Lecture 14. Encryption

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1 Lecture 4. Ecryptio T. H. Corme, C. E. Leiserso d R. L. Rivest Itroductio to Algorithms, 3rd Editio, MIT Press, 2009 Sugkyukw Uiversity Hyuseug Choo choo@skku.edu Copyright Networkig Lbortory

2 Security Risks of Iteret Commuictio Evesdroppig Itermediries liste i o privte coverstios Solutio: Ecryptio (public or privte-key Mipultio Itermediries chge iformtio i privte commuictio Solutio: Methods for preservig dt itegrity (oe-wy hsh fuctios d Messge Authetictio Codes (MACs Impersotio A seder or receiver commuictes uder flse ID Solutio: Authetictio (digitl sigture, etc Algorithms Networkig Lbortory 2/49

3 Termiology A seder (Bob wts to sed messge to receiver (Alice securely wts to mke sure o evesdropper c red messge. Plitext origil messge Ecryptio process of disguisig messge to hide its cotets Ciphertext ecrypted messge Decryptio process of turig ciphertext bck ito plitext Cryptogrphy sciece of keepig messges secure Cryptogrphy sciece of brekig ciphertext Algorithms Networkig Lbortory 3/49

4 Bckgroud: Number-Theoretic Algorithm Useful for public-key ecryptio schemes Esy to fid lrge primes Difficult to fctor products of lrge primes Algorithms Networkig Lbortory 4/49

5 Size of Iputs Few iputs of lrge itegers Size of iput = #bits A lgorithm with iteger iputs, 2,, k is polyomil time lgorithm if it rus time polyomil i lg, lg 2,, lg k ; i.e., polyomil i the legths of the biryecoded iputs Algorithms Networkig Lbortory 5/49

6 Cost of Opertios Arithmetic o lrge itegers tkes time Cost is mesured i terms of bit opertios 2 Multiplyig two -bit itegers tkes ( bit opertios Fster methods do exist, but we will use the others i this lecture Algorithms Networkig Lbortory 6/49

7 Review of Number Theory Z set of itegers N set of turl {, - 2, -, 0,, 2} umbers {0,, 2, } For two itegers d d, d (d divides if = kd, k Z. I this cse, is multiple of d, d d is divisor of (if d >= 0. Every iteger divides 0. Exmples: 2 8, 3 9, 2 0 Every iteger hs the trivil divisors d. Notrivil divisors re clled fctors. Exmples: 2 is fctor of 8 d 0, 3 is fctor of 9. A iteger > with oly trivil divisors is prime umber; otherwise, is composite. The itegers {, -2, -, 0,} re either prime or composite. There re ifiitely my prime umbers. Algorithms Networkig Lbortory 7/49

8 Divisio Theorem For y iteger d positive iteger, there re uique iteger q d r such tht 0 r d = q + r q ( r ( If / is the of the divisio is the / ( or - / ( (b, the Exmple b ( Algorithms Networkig Lbortory 8/49

9 Equivlece If ( = (b, the is equivlet to b, ulo, deoted b (. A equivlece clss ulo cotiig iteger is { k k Z}. The Exmple :if 8, 3, the q _, r _, d some b re. equivlec e clsses ulo re Algorithms Networkig Lbortory 9/49

10 Commo Divisors If d d d b, the d is commo divisor of d b. The gretest commo divisor gcd(, b is the lrgest such divisor d. 0 gcd(, b b x mi (, otherwise For exmple, some commo divisors of 2 d 8 re, 2, 3 d 6. The gretest commo divisor of 2 d 8 is 6. b if if if b 0, 0, b b Algorithms Networkig Lbortory 0/49

11 Euclid s Theorem If d b re y itegers, ot both zero, the gcd(,b is the smllest positive elemet of the set {x by: x, yz} of lier combitios of d b. Exmple: gcd(9, 5 = 3 9x + 5y = 3 x = 2, y = - Algorithms Networkig Lbortory /49

12 Reltive Primes Two itegers d b re reltively prime if gcd(, b =. Itegers, 2,..., k re pirwise reltively prime if gcd( i, j for ll i j. Exmple: 8, 9, d 25 re pirwise reltively prime. Algorithms Networkig Lbortory 2/49

