Coding and Cryptography notes Coding and Cryptography notes, Exams of System Programming

Download Coding and Cryptography notes Coding and Cryptography notes and more Exams System Programming in PDF only on Docsity! WED LUNCH Tome lly Sectek mescages have been needed in Af G ond military appets expedally cluring el farts ines Ciyptoegiapley has if awn. foots in the Eqptian “perod. 4ooe yea agqo,.and up b the 460% tape yaphay was. inainly nede dt the Prfechion of lonal Saute, Since then because a compidery gna _whemet: ek fommerte , the privete sector has become the greater User Of cryphe systems. come in _|446 When Digpie ard. Helman. nto duc (eve luhion wr concept ox publi. ip te aphy, ‘Ln mee Rivest , Shombo. eee C4 The pict produced public hey system aled R&R, The RA to systema 1s based on 4 i pabracte iby of packeting lame inte qers, =n Lys : at i9gs, Elaomal found days ef cryptasy stems bued on Ascete i a Systewt problems. - lam ie 193%, N. Koblite inverted a mettod biel. user elliptic curves . . \ ie tem bay The. discipline devoted eh setiery Sy tn és called Kp rolo. ‘ G 4 ‘ayp 1S art loay thot deals tl dep a It adbdion eS eiipiaas ee Gyptoanalycr? = beasecl on beshihg (oe dipeshey) tes a plainkex , in 4 dphovtex - most stiking aid beanie ode history ye egplegaphy Z e A Gphetr js 4 method fx cottea hg @ leas message, cullef a  The eg detem nes the tuangpormation dak iB used. “The _prcess Of changin plurtex who Ciphovtew 1s called SP en pate bow / Toyerse process Y dan. Ng ti pea tex ck fo plubte 1S called t Ayayplnen Or this is ; 4 The i Cy ice ] We » be in with ‘seer Systems aed oe mochelar a anihmelc ‘ “The __ first e theie Inelhade has i foots we Jubus Caeser, We start by tranclatng bers inho ranvbesy ee B P DSB Fasg Haske 3 z ON | Bo sais Sie eb, 2x5 ix system (plaintex ) tranizonmns one letler of the 4 pluie obyia Pipe, They Ge ace alto aalled men he dghers. lek Pe pumerit. Value of G_ planter lodter C=pumene value & cortespon digg ciphertex ketter Then 4 triveal encay pin oy “ HE CAN. Do” vats b HE cAN Da HE CAN. pa > 4,%,3,0,)3 3, 1% » HEAN Do 3 ciphertex hw‘) C* p+ (mod ae) He GN bo - HECAN do abo Bie Ie  dearsprmnation iS pes le si@ Cron 26) x the solet'on pe- FREI fades ) Euchibean Allgoritt. rm ari cherunbe. ke PLEASE seENP Money | —€=FPHo (mod 26) “aphertex is: LIMKG MamFe BExmw. Meperg4eiykKL m NOPAEST UVWKXYZ a eo 6 $F TD H R Be S &G G IF H Ao 422BAP IS LEQ BHAEETEGOFRPJ ER ; Ophes box FeEXeN zmemk TNHmMG MYzmN pons by the p latte MOT REVEAL THE StcevET Gg af tb (mod Bed aaP=c-h (mod 26) | phd a sur thee fee ad=i Cmod we) Ge ad =\taek au ~atk eal Find the dwense o fa and 26 £€ To solve ite eto, Crew 2b) 2D \AALiH DEUS =i Apply Euchdean — Algarttna worceiny —buchsmeur ey 26 =J-464tt toe fl co 14 = 2-EES = $-2(7-15) + =1.5 72 = Jos dat