This paper describes a new development in the cryptographic digital signature scheme based on Mandelbrot and Julia fractal sets. Recently it has been shown that it is possible to have digital signature scheme based on fractal due to the strong connection between the Mandelbrot and Julia fractal sets

Alia, Mohammad Ahmad

Samsudin, Azman

English

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IJCSNS Intemational Journal of Computer Science and Network Security, VOL.1 No.7, July 2007 99Generalized Scheme For Fractal Based Digitat signature (GFDS)Mohammad Ahmad Alia and Azman Bin Samsudin,School of computer sciences, universiti Sains Malaysia, I lg00 penang, Malaysia.SummaryThis paper describes a new development in thecryptographic digital signature scheme based onMandelbrot and Julia fractal sets. Recently it has beenshown that it is possible to have digital signahrre schemebased on fractal due to the shong connection between theMandelbrot and Julia fractal sets. The link between thetwo fractal sets is used for the conversion of the privatekey to a public key. However in the previous work theverification can be done only by a specific party. In thispaper we intoduce a new variation of the fractal public_key digital scheme (GFDS), whereby the verification ofthe digital signature can be done by any member of thepublic.Kqtwords:Fractals Cryptography, Digital Signature Scheme,Mandelbrot Fractal Set, and Julia Fractal Set.l.IntroductionDigital signature is an electronic verification mechanismbased on the public-key scheme and is considered as atype of the asynmetric cryptography that is focusing onmessage authenticity. The digital signature scheme is usedto provide a guarantee that the original content of amessage is unchanged by unauthorized pafi which isknown as the data integfity, the assurance that the sourceof data is as claimed which is known as messageauthentication, and the assurance that an entity cannotdeny commifinents which is known as non-repudiation [],21. The ourp-ut of rhe signature process is callid the digiiaisignature [3] (see Figure l). In digital signatue based onpublic-key algorithms, the private key ii used to sign ame-ssag:, while the public key is used to verify theauthenticity of the message.ln 1976, Whitfield Diffie and Martin Hellman gave thefust notion of a digital signature scheme although at thattime they only conjectured the existence of such scheme[4, 5]. Soon after that, in 1978 Rivest, Shamir, andAdleman proposed the first digital signature scheme that iscalled RSA digital signature algorithm [6]. Subsequently,few more proposed digital signature algorithms based ondifferent 'hard problems' were soon developed after RSA,such as ElGamal signature scheme [7], Undeniablesignature [8] and others.Figure l: Digital signature scheme.The well known digital signature schemes can beclassified according to the inherited mathematicalproblems. As of now, there are three main different Np-hard problems (Non-Deterministic polynomial) wherewell known digital sigrature schemes have been derivedfrom.l. Integer Factorization (IF) schemes. The security ininteger factorization schemes are based on thecomplexity of the integer factorization problem.Examples of [F scherne implementation are RSAdigital signature scheme t6l and Rabin digiralsignature scheme [9].2. Discrete Logarithm @L) schemes. Discrete logarithmschemes are based on the complexity of the discretelogarithm problem in a specific finite field. Examplesof DL scheme implementation are ElGamal [7], andDSA [|0].3. Elliptic Curve (EC) schemes. The security in ellipticcurve schemes is based on the complexity of theelliptic curve discrete logarithm problem. Examples ofEC scheme is the elliptic curve digital signature [l l].ot v"rltd/N.tok-l273100 IJCSNS Intemational Journal of Computer Science and Network Security, VOL.7 No.7, July 2007This paper proposes a new variation of fractal public-keydigital signature (FPKDS) scheme to sign and verifo thecorresponding message. Similar to the original work [12]'the functionality of the proposed scheme depends on thestrong connection between the Mandelbrot (see Figure 2)and Julia (see Figure 2) sets [13]. Special functions,Mandelfu and Juliafn functions ll4l are used to generatethe corresponding private and the public keys'2. Existing Digital Signature Based on TheMandelbrot and Julia Fractal SetsAccording to ll2l, and with the aid of Figure 4, wedescribe in brief the idea of the fractal digital signaturescheme based on fractal set by using Mandelbrot function"Mandelfn (see Equation l) and Julia functiot Juliafn (seeEquation 2) tl5l.zn= cxf (2,-); z(0) = c; c,zegnez (l)2n= cxf (2,-); 40) = Y; Y,c,zeQneZ (2)In Figure 4, sender and receiver must agree and use thepublic domain value, c. The receiver, Bob, generates e andn as the private keys, while the sender, Alice, generates /cand d as her private keys. Sender and receiver use theirprivate values as well as the value c as inputs to theMandelbrot function to produce the public keys zrt (seeEquation 3) and zpe (see Equation 4). Bob and Aliceexchange their public keys. Alice then obtains Bob'spublic key, zd and uses these values together with herprivate key and the plaintext as inputs to the Julia functionto produce the signature s, illustrated by Equation 5. Alicewill then send a message to Bob. Bob then obtains Alice'spublic key, zp, the signature s and the message n fromAlice which will be used as input values together with hisown private key to the Julia function, to verif,i themessage v illustrated in Equation 6.One limitation of such approach is that, when Alicecreates a message, the message can only be verified by aspecific individual (in this case the individual is Bob). Inthe following Section we introduce a new variation to thisprotocol, such that the verification is not limited to aspecific individual but can be made by any member of thepublic instead.-)lnQ = zr-t x c2 x d: z,c,d e C; n e Z-= zk-l x c2 re;r,c,e e C; k e Z.zk€s: c*-' x (znd)1,e x m;s,c,e,d eC; n,x,k eZ; m eR..v= cn-x x (zp)rd x m;v,c,e,d eC; n,x,k eZ;meR..(3)(4)(s)(6)Figure 2: Mandelbrot fractal image.Figure 3: Julia fractal image274IJCSNS International Journal of Computer Science and Network Security, vOL.1 No.7, July 2007Scndcr (Alice) Recciver @ob)Thc signeturc is valid if v = s.Figure 4: Existing Fractal digital signature algorithm [12].l0l3. Generalized Scheme of Fractal BasedDigital SignatureIn this Section, we describe the idea of the generalizedfractal public-key digital signature scheme based on fractalset, by using Mandelbrot function ,,Mandelfn (seeEquation 7) and Julia function Juliafn (see Equation g)ll5l.In Figure 5, sender Alice, choose her private keys ft and eand generate c, d and n as public keys. Alice uses her keysas inputs to the Mandelbrot function to produce the private\eVs 7d (see Equation 3) and the public key zp (seeEquation 4). In addition, Alice, generates k and. d as herprivate keys, and then generates her public key for thepublic. Also, Alice uses her keys together with themessage as inpu8 to the Julia function to produce thesignature s which is also illustated by Equation 5. Thesignature s will be sent with the message to Bob (amember from the public). After Bob obtains Alice's publickey,z*, c, d, n, the signature s and the message m fromAlice, which he will uses as input to Julia function in theverification process to veriff the message v. This step isillustrated further by Equation 6. Therefore with thisprotocol, the general public is able to veriS Alice,ssignature by using Alice's public key. This is anenhancement over the previous method where theverification can only be done by a specific party whichneeds to be identified by Alice before Alice can create thesignature.4. Key Size AnalysisThe chaotic nature of the fractal functions ensures thesecurity of the proposed GFDS scheme is similar to thefust digital signature scheme based on fractal Mandelbrotand Julia sets [2]. However, to prevent a brute forceattack, the choice of the key size becomes essential. Thekey space in fractal digital signature depends on the size ofthe key. For example in 128 bits key, there are 2t2tpossible key values, as is the case in the symmetricscheme. RSA and DSA keys are basically different fromfractal keys, where the choice of key is influence by theprime number which exist sparsely in a given finite field.275t02 IJCSNS Intemational Joumal of Computer Science and Network Security, VOL.7 No.7, July 2007Figure 5: The new generalized version of fractal public-key digital signature5. Performance EvaluationAs shown in Figure 4 earlier, the first fractal digitalsignature scheme involves a sender and a receiver. Thereceiver must generate the public key from the chosenprivate key and then sends the public key to the sender.The sender will then generate his public key by usingMandelfn function and sends it to the specific receiverwho generated his public key. But in our new scheme (seeFigure 5), the sender generates his private and public keysand tlen sends the public keys together with the signedmessage through the mechanisms of communication tothe public. Thus, any receiver could verifu the validity ofthe message if there is no changing in the content. Thecomparison in terms of algorithm speed