Fractional Fourier Transform Based Key Exchange for Optical Asymmetric Key Cryptography


 Maximillian Ryan
 6 months ago
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1 Fractioal Fourier Trasform ased Ke Echage for Optical Asmmetric Ke Crptograph ALOKA SINHA Departmet of Phsics Idia Istitute of Techolog Delhi Hauz Khas INDIA Abstract : Recetl several optical ecrptio techiques have bee proposed for twodimesioal data. These techiques use radom phase masks jigsaw trasforms digital sigatures ad liear trasforms like the Fourier trasforms the fractioal Fourier trasform ad the Fresel trasform. The stregth of these ecrptio techiques is depedet o the size of the ke but is strogl limited b the securit liked with the echage of the secret ke. We propose a ew techique based o the DiffieHellma protocol i which the ke ca be echaged with high securit. The DiffieHellma protocol allows two users to echage a secret ke over a isecure chael without a prior secrets. Fractioal Fourier trasforms have bee used for the secure ke trasfer. Results of computer simulatio are preseted to verif the proposed idea ad aalse the robustess of the proposed techique. Kewords: Fractioal Fourier Trasform Optical ecrptio Public ke ecrptio DiffieHellma protocol Fourier Trasform crptograph 1 Itroductio Iformatio securit is becomig more ad more importat with the progress i the echage of data for electroic commerce. Optical iformatio processig sstems have attracted a lot of attetio i recet times for iformatio ad data securit applicatios because of their iheret parallelism ad ver high processig speed. Recetl a umber of optical securit algorithms ad optical set ups have bee proposed [16]. Crptograph is the stud or sciece of secret writig ad a crptosstem is a sstem i which either iformatio is trasformed ito secret writig called ciphertet. Ciphertet is trasformed back to the origial iformatio called plai tet [79]. The trasformatio process just metioed is cotrolled b what is called a ke. Hece to trasform i either directio the correct ke is eeded. Simmos classifies crptosstems as either smmetric i.e. secret ke or asmmetric i.e. public ke. I a secretke crptosstem private coversatio betwee two persos is established b usig oe ke kow to both of them. This ke is used for trasformatio to cipher tet as well as trasformatio back to plai tet. A disadvatage of this scheme is the fact that the secrec of commuicatio depeds upo the trustworthiess or reliabilit of the two persos. Aother disadvatage is that i a multichael sceario oe has to keep a lot of kes secret to maitai private commuicatio with differet people. I a public ke crptosstem there eist two separate kes kow as the ecipherig ke ad the decipherig ke. These kes decipher i such a wa that kowig oe of the kes it is computatioall ifeasible to determie the other ke. This cocept was first itroduced i ISSN: Issue Volume 7 Februar 008
2 1976 b DiffieHellma i their semial paper [10]. The DiffieHellma ke agreemet protocol was developed b Diffie ad Hellma i 1976 [10]. The protocol allows two users to echage a secret ke over a isecure chael without a prior secrets. The protocol has two sstem parameters p ad g. The are both public ad ma be used b all the users i a sstem. Parameter p is a prime umber ad parameter g (usuall called a geerator is a iteger less tha p which is capable of geeratig ever elemet from 1 to p1 whe multiplied b itself a certai umber of times modulo the prime p. A primar image ecrptio techique ivolves a process i which the primar image is ecoded with two radom phase masks. Oe mask is placed i the iput plae ad the other oe is placed i the spatial frequec plae. This is kow as the double radom phase ecodig sstem [1]. Fractioal Fourier trasforms (FRT are beig used etesivel i the field of optical securit [11]. Uikrisha et al. have used the radom phase masks i the fractioal Fourier domais to ecrpt the images []. Siha ad Sigh have proposed a ew techique to ecrpt a image b usig the digital sigature of the image [3]. A ew techique based o the jigsaw trasform