Seda.a tl - | = 3(14-2:4) -a-t 2 = Ro} , = 3-(9-9-7 = 3:14 ~90ab—/I1) = Il-|4-8-26 1 fi -26°2 as l ie) xc=h (mod 26) -€ Te (mode) =i. 14x21 tavefl a) “ha atin analy Studies OF frequenus of Cturencies Of ktters th the ; Endlisle alphabet show that, hordes, the mock common lofless one TN, RD, Ay... ois nprrmation can be used tp decay pe A ulphes hs in Afpye fansfow akon Example Suppose we know thet a shupt dphes C2PtK (moe was (sed te encrypt fhe dphertext \WFKMP cESPE CITE dPAFW GECPY NTASP CTYRX PppdeR PD  making a frequency tulle) that The ontly oLLuan. gh ile ae wess “is Theh £ wes transform J hn IS= 44K Gnod 26) = | 2 k= (nod 2g) C2Pti (mod 26) Pech [mee ac) & RTHEO RY lsu SEFUL FOREN CIPHE | mM4M EssAG €E¢ makes Senfe on ceggouep ng ia pext, thar. pe Know tha en apne tangprmaton BaPth (mod 26) was used ty enapher The Gphertex, YRTPS ukki7 ‘YGaFv ElYus ulveu WRIRK Pe, Sutcc RYXD JUuRTu letters, we phd “Kak the mat ean es, te Je “< (fom ze M4 Oem tes letter pre L and tha lb goes ty L and T oes to UY. os = aPth (mod 26) that Hath e\| (mod 2c) and ath =20 (mod 26) jes Sa = 9 (mad 24) gints 1s (mod 26)=F qisi4A =a 2 RY Cmod: 2) ; \)aes = lh (med 264  moe inleechng cause: The plaindex gw Mnscribect normally inte a jects pleassined nainoeg & coluwns 4 The key wa Squane gy hunbess whe Aetenmn ordleg Of the columpy ant & is bated On 4a chee Requmsel Suppose the key word 1S SoRCERY 62F125 + Coumbesed jy alphabehial ordes of prcedince) Vie enetpher fhe plunte LASER BEAMS CAN BE MODULATE TO MORE INTELLIGENCE THAN RAdTO WAVES SaRCERY E341AS5 bbs ER BE ppegyec AN B E moduL aA FEpPTOCA I BR YmorRE SMTEUULT gF NCETH AN RADE O WAVES Enuphenng Consist gy vea ding the plabter Verticallges exdes oy the nuwlered Glumns. Cprecter EcbT™m™ MECAE RKAVOO JEDSA mn = ENASS  CRNPTOGRAPHY I's as Paths - I bS AOA Beets 2S fella) S-a—G gersy =? (mod 2d) +O (Sxtaoy =aS (mod (2) _- & @x+204=2a2 mod /3 ) - @© $ee-3 (mod 13) ax =10 (mod 13) ¥ 2 ap = 20 (moda) Smle Fea (mod iz) SE (mo ig)  Shad fais »_@ x2. = Q)x2 ies +42 1 (mod (3) OS Ys ai Cmael 13) o 4 24 (mod 3) Sica the Sold’ ans Ate (154) = (2,7) ae whee ~=F Gnd 424 (modin) | he Solve baer systems , it Ig bette, 4 Use 1 Let A, B be, nak metnices with Wea er by Wapechvelu, We say That ep A= @ Cnod mi). if Aj & bz (mod m) joall (83), leten, 1ejsk, Examgle oe i" \e e ) (mod i) The jollowiny hol ds bs Ty and Bo mer mahns with prep mad E wares kxP matnox, DR a Pn, Ae =Bc etnot ov) DA= DB (mod _m) Je system Of Ong iuences Ay C1 +. Ag ty +-..F AnXn = L, Bie Sint ain ha dh eT ayy No Bald Am XT, 2%, 4+ ..