and key size isequivalent to the first digital signature scherne based onMandelbrot and Julia fractal sets.6. Statistical AnalYsisA powerful statistical analysis can be use to test thehistogram for the signatures and check correlationcoefficient of the statistical analysis [16]. It is well knownthat many previous digital signature schemes have beensuccessfirlly analyzed by using statistical analysis. Anideal digital signature should be resistant to the bruteforce ofany statistical attack. Statistical analysis has beenperformed to prove the strength of the proposed digitalsignature scheme based on Mandelbrot and Julia fractalsets by calculating the histogram analysis and thecorrelation coefftcient analysis of t'wo adjacent points.6. 1 Histograms AnalysisThe histogram image illustrates how the values of theoriginal message signature and the changed messagesignature are distributed by graphing the number ofrepeated characters. The histograms have been applied toseveral sigratures. In this Subsection we use thehistogram as one of the tools of statistical analysis toclariff the strength of the digital signature scheme basedon Mandelbrot and Julia fractal sets. This strength isshown tlrougft the histogram distribution of the originalmessage signature and the changed message signature.Figure 6, shows the difference between the signarurevalues before and after the changing in the messagecontent. These changes have been applied on theproposed fractal digital signatwe. As shown by Figure 7'the histogram presents the distribution for the originalmessage signature value.276IJCSNS lntemational Journal of Computer Science and Network Security, vOL.7 No.7, July 2007 103The original message before the changing:CRYPTOGRAPHY_IS-THE_S CIENCE-OF-KEEPINGTHE-MESSAGES_AND_ENS URING-AUTHENTICATIONMessage after the change (removing the first charactertct)RYPTOGRAPHY_I S-THE_SCIENCE-OF-KEEPINGTHE_MESSAGELAND-ENSI.JRING_AUTHENTICATIONFigure 6: Al example for the GFDS.Figure 8, shows the histogram for the changed messagesignature value. The change between the previousmessages is only the first character .c, which is removedfrom the original message, which caused the differencesin the value of signatures. As shown, those Figures showbig differences in the distribution for the signatureshistograms.6.2 Conelation Coefficient AnalysisCorrelation coefficient analysis is used in thisSubsection to find the correlation between two distributedhistogram adjacent points for the signatures that has beenused to perform the statistical analysis on the proposedfractal digital signature scheme. The process for thecorrelation coefficient analysis depends on the randomselection of the original message signature and thechanged message signature values as pairs of twoadjacent points. Calculation for the correlati,on coefficientis done by using Equations 7, 8, g, l0 and I l.(10)(l l)The correlation result (r) between two adjacent points fordistributed signature values was calculated by Equation g[7] which was applied in the proposed digital signaturescheme based on Mandelbrot and Julia fractal sets. Thecorrelation coefFrcient value is 0.109138. However, thecorrelation between the two adjacent points for originalmessage signature and the changed message signature isvery close to zero. This represents a non correlation andcan be defined as uncorrelated. The uncorrelated resulthighlights the diffrculty of attacking the sigrrature bychanging the message content.Signatun nluesFigure 7: Histogram signature ofthe original massage.Si@unvaluesFigure 8: Hisogram signature ofthe changed massage.D(x) =*I,r . E(x))'cov(r,y) = *I,r - E(x))(y, - E(y))cov(x,y) : E(x - E(x)(y - n0))fry=(7)(8)where x and y are the values of two adjacent pixels in theimage. In numerical computations, the following discreteformulas were used:s("r=**+fi$ry|ng 888 8E b rIlKotmFO EO I OF! OF O ON-NN NO O Ol(e)277104 IJCSNS International Journal of Computer Science and Network Security, VOL.7 No'7, July 20077. ConclusionThis paper shows the possibility of establishing a fractalbased digital signature with public verification, derivedfrom the logical connection between the Mandelbrot andJulia fractal sets. The security protection ofthe proposedgeneralized fractal digital signature depends on thechanges of the public keys values (zd, zee) which areaffected by the signer's private keys (a ft). The GFDSpresents the possibility of verifying the message by anymernber of the public which is enhancement over theprevious fractal digital signature, where the siglature vanonly be verified by a specific verifier.AcknowledgmentsThe authors would like to thank the Universiti SainsMalaysia (USIO for supporting this study.Referencestll W. B. Schulu, "Electronic Records; ElectronicSignatures," Federal Register ,vol' 62, no. 54 , pp. 13430-13466,2007.t2] A., Menezes, P. Van Oorschot, and S. Vanstone,"Handbook of Applied Cryptography," CRC Press, pp' 4 -15,516, 1996.