13 Uique Fctoriztio Theorem 33.7 For ll primes p d ll itegers d b, if p b, the p or p b. Theorem 33.8 A composite iteger c be writte i e e2 exctly oe wy s product of the form = p p2 where the p i re prime, p p 2 p r, d the ei re positive itegers. p e r r, Exmples: Algorithms Networkig Lbortory 3/49

14 Fidig the gcd Give prime fctoriztios of positive itegers d b, e e2 e p p2 pr, where some e, f my be 0. r f f2 f b p p p r The gcd(,b = i i p mi( e, f p mi( e 2, f 2 p r mi( e r, f r 2 2 r. Exmple: gcd( 255, However, fctorig is ot polyomil time lgorithm. 0 Algorithms Networkig Lbortory 4/49

15 Euclid s Algorithm (/2 For y o-egtive iteger d y positive iteger b, gcd(, b = gcd(b, b Euclid(, b ;secod rgumet is strictly decresig if b 0 2 the retur 3 else retur Euclid(b, b Algorithms Networkig Lbortory 5/49

16 Algorithms Networkig Lbortory 6/49 Euclid s Algorithm (2/2 Exmple gcd(2322, Therefore, 6 0 gcd(6, 6 6 gcd(30, gcd(30, 30 gcd(66, gcd(66, 66 gcd(294, gcd(294, 294 gcd(360, gcd(360, 360 gcd(654, gcd(654, 654 gcd(2322, b 2333, Let

17 Alysis For y iteger k, if b 0 d b F k, the Euclid(,b mkes fewer th k recursive clls. k Remember tht 5 Fk Fk Fk 2, Fk, 5 2 k k b Fk, b 5 5 k 5 5 b, k log b k O(lg b recursive clls O( rithmetic opertios 3 O( bit opertios Algorithms Networkig Lbortory 7/49

18 Exteded Euclid (/2 Sice gcd(, b x by, x, y Z, fidig x d y will be useful for computig ulr multiplictive iverses Exteded if (d, b 0 x, y retur(d, - Euclid(, the retur (,, 0 (d', x', y' Exteded (d', x, y y', y' Ruig time sme s Euclid lgorithm. b x' - - Euclid(b, b b Algorithms Networkig Lbortory 8/49

19 Exteded Euclid (2/2 Exmple (d, x, y Exteded - Euclid(6, 3 (d', x', y' Exteded - Euclid(3, (3,, d 3, x 0, y 0-6/3 0 0 d 3, x 0, y 6x 3y d Algorithms Networkig Lbortory 9/49

20 Algorithms Networkig Lbortory 20/49 Correctess of Exteded-Euclid y' b x'- y y' x by x d' d terms Rerrge ; y' b - b(x' y' Theorem Divisio ; by' b - ( bx' d Algorithm s Euclid' ; b gcd (, d s Theorem ;Euclid' by' ( bx' b gcd(b, d'

21 Modulr Arithmetic ( (b ( b ( (b b - b b (iverse uses idetities 2c ( c 2 d 2c ( c 2 Algorithms Networkig Lbortory 2/49

22 Solvig Modulr Lier Equtios x b ( Give, b, 0; Let d gcd(, Solvble iff d b fid x. Theorem If d b d d = x + y (s computed by Exteded-Euclid the oe solutio is x 0 = x (b / d. Theorem Give oe solutio x 0, there re exctly d distict solutios, ulo, give by x i = x 0 + i( / d for i = 0, 2, 3,, d-. Algorithms Networkig Lbortory 22/49

23 Pseudo code ModulrLi erequtio Solver(, b, (d, x', y' Exteded - Euclid(, if (d b the x 0 x' (b/d for i 0 to d - prit (x i(/d else prit "o solutios" Note: Solvig x ( gives 0 - ;O(lg gcd(, ; rithmetic opertios (oly oe solutio. Algorithms Networkig Lbortory 23/49

24 Chiese Remider Theorem Fid itegers x tht leve remider 2, 3, 2 whe divided by 3, 5, 7 respectively. [Su-Tsu, 00 A.D.] Theorem Let, where re pirwise 2 k i reltively prime d cosider the correspodece (, 2,..., k, where Z, i Z, d i i for i i,..., k. Algorithms Networkig Lbortory 24/49

