everywhere i combiatorics. There are may coutig problems i combiatorics whose solutio is give by atala umbers. Some examples are 1.Fidig the umber of lattice paths of moutai rages with steps (1,1) ad (1,-1) that ever fall below the x-axis which are called Dyck paths.. Successive applicatios of biary operatios ca be represeted i terms of full biary tree. There the atala umber is the umber of full biary trees with +1 leaves. 3. I parethesizatio problem usig postfix expressios ad cyclic shifts of such exprssessios. 4. I the theory of partitioig i ombiatoraics. 5. I Abstract algebra ad sports 6. I Pascal s triagles ad iomputer Sciece. Properties of atala umbers:- Parity of atala umbers:-odd ad Eve atala umbers [6] Theorem:- For > 0, is odd if ad oly if is a Merseer umber. Proof:- It follows from the Seger s recurrece relatio that ( , if is eve 1 ( , otherwise osequetly for > 0, is odd if ad oly if both ad 1 1 umber is odd if ad oly if ad 3 are both odd or Primality of atala umbers:- Theorem:- The oly prime atala umbers[] are ad 3. (+) =(4+) 4 are odd. From this it implies that the atala 1 is zero. Proof :- From the formula (1) 1. Let be prime for some. It follows that if 16 P a g e

3 D. Sravaa Kumar, H. Sueetha, A. hadrasekhar / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: > 3, < 1. So,. osequetly, so 1 k for some positive iteger k. The 1 (4+)=k(+) where 1 k 3 ad thus 4. Therefore, it follows that ad 3 are the oly atala umbers that are prime. L.W. Shapiro of Howard Uiversity itroduced triagular array of umbers B(,r) alled atala triagle. 1, if r = 1 = B(, r) B(-1,r-1) +B(-1,r)+B(-1,r+1), if 1 r 0, otherwise Geeralizatio of atala umbers:- atala umbers are special class of umbers 1 (k+1) (,k) defied by (,k) = where k 0 k+1 learly (,1) =. Some atala Numbers for =1 to 0are give below i the table for ready referece. As the value icreases the umber of digits i icrease rapidly. For example umber of decimal digits i 100 = Applicatio of Number Theory i ryptography:- Applied Number theory has so may applicatios i cryptography [8,9]. ryptographic algorithms were desiged usig Fiboacci umbers ad Lucas umbers. The geeral idea of the Fiboacci cryptography is similar to the Fiboacci codig ad based o the applicatio of the geeralized Fiboacci matrices. Lucas sequeces ca also be used for public key cryptosystems ad sigature systems similar to RSA, but usig Lucas sequeces modulo a composite umber istead of expoetiatio. Proposed Method 1:- Both the commuicatig parties before commuicatig the messages agree upo to use a large radom umber. The they write the correspodig atala umber which is a very large atural umber. Step 1. The seder divides the text message ito data blocks of m characters each, where m is a atural umber equal to the umber of decimal digits i the atala umber. The size of message always may ot be of m characters or multiples of m characters. If the message cotais less tha m characters the the seder adds three hash (###) characters at the ed of the message represetig that the message space is over ad the remaiig characters may be filled at radom. Step :- osider the first data block. All the m characters of the first data block are coded to equivalet biary umbers usig ASII code table. Let the ASII biary code of the th character M be b 7 b 6 b 5 b 4 b 3 b b 1 b P a g e

4 D. Sravaa Kumar, H. Sueetha, A. hadrasekhar / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: Step 3:- Let the th digit of the key i.e., the atala umber be K. The compute K K 4(mod8). Logical NOT Operatios are applied o b l ad b ll ll k k Example: Let the M has the biary code ad let K be 9 the compute l ll K K (mod8) l bits of the umber. K K (mod8) 9(mod8) 1ad K K 4(mod8) 13(mod8) 5. The the logical NOT operatio is applied o b 1 ad b 5. The the umber M is ecrypted as This ecrypted umber is coded back to text characters usig ASII code table. Step 4:- The procedure explaied i steps ad 3 is applied to all the 8 bit biary codes of the characters i the th data block. This procedure is applied to all data blocks. Step 5 :- The this cipher text is commuicated to the receiver i public chael. Decryptio:- The receiver after receivig the cipher text decrypts the cipher text as follows: Step 1:- The receiver divides the cipher text