has bee recetl proposed for image ecrptio [4]. Aother ew method of image ecrptio has bee proposed usig the radom phase mask ad the jigsaw ecrptio i the Fresel domai [5]. A lesless optical securit sstem based o the computer geerated phase ol masks has also bee proposed recetl [6]. All of these techiques deped o the securit of the secret ke which has to be trasmitted to the recipiet for the decrptio of the data. The iheret drawback of all these techiques is that the all require secure echage of kes. Thus however secure the ecrptio procedure mabe the are still susceptible to the ke beig itercepted durig the ke trasfer process. I this paper we propose a secure method to echage the secret kes (radom phase mask required for ecrptio. FRT has bee used to desig a algorithm for the trasfer of the ke usig the DH protocol [10]. Fractioal Fourier Trasform FRT is the geeralizatio of the covetioal Fourier trasform i the fractioal order [ 11]. The twodimesioal FRT of a fuctio f( with a separable kerel ca be defied as F α α [ f ( ]( u v = Kα ( ; u v f ( dd α with the kerel where K α ad K ( u α α ( ; u v = Kα ( u Kα ( v = A ep [ iπ ( cot u csc + u cot ] δ ( u if α = π δ ( + u if α = ( + 1 π [ i( π sg( / 4 / ] si( ep A = (1 ( if α π (3 where α π / is the agle correspodig = to the trasform alog the ais. The kerel alog the ais Kα ( v ca be obtaied similarl b simpl substitutig for ad v for u respectivel. FRT will reduce to the covetioal Fourier trasform whe α = α = 1. 3 Proposed Techique I the proposed techique FRT together with the DiffieHellma protocol has bee used for ke echage over isecure chaels. Let us assume that two persos A ad wish to echage a secret ke. First the agree o a commo image (T to begi with. This startig image is kow publicl. Perso A secretl chooses two umbers A1 ad A that lie betwee (0 1. Similarl perso secretl chooses two umbers 1 ad that lie betwee (0 1. Perso A the ISSN: Issue Volume 7 Februar 008
3 WSEAS TRANSACTIONS o COMPUTERS ecrpts the image T usig the fractioal parameters { A1 A } to obtai ( ( A1 K A A T. (4 Similarl perso ecrpts the image T usig fractioal parameters { 1 } to obtai ( ( 1 K T. (5 A ad ow echage the ecoded images K A ad K over a isecure chael. Perso A ow takes the ecrpted image of perso ad further performs the FRT usig the secret parameters {A1 A } to obtai K A ( A1 A ( K ( A1 A ( 1 FRT FRT ( T = ( ( A1 + 1 A+ ( T. (6 Meawhile perso takes the ecrpted image of perso A ad performs the FRT usig his secret parameters { 1 } to obtai K A ( 1 ( K A ( 1 ( A1 A FRT FRT ( T = ( ( 1 + A1 + A ( T. (7 From (6 ad (7 it ca be see that K A = K A i.e. perso A ad perso have echaged their ke secretl. The echaged secret ke is K A = K A which is ot kow to a eavesdropper listeig i o the isecure chael. The method of echage of the secret ke is highl secure because the echaged secret ke is ukow to either part i.e. perso A or perso prior to the start of the echage procedure. It gets geerated i the process of ke echage. The actual value of the secret ke (radom phase mask depeds o the radom parameters { A1 A } chose b perso A ad the radom parameters { 1 } chose b perso idepedetl. 4 Simulatio Results Computer simulatios have bee doe i support of the proposed techique. The primar image chose for simulatios is of Lea of size piels as show i Fig. 1. Let perso A choose the fractioal orders ( ad perso choose the fractioal parameters ( Figs. (a ad (b represets the ecrpted images K A ad K. Figs. 3(a ad 3(b represet the images K A ad K A after perso A ad perso have carried out the steps as outlied i Equatio (6 ad (7. This is the echaged ke. The problem of breakig ito the proposed ke echage techique equates to fidig K A (or K A from the kowledge of K A ( + + ( ad K. A1 1 A That is fidig FRT ( 1 from FRT A A ( 1 ( ad FRT (. Let a error of Δ i the fractioal order cause a large eough error betwee the actual ke (K A ad the estimated ke usig the icorrect fractioal parameters. I order for the hacker to determie the correct value of A1 it would require (/Δ attempts sice A1 ca