+ Ant, =b, Ge is equivodet™ to the matnx Congruene A (mock +) If a)e() moa a Ss 6 4 6 # Ran L td D sider chagraphic Ciphees wher each block % two ¢ baintex & tepluced by a block two ledttes oe apherter GOLD Zs BURIED 1N ORGNO, we split the as follans Go. OL se uk LE OT NO Ro NO to numerical equivalents $6 uN, 3 8% Lao 4843 8,18 HIF Be The key C2 Sh tA, (modas), 0FG 226 #P +15 B (mod 26), 02 6,<Aab  pont blck 14 3 goes hb: 6-26 beaage 6 2 S:1I414-4 ZE (mod 26) Gs AtIS-+ 225 mod 26) Continue — and enciphet te ehle béche 4 cbt the 6 AS 1S 2, AB IB a) a 37, 8S ASH My 2) ZF AMM Translechay back +. lettess 429cxK NvebT tXEOV CReLS RC for Aeciphe-iny, we piost pres the Aaigrephic ciphee Sgstew in maki notation : (2) -(° *) (7) God 26) 4 1S7VR IF -4 dt Pn eneiphes wake ghow that is P| has mene | (moet 2c’) \> et uc Thas be “fF ta) (mod 2¢ ) a Ln general gall Hil Cjphes system mM be obtain upherterts using CEAP (mod 26 ) Whee A is “an nyn mathx , Aek is tduhvely ’ ® Cc Ef | and Rx } Gi B ACZAAP eT (me 26) > pthc (moel-od  4 arctan hes Seay a STOP PAYMENT dex nunbes became a = — oa Ak RO B + By Ba, Ri ETNNE PAcwU LA R |  ex | g _ Omp, Eve Ain example ea pproveably seeude Cup b system One-tme pad, which iz a for yy s. rkeqma The iz Supposedly Useol per the hotiine beh, and Wich ng fon Whom A Messanlo P needs to ee sent, it is ] iv trans ladecl into a bi sting bee, Sey vence of og Usiig the bneay Cys velen#S Of the letter nun on Bc peFGHETkemMmN oP ARSTUD © ' a By -€ C212 F294 a hele ws © tip 4 Fo Ay h fandom Seg ilenc€e K g be hi) Is Z ated | equine Kap fit shangs genet Sequerrce (s then abded , Le b bit, fo te biha Representahion Y the plaintext MW (Note that the { K must cul the Crttve lenge M1), Then C2 Mth (mod a) for each pair t at xorg © We Use the fandom Sequence are {OO ) K loio1on ©1020 te Cnuphes the word PEG Pes IS Se AQAA aA oo eotity LT>¢ = (jo00), = oro00 426 = (ep), = oono  “ig & (61000 oollo oil oo Wooo 00/02 Wilo > we Suc l, th. ue o2rcm, ~° (mod 2) = 21 =25( mod 23) z 7 (mod. 2 As (mod ay) ybove Cam i obtained ye dla, m) =i, thon : , Lf . MP a pune, aA el (mod p ) herrenn )  b(as) = $(2*%) 22.946 =a So =lat¢ fe Beo =(2")" 27 2O)2% 29 (med 28) Ane Bl eb ncaa soste E scam le : Compute E, o4zt <6 FS where ao cr Cred él 5 21 (mod 260) 2) 436 726 = (ini Noo cop), a, Bos to =2? 42842 428 2 \geto 2: a dato Ba ce 6 (neh dao ar £670 5°z 25 pl aero 5 seus tae a. Wet 62 (aeJien. a a sil Ginik a/ atl aj ito or zat 261 2( ot) tle find 3a (mod 2) wet solve RABY Z| (med 132) i (a3, 122) #1 >> 23% = } 4324 So 232-ay= | Sool a | l26-hs Za = Gea3dtit )\ 2 t-Geem 2/46 [ vce i IFa=26+5 = 3/22 -[lF)- 6 =|.6 #1 = 2.23-fae Non-€e,0 NEG why x rt Compost ALK  oS Fh RI 4 3 2g gy \8g 253TH 34S Ay Sh my is pune then m8 prime, Precip fap ¢ I" ape led MesTene primes fey 4-94 pi2os34 C2/% aa satiate Bes led 4-56 ) inte cer) =(P-1) (4-1), =42- $8 72436 ig 21 (mod age) > jed-ayze4 =! 