Extension of Wiener s Approach on Security on Aadhaar Card based ATM System
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1 Inrnaional Journal of Compur Applicaions ( ) Volum 55 o, Dcmbr 06 Exnsion of Winr s Approach on Scuriy on Aahaar Car bas ATM Sysm Rashmi Singh Dparmn of Compur Scinc Babasahb Bhimrao Ambkar Univrsiy (A Cnral Univrsiy) Lucknow (U.P.) 605, Inia ABSTRACT Crypography is ncssary for h scuriy of h aa which may in h form of x, auio an vio. In gnral Symmric an Asymmric crypographical mhos ar us for h abov purpos. In which all yps of aa is convr ino h binary numbr sysm. Du o non availabiliy of h crypographical sysm, hackrs may hav hack h aa. For h srong scuriy, various rsarchr hav us RSA cryposysm which is wily us in h igial signaur. In h prsn papr, an xnsion of RSA crypography is wll xplain for h hr variabls. An implmnaion of h propos RSA sysm for hr variabls is monsra hrough h implmnaion of h Winr s xnsion. Various horms wih proof ar givn in h papr an implmnaion is givn on h Aahaar car bas ATM sysm. Kywors Crypography, Digial Signaur, Hackrs, Symmric, Asymmric. ITRODUCTIO Th cryposysm is frqunly us for proviing scuriy an auhniciy of igial aa an scur rmo login sssion. In h prsn im, Rivs-Shamir-Alman (RSA) approach is wily us in various commrcial sysms []. Los of inusris ar using scur RSA igial signaur for onlin ransacion. In crypography, Kobliz [] has sui various approachs of sning mssags in iffrn form in such a way ha only auhoriz usr can rmov isinguish an h original mssag. Winr s mho of coninu fracion fins h nw waknsss in RSA an affcing facors of wak kys o improv h scuriy []. Th aack of Winr s approach on RSA cryposysm wih a small ciphring xponn xpns o sysm using ohr groups such as llipic curvs []. Slcing an RSA moulus wih a lil iffrnc of is prim facors provis vlopmns on h xponn aack of Winr s approach [5]. A crypanalyic aack gnras h us of shor RSA scr xponn an is wll fin in liraur. This crypanalyic aack gnras h us of an algorihm bas on coninu fracions ha inifis h numraor an nominaor of a fracion in polynomial im whn a clos nough sima of h fracion is known. This aack acs as a no hra o h normal cas of RSA whr h scr xponn is almos having h sam siz as h moulus [6]. Thr is a pracical survy on principls an implmnaion of crypo graphical mhos for scuriy ovr nwork an pracical applicaions ha hav bn implmn o provi scuriy ovr h nwork [7]. On h basis of analysis, hr ar wo yps of concurrn voluion in crypography. Various applicaions of lprocssing hav givn avanc n for nw yps of crypo graphic sysm which ruc h n for scur ky isribuion channl an provi h inical signaur [8]. In h RSA public ky cryposysm, if h priva xponn is lss han 0.9, hn h sysm is inscur. This is h firs avancmn ovr an ol ffc of Winr which shows whn is lss han 0.5, hn h RSA cryposysm is no scur [9]. In h Winr s hory h vrificaion of h opimum pricor crass h rsul of an ingral quaion, which is improv form of h winr-hopf quaion. Th us of h hory is rprsn by iffrn pracical xampls [0]. Encryp Imag-Bas Rvrsibl Daa Hiing (EIRDH) is a common opinion of informaion hiing. In his procss, hr ar hr niis which ak par as imag provir, aa hir an rcivr. A nw chniqu has bn propos for ncryp imag-bas rvrsibl aa hiing using public ky crypography from various xpansions. Th basics of h propos chniqu ar o prprocss an imag wih h qualiy of various xpansions bfor ncrypion of an imag. This chniqu givs goo payloa an improvs qualiy of imag []. In h aa ransmission, scuriy is h mos significan an imporan issu ovr nwork. In aa ransmission, Doxyribo uclic Aci (DA) crypography plays a crucial rol an his concp is no us only o sor aa bu also prform compuaions. In wirlss nwork DA crypography wih Scur Sock Layr (SSL) provi a scur channl wih scur inrchang of informaion []. Som varians of RSA an analysis of crypanalyic aack agains hs varians ar fficin RSA; pnn RSA provi smanic scuriy o h original RSA an hir varians is Carmichal RSA uss h Carmichal funcion []. Th RSA algorihm inclu wih h opraion