lie i the rage (1 1. Idepedetl to determie the correct value of A it would require (/Δ attempts. Sice the correct combiatio of { A1 A } is required the total umber of times the hacker has to calculate the FRT will be (/Δ (/Δ = (/Δ. Let the computatioal requiremet to calculate FRT be m. The the umber of attempts to guess the correct ke will be N = m (8 Δ The computatioal compleit m depeds o the size of the image. We kow that the computatioal compleit of FRT is proportioal to FFT which is at best O(N log N where N is the legth of the iput. For D FFT it will be O(N log N. Thus the umber of attempts is ver large ad the method is highl robust to blid decrptio. The mea squared error (MSE betwee the decrpted ad the origial image has also bee evaluated to test the performace of the ke echage techique. The MSE is evaluated as a fuctio of the errors i the decrpted fractioal orders. If o(ij ad r(ij deote the values of the origial ad the recovered image at the piel (ij respectivel the the total MSE ca be defied as follows: N N 1 MSE = r o = r( i j o( i j N N i= 1 j= 1 (9 ISSN: Issue Volume 7 Februar 008
4 where N N is the size of the images. I the MSE calculatios each of the results was averaged over 10 trials. We have plotted the MSE versus the error i the fractioal order i Fig. 4. It ca be see from the figure that a error of Δ > 0.05 i the fractioal order causes a ver high MSE betwee the actual ke (K A ad the estimated ke usig icorrect fractioal parameters. For a tpical image of size ad Δ = 0.05 the computatioal requiremet is of the order of 10 9 ad for a image of size the computatioal requiremet is of the order of Coclusios I this paper for the first time a secure ke trasfer protocol has bee proposed that ca be carried out opticall. This is based o the FRT which is a ideal cadidate to be implemeted opticall. The simulatio results have show the validit of this ew algorithm. This secure ke echage algorithm i cojuctio with the ecrptio algorithms ca be used for a reall secure ad robust optical ecrptio techique. Practical applicatios would iclude optical digital sigature ad secure commuicatio over optical chaels. This work ma be further eteded b developig a optical setup to implemet the proposed techique. [5]. M. Heell J.T. Sherida Radom phase ad jigsaw ecrptio i the Fresel domai Optical Egieerig Vol.43 No pp [6] G. Situ J. Zhag A lesless optical securit sstem based o computer geerated phase ol masks Optics Commuicatios Vol.3 No [7]. Scheider Applied CrptographJoh Wile ad Sos [8] KuYua Chao JaChe Li Fault Tolerat ad NoEpaded Visual Crptograph for Color Images WSEAS Trasactio o Iformatio Sciece ad Applicatios Vol. 3 No pp [9]. Otiveros I. Soto R. Carrasco A New Crptograph Algorithm usig Cab Curves ad LDPC for Wireless Commuicatio Sstems WSEAS Trasactio o Mathematics Vol. 6 No. 007 pp [10] W. Diffie M. Hellma New directios i crptograph IEEE Trasactios o Iformatio Theor Vol. No [11] H.M. Ozaktas Z. Zalevsk M. A. Kuta The Fractioal Fourier Trasform with Applicatios i Optics ad Sigal ProcessigWile 001. Refereces [1] P.Refregier. Javidi Optical image ecrptio based o iput plae ad Fourier plae radom ecodig Optics Letters Vol.0No pp [] G. Uikrisha J. Joseph K. Sigh Optical ecrptio b double radom phase ecodig i the fractioal Fourier domai Optics Letters Vol.5 No.1 000pp [3] Kehar Sigh A techique for image ecrptio usig digital sigatures Optics Commuicatios Vol. 18 No pp [4]. M. Heell J.T. Sherida Optical image ecrptio b radom shiftig i fractioal Fourier domais Optics LettersVol.8 No pp ISSN: Issue Volume 7 Februar 008
5 (a Fig. 1 The origial image of Lea of size 5656 piels for ecrptio. (b (a (b Fig. 3(a ad Fig. 3(b represet the images K A ad K A (the echaged ke after perso A ad perso have carried out steps outlied i equatios (6 ad (7 respectivel Fig. (a ad Fig. (b represet the ecrpted images K A ad K respectivel. MSE Error i Fractioal Order Fig. 4 MSE betwee the decrpted ad the ecrpted image as a fuctio of the errors i the fractioal parameters i FRT1. ISSN: Issue Volume 7 Februar 008