2430 = 344 2) d= 43% Polga0 > Cz isa0 2 {mod 3533) 2 ogrerl => Cc 245 (mod 2637) € =0045 bln) =? nePY Gs r (mod ey Ss pact (med n) a cle pe! Cmodn) bat ed abt kelr) aA ors) ace” zp “Gned in) 2 cf =F Cwiodw Since a \ loassal n\ \ eu $n) = (pot) (ga ef = ( (med tp P Wb bec  FistHy din) = ap? cq) =(P-() (y wy =4f 1, 4C4en)) = $4 4,) ~ 94) $ oF O64) = ach-) £4, -1) > = 2P 4, ay nis hasan in Pag way Hen aluost Cue Gren ption exponent € wilh have a vem large elmo den) ) Recall the if He ae A Wz c, avid. ey is x4 such that 6, xt&s =A un ne Ewe Odean Algoc then “Thus ra (é,,€,) =|, we can wate ext wel pos some zxyeR. Sup pose i Users S hate the Sanne fs A mea aA ppcent eyponew ts G > £2 Suppose he Sanne Message it seek © each, xe hb the plaintent em Can ba (covered vesthg Me mer tes = (m&:)* (m®)¥ (meet n) A, Geks mem" (mod n ) dy Je ts mem? (mod ‘al => mm Gm be found @eR Ca¥ 4cd (3,474 H=4.( +! H—-34 =l A) tt) =! Bt) : semels 4 A dake oa Ny 5-9 AP the word we BIG AN (med Ny) Ae er C€*#S" (mod Ng ) dy is Such thab xda=' (med p(a2049)) 3h 21 (med 19)2) 2 &da.-loiry = | 3 (-334)-jo1a (4) =I de = - 334 = 6645 ( mod jolz ) 0108, 0¢/2,003 S108! 64S ) ; r F© (mod (asi) > Ssiol 0103 => S £108 67g x > $2 ¥53 [i ola > SZ 61a (mod 106) 001g => § =y a4 (mod og.) > S386) ol xsd Ylow S-l0| => C104" Cd i324 ) 29529 (mod ig27) 453 3 C2 4637 (mod gaa) (560 (mod 1824) Ono stead) 326 Cz 3367 (mod jaay) = ose Ciphacteact id Abney u 252 ([Sbe i362 . A, F Cec * Cmoct rape) ae? (mod igae) = /2/  tT 7 lake Ny = S14 Ng = 624 eSa4 @ len lrigte « : ¢ jatounte ¥ Is Elementiay jactonhg Methods the Seau rity of fiSA ancl elated ope cf oF Ye a Ir cil ) ott depend a the “eres uty ip factoring “ The sthnple A> method ii HleA TR VIAL We succeaiely ty tb dide op by hh ovder, - We typlace a Composite ihdeges nr has a prie Pe Un’ nt r 5 n by p % Fla, Thon we \ysp Exa wple Ee jfache ally but Flare Ute El V3 <6 > 31 must be prime HU BIR F Example “Take = #32677 Lin = 8 oS “Ty 2-235 % #fn Bute 4 jn n=Fxlote A Liotewr| = 323 N= FAXtXIA TES Lvi¢gs% | = ia Ty 24, i, 13, i+ > A } /49153 buh Itfe4s3 14.153 219- F8F LIFs?| ~ 98 Ti 11,23 ,. 4 Ahaer > 43} musk bez, a prme . Hane lé4 ules anur  Ti Ma Lity Tesh (2 Fctontadin, ) Tiel Anition Test ps bth. poe all p ore BK Ao such» a, then on ib pome. Ke coll Feemat ts LiHte Theaxew 4 Ty Pr is prime and Ge ® gt tp, 7 =] anes Gusd: Ad Tk the cowerse. tne 2 Vad Je giben neg with (nia) =f srk gee | eee gh als { The comerse iz palee in general. @-4.. 37a | @Cmed a) but 4) is not P eq. 2 = 1. (med Sint) vohuds 2gho lla x= 27° (med aol) 22 x = ott? Ge % « “Qbite & 340 =10.7¢ +0 340 = Zo.ll tlo x =| Geet 4) Ip =8t2 x =((@7%) j2!.8 att, c Cael X51 BL b4+ 6.4 4. -l mad am “Thasiin C Jewaeate Cowegse Fecunvaks itt Suyppose fa whes ef 4: tach. Ste Se 5 —- oe bac for eacky pve L Aivicling ere , 4a e then n is prime ‘ Yoo We show that ord ac=ne-l, “The Congr “e that k= A> and leé & be pame witl, Blk. 1) and at