CSCBNO. Asymmetric Encryption

Transcription

1 CSCBNO Asymmetric Encryption

2 Posted today issue YHW # 1 use all 3 types HW # 2 If no tell due me soln why Friday

3 Last AES ( also a bunch of Zn ) Since working in Zzsg or 2512 addition is XOR Weakness Need a common ( secret ) key

4 Cryptography More interesting How do we secret agree key? on a Bestwayi Physically exchange things However for impractical like web MARC or Public Key Encryption decryption function D E Goals ( ECM ) ) M ( and ECDCM ))M ) 2 E + D are fast 3 Given E hard to derived

5 New Diffetttlmankgexohange " From directions in cryptography by " Dike + Hellman in 1976 Daily conspiracy tidbit CFtksdeayoedk#Keylagi Start with Zp These (pprime or power Prime ) have groups multiplicative inverses ie 2xl mod 5 3 is inverse of a

6 The protocol Alice + Bob & p Sap _SEZP Alice Bob chooses secret Alice Alice computes are both public chooses secret a < b<&p p#bb A samodp Bby#BsBsbmodp Bob K Bamodp computes Abmodp K a Ab ( say gab p

7 Example a s2 p 29 a Alice Bob picks a 3 picks b 7 At 23 mod 29 E 8 Alice sends 1327 mod Bob sends 8 > mod 29 " FR 123 mod 2g

8 Whip Common key is k5b modp Public info p s A samodp and 13 sb mid p What can an a Hacker AB Attacker " hidden satb (5) 64 X must " exponent End try?

9 Has? At its root the to 1 why this is default is the discrete log probbm_ Remember logo logarithms? 1000 logz 1024 Here discrete version Gen A find logs A logs SA modulo_f

10 Had hard? This problem is connected to factoring Not But no efficient algorithms are known Npatttrpadyhomial Other key exchange work in other groups ( like elhptc curves ) algorithms

11 OI ' R# Rivest Its hardness is factor ua any $as Shamir & Adleman bed to algorithm to factor a number would break RSA First more number theory! Euler 's tohent function ( n ) # of positive integers Cn) in that are relatively prone ' ' Ej (a) 53g " 4 ptpt is p Prime OIC F G) " 2634 ' If

12 What about non primes? Interesting special case when So prqpnmie Iq What is not relatively #pqjk3s prim divisors with Zp 3p pq ( p&z p " n qp iqp? CQD qt ) Cpt )

13 Why we care Remember a number K has to be relatively Prime ton in order to have a multapkcakue inverse in Zn Ey 5 in Euleisthm Wtq wgs XEZN st mod li 22 net gcdcx inverse and n)l Then X# 1 modn

14 To compute inverses gcd need Then Eat Euler 's thm x0ih1 mod ' ) L modn More inverses generally in 7 are a bit n IoCn)_ algorithms n more complex

15 ed why we care! ) Select Steps I Let npq OI In ) Got ) lqt ) BacktoRSA_fe oslaergpeagpnmesptq Select &d e st Te9na&ueh#hptqr 1 mod OICN) G How to get d?

16 Computing in verses Remember Euclidean gcd Can EA to augment here 1 alg godcxetn#d give ixtjngcd Can ) D Now if gcdcx ixtjn 1 W n ) 1 in Zn i / mod n

17 Extended Euclidean Algorithm Euclidean Computed algorithm D gcdca b) Let by doing gcdca b) gcdlb a mod b) a amotb b@+r deb+j0r_ for some q Z whoey We will modify Euc Alg so that each call returns not He but also just I ty gcd inverse wotb

18 Some Goal ugly math Had r a mod b and v a a If D ibtjr qbtr qb ibtj ( a qb) i@atciq ) b j here is a 's inverse mod b

19 Eds { EEIEIETEE#iang of q integer part ( al j ) Extended Eucttg ( b return ( dij r ) Runtime g n )

20 ed 1) So RSA ( finally! ) Bob 2 large primes prq Jet npq Selects )Cp OI In Cqt ) &d e st e and Ek ) are relatively Prime I mod OTCN) Extended Ee Alg Not (e) ) n is public key d is private key ( also page )

21 ' Encrytng_ Alice gets Can ) She takes a message M with 0 < M< n ( chops into pieces ) Then C Me mod n ( Remember public n ) was kg ) Alice sends C to Bob

22 Bob Decrypting gets C C Me mod n Bob calculate Cd mod n Claim Why? Ted mod + n M (A) dmodn Met mod n Know ed 1 mod Eon ) M@dIMlkoIMtDmodnfMottDkoMmd It " Eukrthm Mmodn M n

23 se Why Bob can decrypt! secure? He knows ( secret ) d Alter Eve 's goal Fgwe out d! How? needed d Bob is e 's inverse mod n ftpsina knows Attacker n ( but riot OICND How to find OICN)?

24 Soe whole thing is secure as Eve can't long Ecn ) or prq Badly Factoring Not NP Hard get is as Best algorithm Number feld sieve O(e lgny3a gkgn5 'T

25 RSA Somepraotats can be used to encrypt ( but usually entire message isn't ) Slow Easier ( compared to XOEMG ) to break than AES or other symmetric protocols Also I was (M n)l! saved Here am will be prune to assuming napg relatively since p org

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