of larg numbrs. Th ky lngh is incras for proviing significan scuriy []. w varians of RSA ky gnraion algorihm giv wo iffrn ky pairs having sam public an priva xponns. Du o hs varians i is known as ual RSA an h us of ual RSA crass h sorag ssnial for ky [5]. Th inusry sanar igial signaur schm is us in iffrn omains in various inusris o obain h scuriy lvl in various applicaions [8]. Th prsn work is bas on h hr variabls RSA cryposysm an implmn in ATM ransacion which is bas on Aahaar car an monsra by h hlp of h Winr s algorihm. This papr als wih h implmnaion of h Winr algorihm on h scuriy sysm. Th scuriy sysm is bas on h cash wihrawal from h ATM machin an ransacion is bas on h uniqu inificaion i.. Aahaar car numbr. Th purpos of his papr is o provi h auhniciy in h ransacion from h ATM. 7
2 Inrnaional Journal of Compur Applicaions ( ) Volum 55 o, Dcmbr 06. MATERIAL AD METHODS. Winr s Aack on moifi RSA Cryposysm L us consir h hr prim numbr as p, q, r which ar us for h Winr s aack on h RSA cryposysm which is xplain blow for h hr prim numbrs: W hav pqr, for r p r b h moulus for RSA, is h public nciphring xponn an is h ciphring xponn. If, hn is a convrgn of, whr, () 6 () (p)(q )(r ) (pq q p)(r ) pqr qr pr r pq q p (qr pr pq) (p q r ) (( p q r) ) ( A) Whr A ( pq qr rp) W hav (), hr allows h fracion convrgn of an () Thn () Divi by () (p q r) A (pqr) A ( p q r ) A ( p q r A) p q r A Also o ha ( ( () ) 0 Sinc > () an p q r A p q r A pqr o b n for 6 6 Hnc From h approximaion horm is a convrgn of. L us consir a numrical xampl, l p,q 5,r, hn pqr =65, () (p)(q )(r ) = 80 whr (n) ow gc (,n) =, whr is public ky an fin Such ha *= mo (n) = (mo 80) l = 7 = 7 7 (mo 80) = public ky= (7, 65) priva ky= (, 65) C= M (mo n ) M= C (mo n ) 7 C= (5) (mo65) C= 60 M= (60) (mo65) M=5. Implmnaion of Winr s Exnsion on moifi RSA L an for any convrgn 6 ' of, ak ()= ' '(), x = an y =. If, y', hn h priva ky (p,q,r,) ( y', y',,') Proof: l, y' = ( y' ) = ( y')( y') an by finiion of y Sinc ( y')( y'), hnc ar facors of. Thn ( y')( y'), p, q, r or ow as q < p an r =, w hav four cass (i) y' an y' (ii) p y', q y' an r (iii) p, q y' an r y' 8
3 Inrnaional Journal of Compur Applicaions ( ) Volum 55 o, Dcmbr 06 (iv) p y', q an r y' () Whr, y' Cas : ow w will show ha cas (i) is no possibl: For, if y' an y' hn () Sinc, hn ' This implis ha ' 0 Thn ' 0 '() Thus =, hrfor cas (i) is no possibl, sinc > Hnc, h cas (ii) is possibl. Cas : q y' an p y', r Whn vr To show =, y' By finiion of x, w hav () '() Thn '() (p q r) () ' ow '(), w hav ' mo '() Thn ' mo () Which givs ' mo () () Sinc h squnc of nominaors of convrgn ar sricly incrasing Thn ' (),' () from () w hav = () Sinc h squnc of nominaors of convrgn ar sricly incrasing,. () From () an () Thn h priva ky (q, p, r, ) = (x - y, x + y,, ) Sinc (), ' () an from () w hav Cas : q y', r y' an p whr vr To show =, y' By finiion of x '() '() () (p q r) ' ow '(), w hav ' mo '() Thn ' mo () Which givs ' mo () () ow h convrgn of is ihr ' or occur afr '' Sinc h squnc of nominaors of convrgn ar sricly incrasing,. Thn Sinc (), ' () an from () w hav = () From () an () h priva ky is (q, p, r, ) =(x - y,, x + y, ). Cas : q, r y', p y' whr vr To show =, y' By finiion of x '() '() (p q r) () () ' ow '(), w hav ' mo '() Thn ' mo () Which givs ' mo () () ow h convrgn of is ihr ' = () From () an () or occur afr. ' ' h priva ky is (q, p, r, ) = (, x + y, x - y, ).. Winr s Exnsion on RSA L pqr for r p r b h mouls of RSA wih h nciphring xponn an ciphring xponn. For = pqr, If, hn is a convrgn of. 9
4 Proof: By h finiion of, hr xiss a posiiv ingr such ha (). I can b wrin as Inrnaional Journal of Compur Applicaions ( ) Volum 55 o, Dcmbr 06 () () () () () Also no For r p r h bouns for () ar 8 8, sinc. () (5) (5) Thrfor, W hav Sinc (p q r )(p q r ) 0, p q r 0 Thrfor 0 p q r (6) ow W hav () () an 8 6 an, w hav () () () () () () As () an () (), hn, sinc >0 () ( ) () () (() ) () () () () ( ) () () () ( ) () () Using (5) an subsiuing p q r () in (6) (), sinc p q r 9 by (6) Thrfor, is a convrgn of hols. whr Lmma : If r p r an () (p)(q )(r ) hn. Proof: w hav () (p)(q )(r ) pqr (pr pq qr) r q p A (p q r )(p q r ) 0 Thn p q r hn p q shoul b lss han 0 r p r () p q r A as p q r 0 Thorm : (Winr s Exnsion on RSA ovr P (Zn)) L pqr ar prim numbr such ha r p r wih h nciphring xponns an ciphring xponns such () (p q) 0