25 Algorithms Networkig Lbortory 25/49 Chiese Remider Theorem Theorem (cot. ( c c ( m (m c k,.., for i / m,..., (,..., ( From,..., ( From b,..., b ( b b (,, b (( b ( b (,, b (( b ( the b,..., b, (b b d,,, ( If k k i - i i i i k i k k k k k k k k k k k k 2 k 2

26 Exmple Give 2 ( 5, 3 ( Fid x ( 55 m m m c 2,, 5, m (m 2 m ( ( ( 55 - ( 47 ( 55 Thus, we c work i ulo or ulo - c2 m2(m2 2 i 5(9 45 Algorithms Networkig Lbortory 26/49

27 Corollry If re pirwise reltively prime d, 2,, k, 2,, k the for ll itegers x d x i iff x. Algorithms Networkig Lbortory 27/49

28 Euler s phi fuctio Euler s phi fuctio Z * { Z :gcd(, } (p p-if p is prime. ( is the size of, the multiplictive group. Euler s Theorem For y iteger, ( ( for ll Z *. Fermt s Theorem If p is prime, the p- ( p for ll Z * p. Algorithms Networkig Lbortory 28/49

29 Repeted Squrig Compre: b, where d b re oegtive iteger d is positive iteger. Let b = <b k, b k-,, b, b 0 > Compute c by doublig c for ech i d icremetig c whe b i =. Algorithms Networkig Lbortory 29/49

30 Pseudo code(/2 Modulr - Expoeti tio(, b, c 0 d let b b b for i k dowto 0 c 2c d (d d if b i k k-,..., b be biry ecodig the c c d (d retur d, b 0 of b Algorithms Networkig Lbortory 30/49

31 Pseudo code(2/2 Modulr-Expoetitio(=5, b=50, =6 b = 00 i = c = d = Algorithms Networkig Lbortory 3/49

32 Alysis If, b, re -bit umbers, there re O( rithmetic 3 opertios d O( bit opertios. Algorithms Networkig Lbortory 32/49

33 Ecryptio Symmetric Cryptogrphy Privte Key Alice d Bob shre key K the dversry does ot kow Alice d Bob gree o cryptosysytem d key Bob ecrypts plitext usig key, seds ciphertext to Alice Alice decrypts ciphertext with sme key d reds the messge Advtge: fst Disdvtge: keys must be distributed secretly Disdvtge: if key is compromised, ll is lost Disdvtge: umber of eeded keys is 2 Algorithms Networkig Lbortory 33/49

34 Ecryptio Public-Key Cryptogrphy Ecryptio key is public Decryptio key is privte (secret Privte key cot be clculted from public key i resoble mout of time Algorithms Networkig Lbortory 34/49

35 RSA Public-Key Cryptosystem Rivest, Shmir, d Adlem, 977 Most commoly used ecryptio d uthetictio lgorithm tody Used i Netscpe, Microsoft browsers, Iteret d computig stdrds Sed ecrypted messges Apped uforgeble digitl sigture Bsed o ese of fidig lrge primes d difficulty of fctorig their products Algorithms Networkig Lbortory 35/49

36 Public Key Cryptogrphy Ech prticipt hs Public key relesed to others Secret key kept secret Exmple Public d Secret fuctios re iverses M = M = Alice ( P S A ( P A (M P A ( S A (M A, Bob ( P Must be ble to revel while remis ucomputble (or t lest very to compute Security depeds o method of computig keys, S RSA fctorig lrge itegers McEliecee decodig lier code (NP-Complete El Gml discrete logrithm problem Chor-Rivest kpsck (NP-complete A P A B, S S A B Algorithms Networkig Lbortory 36/49

37 Protocol for Sedig Ecryptio Messge M Bob looks up Alice s public key Bob computes ciphertext messge origil messge M for his Bob seds C to Alice (evesdroppers do ot hve Alice computes S ( C S ( P (M M A A A P A C P (M A S A Algorithms Networkig Lbortory 37/49

38 Protocol for Sedig Siged Messge M Alice computes digitl sigture S A (M' Alice seds (M, to Bob Bob check tht M ' P ( A P A ( S A (M ' M ' Messge M is ot ecrypted Algorithms Networkig Lbortory 38/49