ord alk £ an a = ete tle “44 i (med n) by hope si bite Aad ahd Aor an. ork, A ae Bo | ord, hence Qa is prime q Na+2 Gs on pitrae: Th Mee $6 = 2-23 a= 2,3, 4 \ até=<! (mod #2) he glk si (mod +7) we an Show aa Fit fay l Pas < sat (modt?) 6% se, = -1 Col er) “= 4 ( (mod er) nett kk pume . V CP, len bey Fina tty dest ) Be erie 2 yt i hat = PR where CF REI 2k» The integer R's pave i qo an integers q "Be n) zt whines FIP and ah'sy eat aw}  Nota tis t n-l = FR . Shen F Rep restarts the factese Dt into pilmes, Gok oh tbe Hees litte foctrad ma pomes we Thas p-l2FR A fartonsahen ta Example Nn=zasgol n=l = oz Ye © : A pettic! fuetriahyr of nw ix n-1 =a38p itt F zAa@a 2 2*7" Gn bl. ken Take G23 CExerate nase bi yor Hot eel R. a=3 ahs Pre “hays Tt) CPreth-s Bimad ext) Tesseyn keg” ont ad Say pose 4 Grn 4e ze such 4 b a |= = they, n 8 pr ve Excmpl g 4 N=iz.ar*H = 33249 tie Porth x fest te show ahi n prime ov 3328 /%, eee z = 8 22 = -l) (med 3209) = n as prime Dpercke lo est bs ak Leisiencl Crptarag = Y 0 ¢ The eres op w modufp m | chnoted lear © x $¢ a®%s1 (modula m), Nofe that al, 4 | br) » by curlers a oC + fo compute — ehicurete logs i to tabutete Be fy lex & P-! R sy (mod an) DeB=ii (2.23 a\ a 3 ¥ 5 6 + $ 4 Me ori te et Ge NI hasanye Hhe +tible jn. ondes ay nheasing Velues Of || mod P) : Nw of 4 Steps using the method B poporkonal hich fs nol pact cad few large P. 26 (med nh) > rehdig ae erviow __ i ¢4 excher Rotoew k the fiat App (cath of the Ais rele toy problem pboarcplay - se Vind. and Bongant with b exchange a Ser f choose — hummbers 4 andein whith ae made pobbe. dl Chooses a andor num her A, Compute $ 14) (mod n), and Sends alt Bongam  (2) Bongcun chooses a random numbeg. te Bae veg? tmed np). and . sods Yb PP (3) Bongaams comp ues the Key k= Us (qa)? at &) Aycnda computes the key keV (4P Note Kec they ompake the- same hey 4 (a9"=(45)% <4 Cnod a} i Shes Spy knows 4 cind n, as Wwe ats fie ui and Vv), ly the Spy Can solve fhe Aiscreke bs pro ble a=thd, U gneb the pq can the ropuke a Altematawery given FP Fn nd 9 4g the § compude. 4% mod n) The called he Dipfie- Hillinan probe Pe conjectured to be eg aviglenr to fhe cArecw Emerulse 7 0) Using the Digyie-tHillman — **Y agree mad f COmMman key hak can liz Usea by Ayam Wi wit, hays azat | b=31, and modulus 9zS., . Soludtiva uae (mo los) uw £4¢) Umeel 102) Se. Pyanela — Senells uzat + Bongam Bongam Computes» Keay? 290 (uocl. to9) €2 , 122, 427, b=2. E mot sa) 2 (mot <8? 