5 Inrnaional Journal of Compur Applicaions ( ) Volum 55 o, Dcmbr 06 ha ().If = p-q-r =,, hn is a convrgn of. Proof: W hav () () () () () () >0 as () () () () ( ) () as () () ()( ) p q r A () () ()( ) (p q r ) A () ( ) A () p q r A as () () () () Thrfor () o ha (). Sinc p q r larg. Also o 8< for all Thrfor, for an A glcing 9 by assuming is 8, such ha. an () an 8 W g 6 An as for all, w hav Thrfor is a convrgn of for. Thorm : (Implmnaion of Winr s xnsion): L for p q an for any convrgn ' of ' '() ak '(), an y' (). If, y' ϵ, hn '() () an h priva ky is (p,q,r,) ( y', y',,). Proof: For y = (), ( y').( y'). If x, y ϵ, hn h possibl cass ar (i) ( y') an ( y') (ii) ( y') q, ( y') p an r=,as pqr an r<p (iii) ( y') q, p an r ( y') (iv) ( y') p, ( y') r an q= Cas (i): For ( y') an ( y'), w hav,hn '() as Thus () '() ', an (). Thrfor ' ', for som Which is a conraicion, as w ar choosing a larg. Hnc cas () is no possibl.
6 Inrnaional Journal of Compur Applicaions ( ) Volum 55 o, Dcmbr 06 Cas : ( y') q, ( y') p an r= By fining of x, w hav '() '() p q r () ow as ' mo '() an '() (), =. Thrfor, for '(),, y' ϵ, h priva ky (p,q,r,) ( y', y',,). Cas : ( y') q, p an r ( y') w hav '() '() p q r () ow as ' mo '() an '() (), =. Thrfor, for '(),, y', h priva ky (p,q,r,) (, y', y',). Cas : ( y') p, ( y') r an q W hav '() '() p q r () ow as ' mo '() an '() (), ' Thrfor, for '(),, y', h priva ky (p,q,r,) ( y',, y',).. RESULTS AD DISCUSSIO. Dmonsraion of Propos Algorihm Th following abl shows h convrsion of Aahaar Car umbr (Plain Tx) ino h ciphr x wih h hlp of public ky an hn convring h ciphr x ino h plain x i.. Aahaar Car umbr by using h priva ky. This algorihm aks h following sps: Sp :- Brak h Aahaar Car umbr (Plain Tx) ino singl igi numbr an hn on ha singl igi numbr furhr opraion will ak plac. Sp :- In his sp h ASCII valu of h iniviual igi of Aahaar numbr (Plain x) is calcula. Sp :- ow Ciphr x is calcula by using public ky, C=M (mo n), whr M is h plain x i.. h iniviual igi of Aahaar numbr, is h public ky an n = pqr (whr p, q an r ar prim numbrs). Sp :- In his sp ciphr x which is calcula in prvious sp is convr in o plain x by using priva ky, M=C (mo n) Whr C is h Ciphr x, is h priva ky an n= pqr. In his sp original igi of Aahaar Car numbr (plain x) is rriv i.. M. On h basis of abov l h Aahaar Car umbr 5955 which is us as a Plain Tx by h priva an public kys. Th plain x is ncryp an cryp. Th compl compuaions ar givn in following abl. Tabl : Dmonsraion of Propos Algorihm on Aahaar Car umbr Aahaar Car o.(plai n Tx) ASC II valu C=M mo n M=C mo n Plai n Tx COCLUSIO Th ia of Winr approach provis h scuriy of Aahaar Car bas ATM sysm an i is mor scur for ransacion hrough ATM machin. In his papr Winr s ia for crain rsricions allow o obain a convrgn rsul ha is us for fining h various facors us in h algorihm for RSA cryposysm wih nciphring xponn an ciphring xponn. W monsra h propos algorihm on ATM car scuriy by using Aahaar car which may provi mor scuriy wih br convrgnc of rsuls. Th rsuls ar pic for consiring h priva an public kys from h hr variabls implmn RSA cryposysm. 5. REFERECES [] D. Bonh, Twny Yars of Aacks on h RSA Cryposysm, oics of h Amrican Mahmaical Sociy (AMS), Vol. 6, o., pp. 0-, 999. [] D.. Kobliz, A Cours in umbr Thory an Crypography, Springr-Vrlag w York Brlin Hilbrg Lonon Paris Tokyo Hong Kong Barclona Buaps, ISB , Scon Eiion 987. [] S. Maira an S. Sarkar, Rvisiing Winr s Aack w Wak Kys in RSA, Springr-Vrlag Brlin Hilbrg, Vol. 5, pp. 8-, 008. [] R. G. E. Pinch, Exning Th Winr s Aack o RSA- Typ Cryposysm, Elcronics Lrs (995), [5] B. Wgr, Crypanalysis of RSA wih Small Prim Diffrnc, AAECC, 7 8 (00).