[3] Public Law, "Elechonic Sigrratures in Global and NationalCommerce Acr," Ileekly Compilation of PresidentialDocuments. vol. PUBLIC LAW 106-229, no. 36, pp'464-476,2000.t4l W. Diffie, and M. E. Helknan, 'New Directions inCryptography," IEEE Transactions on Information Theory,vol. IT -22,no.6, PP. 644'654' 197 6't5l A. A. Lysyanskaya, "signature Schemes and Applicationsto Cryptographic Protocol Design," PhD thesis, MIT,Massachusetts Institute of Technology , pp. l-3' 2002'[6] R. Rivest, A. Shamir, and L. Adleman, "A Method forObtaining Digital Signatures and Public-KeyCryptosystems," Communications of the AC M, vol' 21, no'2.pp. t20-r26, 1978.[7] T. ElGamat, "A public key cryptosystem and a signaturescheme based on discrete logarithms," IEEE Transactionson,'rol. 31, no. 4, PP. 469 - 472' 1985.l8l C. Davi4 and H. V. Antwerpen, "Undeniable Signatures,"Crypto'89, LNCS 435, Spinger-Verlag, Berlin' pp 212-216, t990.[9] IC Kaoru and W. Ogata, "Efficient Rabin-type DigitalSignature Scheme," Designs, Codes and Cryptography'Springer Netherlands, vol. 16, no, I , pp- 53-64, 1999'[0] J. Bunows, "Digital Signature Standard (DSS)'" FederalInformation Processing Standards Publication 186'Computer Systems Laboratory' National Institute ofSnndards and Technologt, Fips Pub I 86' pp. l-5, 1994'[l] D. Johnson, A. Menezes, and S. Vanstone, "The EllipticCurve Digital Signature Algorithm (ECDSA)," Certicom,2001.tl2l M. Alia, and A. Samsudin, 'A New Digital SignatureScheme Based on Mandelbrot and Julia Fractal Set,"American Joumal of Applied Sciences, vol. 4 , no' I I ' pp.850-858,2007.[3] B. Mandelbrot, "Fractal Geometry of Nanre," SanFrancisco: W. H. Freeman, 1982.[4] N Giffin, "Fractint," The IJniversity of Bnfish ColumbiaCampus in Vancouver B.C. Canada, TRruMF project' pp'l-7,2006.t15l M. Alia, and A. Samsudin, "New Key Exchange ProtocolBased on Mandelbrot and Julia Fractal Set," InternationalJournal of Computer Science and Network Security, vol. 7 ,no. 2, pp. 302-307,2007.[6] S. Behnia, A. Akhshani, A. Akhavan,, and H. Mahmodi'"Applications of Tripled Chaotic Maps in Cryptography"'arXiv.org, The Cornell University,2007 'tlTl H. E. H. Ahmad, H. M. Kalash, and O. S' FaragAllah, "AnEfficient Chaos-Based Feedback Stream Cipher (ECBFSC)for Image Encryption and Decryption," Informatica, vol'4,no. 3 l, pp. l2l-129,2007 .Mohammad A. Alia received the B.S'degree in Computer Sciences and M.S.degree in Information Technology fromAl-Zaytoonah University in 2000 and2003, respectively. During 20@-2004, hestayed at Al-Zaytoonah University-Jordanas an instructor of Computer Sciences andInformation Technology. Then, he workedas a lechrer at Al-Quds University in Saudi Arabia form 2004 -2005. Currently he is a PhD student at the School ofComputerSciences, Universiti Sains Malaysia.Azman Samsudin is a lecturer at theSchool of Computer Sciences, UniversitiSains Malaysia. He received the B.Sc.degree in Computer Science from theUniversity of Rochester, USA, in 1989.He obtained his M.Sc. and Ph.D. degreesin Computer Science from University ofDenver, USA, in 1993 and 1998,respectively. His research interests are inthe field of Cryptography, Interconnection Switching Networks,and Parallel Distributed Computing.278

A New Digital Signature Scheme Based on Mandelbrot and Julia Fractal Set,&quot;

A public key cryptosystem and a signature scheme based on discrete logarithms,&quot;

Applications of Tripled Chaotic Maps in Cryptography&quot;' arXiv.org,

Efficient Rabin-type Digital Signature Scheme,&quot; Designs, Codes and Cryptography' Springer Netherlands,

Elechonic Sigrratures in Global and National Commerce Acr,&quot;

Electronic Records; Electronic Signatures,&quot; Federal Register ,vol'

Fractal Geometry of Nanre,&quot;

Fractint,&quot; The IJniversity of Bnfish Columbia Campus in Vancouver B.C. Canada, TRruMF project'

Handbook of Applied Cryptography,&quot;

He obtained his M.Sc. and Ph.D. degrees in Computer Science from

New Directions in Cryptography,&quot;

New Key Exchange Protocol Based on Mandelbrot and Julia Fractal Set,&quot;

Generalized Scheme For Fractal Based Digital Signature (GFDS).

http://eprints.usm.my/9434/1/Generalized_scheme_For_Fractal_Based_Digital_Signature_(GFDS).pdf

Generalized Scheme For Fractal Based Digital Signature (GFDS).

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