26 8 (mod $3) Bi, (mod S2\. m were ateaa Cmod 34) 2 Up Sleanvitss. dnelbod fr fling Apuere legs a 11) | 1 nding. Daa by K =9* mot?) Wt wich 4 find ind, 4s anu WM Ne Can wote 2 =MIgse, whee )2 fo! A = 4m Bp EL | and vex Cmodm) gers y badd?) and ths inwp les that (2 4g Cmod P) --- Bx hs method connvts of makhy a takle of Valles 9p br 0sge Fo and a table gp bales of 4g™ F ny z= w 4 h pest em-l, and Hen sorting the tables and x4 the values YX 4 and p hat satlipy # Than Re vdeo m2 Up] decause phic maker_-fhe howe ee Same length,  desl, 5 Leb P=37, 922, a~azi- 2: bea?) Cmpl az) => b= aa. lmehgs Thus tke key is = (2, 22,39) up pase Bowqcum’s plate wt is Me iq, He Cc fandom number = 7, Then Y, = 2 gt el xaat (mod 37) And 4 2 om 4 =e Iltxaced -dme# 33) Thes Bsgan senets Ey, tm) = (4, i) ts janclea Acc hexs with Ded 1%! ) =/-17 H! (mod ' - rte compicdecdion OF the pl wale key 4 fow ie Key bes Sa. (mod P) is the discrete logcuta Sup pose it aS possible to (2,0 v4 the “A know ledge a a a, 52, den we compute my 2b (mod pl = ; ’ We alieaty knove fan 9”. Cod Tc would wean thot we cow <olve problem ; uk this \ iss .assumed ty be sae Oa Aisucke log pb lem, The tandem — nu ber is importerns fee Value x c shoulsl be uUseel for ach, planter b Bit fest te nolo of elec 2, am elemnerts- P= < Cucd ff) eG) = plaihti.xt Cmedl p) cheek femein ds a Beg fomete ite then ip peter pla the a?'24 Cool?) hace. L iw) =) and a2 en and a a zl Cnod qj Beers jy nn then on pone hon van Andwee — pen stpabie schenre hat, the RSA cand Elqamal schanes, a prvtake key A t Sigh the__meseaue . ‘fle Sngner could deny serdiag Iynactusc key clesmay that — bis piwate key hes been Stolen, Mis: fle uncemble sizmatue selene Y& ba'n-van fen Cirg0). As its name suggests, ibs hag She inyor tant “i thet a valid signadre cannot be dene l by fhe ie Veatfeatin ef the scheme leauilies He co-apercbrrn y | pes. Eeheme lenies its Secaunty fon He by picu by Ce tee digerth« eblem, a: Agenda ws i es bo slog documents Us “4 the Bevan Antyesper scheme. She has pable keys ,b, 0), wher a prime J hus a large ade mod P amd Bo mck Pp). there a is the private hey. We assume et P= ag has * onder q whee <b. is abe: let a be the set yp pots SPY S-4L99%.., 8% Gmod 4) Suppose of x ey is the plainterct Message to be og Then He Aigjted 5 igrnschase is Ysx7 Cmed® “The veri ytaMn isa chakenge = Peayorse potoctl @ Bonga sekets 4wo numbers k, , Ka ) He cOmpuckes z= ykipke _ kk, j and sends at fands ont P zs to Ay anda ) Ayana uns c =2% ae mod P) i (te) Ranges Mais, the Siinatude by checking GC a\sectatal (med?) 622 mot 4 {il co We check the 7 Vali ty 8} He Veal cation 3 imed J, kat med g \ h ki gkalaa) med g g* (med P) Hence 4b a Vala siymatwre go Mess ane wd caphey Cte F a Caphcny yp D Ri. a owra 7) ileead ne ECS hey fandom primes P a G tn RsA, = Tandon sequene 4 O's emcl |’s can be eS & e hey fr a one Lbme pad, i walla leayaes. tan dsm =N2’s te omuluk - ell; Mi qe eqwaives om CIs < lly | D pinniey eee wheel, ert. c, N2%s ome used n Wonke-Casto methacls wee Age mathemadtral eperativns ilipel Lea Fnowsn pscucls random ng generehn B fee conn endrcc| Othe A CPM Lehwsr ine) | posthe integer