7 Inrnaional Journal of Compur Applicaions ( ) Volum 55 o, Dcmbr 06 [6] M. Winr, Crypanalysis of Shor RSA Scr xponns, IEEE Transacion on Informaion Thory, Vol. 6(), ,990. [7] W. Sallings (998) Crypography an work Scuriy, Thir Eiion, 006. [8] W. Diffi an M. E. Hllman w Dircions in Crypography, IEEE Transacion on Informaion Thory, 978. [9] D. Bonch, G. Durf Crypanalysis of RSA wih priva ky lss han 0.9, IEEE Transacion on Informaion Thory, Vol. 6, (), 9-9,000. [0] L.A. Zah an J. R. Ragazzini An Exnsion of Winr s Thory of Pricion, Journal of Appli Physics, IEEE Xplor Vol., (7), ,. [] C.W. Shiu, Y. C. Chn an W. Hong Encryp imagbas rvrsibl aa hiing wih public ky crypography from iffrnc xpansion, Signal procssing: Imag Communicaion, Vol. 9,6-, 05. [] Monika an S. Upahyaya Scur communicaion using DA crypography wih scur sock layr (SSL) proocol in wirlss snsor nworks, h Inrnaional Confrnc on Eco-frinly compuing an communicaion sysms, Procia compur scinc, 70, (05) [] K. Balasubramanian, Varians of RSA an hir crypanalysis, Communicaion an work Tchnologis (ICCT), 0 Inrnaional Confrnc on, Sivakasi, pp. 5-9, 0. [] Chu-Hsing Lin; Jung-Chun Liu; Chng-Chih Li an Po- Wi Chu, "Paralll Moulus Opraions in RSA Encrypion by CPU/GPU Hybri Compuaion," in Informaion Scuriy (ASIA JCIS), 0 inh Asia Join Confrnc on, vol., no., pp.7-75, -5 Sp. 0. [5] Hung-Min Sun; Mu-En Wu; Wi-Chi Ting; Hink, M.J., "Dual RSA an Is Scuriy Analysis," in Informaion Thory, IEEE Transacions on, vol.5, no.8, pp.9-9, Aug [6] R. C. Das, P. P. Purohi, T. Alam an M. Chowhury, "Locaion bas ATM locaor sysm using OpnSrMap," Sofwar, Knowlg, Informaion Managmn an Applicaions (SKIMA), 0 8h Inrnaional Confrnc on, Dhaka,,pp. -6, 0. A. M. Anony, R. Aswahy an K. H. Krhana, "G ATM," Currn Trns in Enginring an Tchnology (ICCTET), 0 Inrnaional Confrnc on, Coimbaor, pp. -, 0,. [7] Roy A. an Karforma S., A survy on Digial Signaurs an is Applicaions. J. of Comp. an I.T. Vol. (an), 5-69,0. [8] Hink M. J., Low, M. K., Task, E., On Som Aacks on Muli-prim RSA, procing in SAC '0 Rvis Paprs from h 9h Annual Inrnaional Workshop on Slc Aras in crypography, pp. 85-0, 00. [9] P.A. kamswari, L. Jyosana, Exning Winr s Exnsion o RSA-Lik Cryposysms ovr Ellipic Curvs, Briish Journal of Mahmaics & Compur Scinc, Vol., pp. -8, 06. IJCA TM :
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