ito data blocks of m characters each. Step :- He decrypts the message by ecryptig the cipher text as explaied i the ecryptio algorithm. If the receiver fids three cosecutive hash (###) characters i the cipher text the he recogizes that the message is over ad the discards the remaiig cipher characters. ad Example:- Ecryptio:-Suppose that the commuicatig parties agree upo to use the radom atural umber =7 i the process of commuicatio. The correspodig atala umber 7 = 49. The message is divided ito blocks of 3 characters each. Let the message be rrr. This message costitutes oly oe text block i.e. rrr, as m=3 i the selected example. K 4(mod8) 4, K 4 4(mod8) = K (mod8), K 4(mod8) = 6 K 9(mod8) 1, K 9 4(mod8) = The biary equivalet of r from the ASII code table is The first character r is coded as which correspods to the character c i the ASII code table. The secod character r is coded to which correspods to the character 6 i the ASII code table. The third character r is coded to which correspods to P i the ASII code table. So, the same character r i the message is coded to differet characters i this proposed method. rrr is ecrypted as c6p. Here a simple case of a small radom atural umber = 7 is cosidered as the example. But, i practice very large radom atural umber should be selected so that cotais large umber of decimal digits. Such selectio esures m is large. For example if = 100, m =57 Decryptio:- The receiver receives the cipher text c6p. He computes 7 ad calculates K 4(mod8) 4, K 4 4(mod8) = K (mod8), K 4(mod8) = 6 K 9(mod8) 1, K 9 4(mod8) = 5 He writes the correspodig biary code for c6p. The 3 1 correspodig biary code of the character c is To decrypt this biary umber he ecrypts it usig the ecryptio algorithm. This results i decrypted code correspodig to the message character r. Similarly the other characters i cipher 6 ad P are decrypted to r ad r. So, the receiver recovers the origial message rrr. Proposed Method II: - Both the commuicatig parties before commuicatig the messages agree upo to use a large radom umber. The they write the correspodig atala umber which is a very large atural umber. Step 1. The seder divides the text message ito data blocks of m characters each, where m is a atural umber equal to the umber of decimal digits i the atala umber. The size of message space may ot be of m characters or multiples of m characters. If the message cotais less tha m characters the the seder adds three hash (###) characters at the ed of the message represetig that the message space is over ad the remaiig characters may be filled at radom. Step :- osider the first data block. All the m characters of the first data block are coded to equivalet biary umbers usig ASII code table. Let the ASII biary code of the th character M be 164 P a g e

5 D. Sravaa Kumar, H. Sueetha, A. hadrasekhar / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: b 7 b 6 b 5 b 4 b 3 b b 1 b 0 Step 3:- Let the th digit of the key i.e., the atala umber be K. The K which is a decimal digit is coverted to 4bit biary umber. The the th digit M which is i the 8 bits biary format is XORed with 841 code of K K. For example, let the biary code of 4th digit of the message be ad the 4 th digit of the key be 5. The is XORed bit wise with the 841 code of 55. ( ) XOR ( ) = Step 4:- The procedure explaied i steps ad 3 is applied to all the characters of each data block which are i biary format. Step 5:- The this cipher text is commuicated to the receiver i public chael. Decryptio: - The receiver after receivig the cipher text decrypts the cipher text as follows: Step 1:- The receiver divides the cipher text ito data blocks of m characters each. Step :- He decrypts the message by ecryptig the cipher text as explaied i the ecryptio algorithm. If the receiver fids three cosecutive hash (###) characters i the decrypted text the he recogizes that the message is over ad the discards the remaiig cipher characters. Example:- Ecryptio:-Suppose that the commuicatig parties agree upo to use the radom atural umber =7 i the process of commuicatio. The correspodig atala umber 7 = 49. Here K 1 =4. The correspodig 841 code is The 841 code of 44 = K =. The correspodig 841 code is The 841 code of = K 3 =9. The correspodig 841 code is The 841 code