39 Protocol for Sedig Siged, Ecrypted Messge M Bob computes digitl sigture ew messge M' M, S B (M, d cretes Bob computes C P (M' d seds C to Alice A Alice computes sigture usig M, S (C A M P B ( d the verifies Algorithms Networkig Lbortory 39/49

40 RSA Cryptosystem Public d secret keys re creted s follows Select t rdom two lrge prime umbers p d q (sy > 00 deciml digits ech Compute = pq Select smll odd iteger e tht is reltively prime to Computed Publish pir P = (e, s RSA Public Key Keep pirs S = (d, s RSA Secret Key ( (p (q d e - e P(M M ( d S(C C ( ( (multiplictive iverse Algorithms Networkig Lbortory 40/49

41 Exmple of RSA Ecryptio. p 4, q pq ( (p -(q - 40* Fid e such tht gcd(e, 2320 d e is smll d odd e 3 works 4.d e - d 547 d e ( P (e, (3, S (d, (547, 249 P(M M P(M M ( 3 3 ( ( Note:oly 249 differet messge re possible Algorithms Networkig Lbortory 4/49

42 Implemetig RSA Bob geertes two lrge primes, p d q probbilistic primlity testig O((lg 3 Bob computes = q d ( (p -(q - Bob chooses rdom e - Bob computes d e ( Exteded Euclide Algorithm O((lg 2 ( e ( such tht gcd(e, ( Bob publishes d e i directory s his public key Algorithms Networkig Lbortory 42/49

43 RSA Computtio Usig public key P = (e, to trsform messges M: P(M e M ( Usig secret key S = (d, to trsform ciphertext C: S(C C d ( Use Modulr-Expoetitio: If e O(, d 2 The Public key requires O( ulr multiplictios, O( bit opertios 3 Secret key requires O( ulr multiplictios, O( bit opertios Algorithms Networkig Lbortory 43/49

44 Correctess d P(S(M (M e P(S(M (M Sice d e The ed k(p -(q - If M 0 ( p, the M If M 0 ( p, the M M(( M( M M ed ed ed - M(M p Similrly p M ( p M ( q [(p -(q -] p- ( for q, thus p d q re prime, pq k(q- e d M M ed p M( Fermt' s Theorem Thus, by the Corollry to the Chiese Remider ed ed p Theorem, M ( If the dversry c fctor ito p d q, the the code is broke, but this is hrd Algorithms Networkig Lbortory 44/49 M ed

45 Primlity Testig Fidig lrge primes Desity Of Primes The prime distributio fuctio specifies umber of primes lim Theorem ( / l for exmple, ( (0 l ( 9 48,254,942 The probbility tht rdomly-chose is prime Thus, try lodd l 2 umbers er to fid prime with high probbility. For exmple, 00-digit umber l Try 5 odd umbers er 000. About / digit umbers re prime. Brek iput messge M ito umericl blocks smller th. Algorithms Networkig Lbortory 45/49

46 Tril Divisio Try ll odd umbers 3,, to test for primlity. 2 ( lg( (2 Ruig Time, but (expoetil This works well oly for smll. Algorithms Networkig Lbortory 46/49

47 Pseudo Z Z+= ozero elemets of Fermt s Theorem, if is prime, the for every. IF some violtes, the is composite. {, 2,..., }. By ( Pseudo test tries formul for =2. If stisfied, declre prime. Does ot lwys work, but the umbers errtly declred prime (bse pseudo primes re rre. Crmichel Numbers re composites tht stisfy formul for * ll. Very rre Z Z Miller-Rbi rdomized primlity test overcomes this deficiecy i the pseudo test (tries rdom s. Algorithms Networkig Lbortory 47/49

48 Iteger Fctoriztios Tril divisio by ll itegers up to B to fctor umber up to Pollrd-Rho fctors umbers up to (usully Works well i prctice o umbers with smll fctors Alysis: To fctor ( p to fid fctor p Try ll prime fctors < -bit composite umber The ru time is bit opertios B rithmetic opertios 4 B 2 Algorithms Networkig Lbortory 48/49

49 Hybrid Crytosystems I prctice, public-key crypto used to secure d distribute sessio keys, which re the used with privte-key crypto to secure messge trffic. Algorithms Networkig Lbortory 49/49

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