mM, tke modulus, ts meth cl uses ya multifler a, and Gn ynegemnA Co 4 No ave compute f Sequertially : Hoag 4c 1 Sita te, He ~ Coto es) 2¢ hi <M a sequence of heg ex Ay ay nny Mar any tue otcun twre na Sequence , then the 2 must kepeak,  Suppor 3. if x Shs zy ; fox. Jé L te, Ye Xa = Ajty,.. ance ther om oe pat each Xo, the Sequence must lepeak ‘ 's os a periad othe Sequence Xo, Ty y%ay,-y Lt infegoc T Such hak Ei pe = i At Oglala Quest ser : 10 Aihadhio He parse & decmng a, fhe sequeue apy nes generates boy umis %&=6 ¢: Doe What is The peste J? Solidieas a=5 Cra Wee 14 . x =6 x, Eox,da 22acie med iglG %, = $(iedta = lo Kosh. ip) a %, 216 (med tq) seis wel Wee XX 216 M26 Ye Fla .28bt ia The SEGUENLE OF ps Bee peu Pocanden V2 figs te1s,),4 1%, 1610 feiod Tie Eweruze Find pedo gy the Sequence gy gscuclonandl Senosked X= QjAq=¢ , C4 end nzag  _phea c= 0 eax, (nolm) %,= Az, Cuadn) Bz (mets) =G? x, (nsdn ) ln © =a" & (nob) a Bcd (a,m)=! Han a*z¢ Cmod n) whenovep ke { mu Mtp le & ad, m. es artln 4 age ehctinck Crp n) the 1x;4 Fepects ag ae f @ terms, Mu prime ant a fi @ pomche Yoot mod m, We will cbbuh a prevderinden Sequence ay (ages aber, sane of, 4 = P(m) = m-t CO have X= aXe +6 Cmod md % = axX,+c (med m) 202, tiitare Cnodm) Xy= AX+C sax, +(ltata)e Cmokm) } Le =akx + Citatary.4+a™')c Cmod m ) Bibheviuke Ct tutarr.., tak) by Ye aul G& D ie (a-l) = (4-1) C lta tars. ta*') (mod m) a* =i (Mod m) Ww j We Can write @ 4 BX = a*®-Ox, +46 Cured m) =(a-t) ‘the Fat Sie rx. Cro dm) 2] (a-1), rely re ne Ofek Cm 4 Then Lars 2 +4 com) God ») hope Locad nv) henre Ler = % Cmokm) — Since Yipes a af Ts (#0 then this innplies Hat the Sequend| pesiod € with fe Tim) ,a conkadyMaa a =o. and Tom) |s as (equired, (b) roo ¢ = CK U ketene perio dee sequence J @) if sek and 39 EX: (med m) ers “ $ ig a Mu Vp le Op the prod. (b) The peso Tom) oy ZZ, mod m op TCRE) ae alm, aa |m,..., Alm , then ie azb Umcd m) Hen. d=b Cmackaiin 7 Fro nm the lag (eum a, We See That to dehemmine the pene d 9 ye te 4 equimce wy ua pie nawdbess. ? Ne sina bets Px bean -ade paling and rex, amet Up Sl ta +ot4. tq (mad pd) Jhon | /) | #1 (mod f) Hen Hie pedod & Drcinooll ord, A Pi a thin the perivd a ie Fuel) a=) mod 1) Hen He perisl op Ye & 2. Pla Then Pla, $2 2iratqat.tacita” Grod p) 2\ tat tat! oe Cmod p”) \y ; Siriap HL (mod et) ‘7 ete. S peniod oF Y. i 1, 1a i adi (mod a) , then a-l hace an Parse J, <Itata*t. pat! ela-ty Cat-) Gned pr) 2 Smallest value a € for Which, “this ain hol 2 iz op Aw Hen tHe penoal is od, a. ore Ye =o ip qt ow if a*-, =p Cmed p) bee Yai ta ta®d tar’ Cod at) r en the peed oe tke Seguence Ye is Pi if asi (mele) E xtumn [ues 3, x  * Facdorit akin techy wes Were permet grin Prurnatudy test RSA cxypobyeleon Chindie. rein deg BResrrema Diente bg bagel apy Senn — Byamred No -pweuke -yyntom nukes