of 99 = Let the message be rrr. This message costitutes oly oe text block i.e. rrr, as m=3 i the selected example. The biary equivalet of r from the ASII code table is The first character r is ecrypted as XOR = This biary code correspods to the cipher character 6 i the ASII code table. The secod character r is ecrypted as XOR = This biary code correspods to the cipher character P i the ASII code table. The third character r is ecrypted as XOR = This biary code correspods to the cipher character ё i the ASII code table. So, the same character is coded to differet characters i this proposed method. rrr is ecrypted as 6P ё. Here a simple case of a small radom atural umber = 7 is cosidered as the example. But, i practice very large radom atural umber should be selected so that cotais large umber of decimal digits. Such selectio esures m is large. For example if = 94, m =54 Decryptio:- The receiver receives the cipher text 6P ё. The ASII biary codes of the characters of cipher message are , ad He computes 7 =49 ad recogizes K 1 =4,K = ad K 3 =9. The correspodig 841 codes are 0100, 0010 ad 1001 respectively. He calculates the strigs required to decrypt the first, secod ad third characters i the cipher message as , ad He decrypts the cipher as follows. 1) XOR = This biary code correspods to the cipher character r i the ASII code table. ) XOR = This biary code correspods to the cipher character r i the ASII code table. 3) XOR = This biary code correspods to the cipher character r i the ASII code table. So, the receiver retrieves the origial message rrr from the cipher text. oclusios:- I the proposed method of ecryptio usig atala umbers a large radom atural umber is agreed upo by the commuicatig parties as the secret key. The decimal digits i the atala umber correspodig to the agreed upo secret key are used to ecrypt/decrypt the message. The proposed two methods of ecryptio use logical NOT ad logical XOR operatios o the bits of the ASII biary codes of the characters i the message. The message is divided ito data blocks of size m characters each, where m is the umber of decimal digits i the atala umber. I the algorithms proposed decryptio is effected by ecryptig the cipher text, i.e. ecryptio of plai text gives cipher text ad ecryptio of cipher text gives the plai text. The ecryptio algorithms proposed usig atala umbers resistat to most of the types of kow attacks such as plai text attacks ad chose cipher text attacks [4]. The idetical characters of the plai text are coded to differet cipher characters. Hece, the brute force attack [5] ad exhaustive key search are difficult to execute. The time for ecipherig or decipherig is idepedet of characters i the data block. The time required to ecipher or decipher a data block is same for all data blocks. Eve though the origial message cotais the characters whose umber is less tha the umber of digits of the atala umber the remaiig characters are filled at radom, so that each data block 165 P a g e

6 D. Sravaa Kumar, H. Sueetha, A. hadrasekhar / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: cotais exactly the same umber of characters. Therefore the cipher proposed here is less proe to timig attacks. The mai advatage of the methods proposed i this paper are higher level of security at relatively low computatioal overhead. Refereces:- [1] Alter.R, Some Remarks ad Results o atala Numbers, proceedigs of Louisiaa oferece o ombiatorics, Graph Theory ad computig (1971). [] Alter.R, ad K.K. Kubota Prime power Divisibility of atala Numbers, Joural of ombiatorial Theory, series A, 15 (1973), [3]. Brualdi, R. A. Itroductory ombiatorics, 4th ed. New York: Elsevier, 1997 [4]. aetti R. Halevi, S. ad Katz, J. hose cipher text security from idetity-based ecryptio, Advaces i ryptography-eurorypt 004, Vol. 307 of LNS, Spriger-Verlag. [5]. Eli Biham, ryptaalysis of multiples modes of operatio, Joural of ryptology, 1998, Vol.11, No.1, pgs [6] Jarvis.F. atala Numbers, Mathematical spectrum 36 (003-04), 9-1. [7] Koshy.T, The Ubiquitous atala Numbers, Mathematics Teacher 100 (October 006). [8] D.R. Stiso: ryptography. Theory ad Practice, R Press, Boca Rato, 00. [9] W. Trappe, L.. Washigto: Itroductio to ryptography with odig Theory, Pretice Hall, Upper Sadle River, P a g e