Online Cryptography Course. Basic key exchange. Trusted 3 rd par7es. Dan Boneh
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1 Online Cryptography Course Dan Boneh Basic key exchange Trusted 3 rd par7es
2 Key management Problem: n users. Storing mutual secret keys is difficult Total: O(n) keys per user
3 A befer solu7on Online Trusted 3 rd Party (TTP) TTP
4 Genera7ng keys: a toy protocol Alice wants a shared key with Bob. Eavesdropping security only. Bob (k B ) Alice (k A ) TTP 7cket Alice wants key with Bob choose random k AB k AB k AB (E,D) a CPA- secure cipher
5 Genera7ng keys: a toy protocol Alice wants a shared key with Bob. Eavesdropping security only. Eavesdropper sees: E(k A, A, B ll k AB ) ; E(k B, A, B ll k AB ) (E,D) is CPA- secure eavesdropper learns nothing about k AB Note: TTP needed for every key exchange, knows all session keys. (basis of Kerberos system)
6 Toy protocol: insecure against ac7ve afacks Example: insecure against replay afacks AFacker records session between Alice and merchant Bob For example a book order AFacker replays session to Bob Bob thinks Alice is ordering another copy of book
7 Key ques7on Can we generate shared keys without an online trusted 3 rd party? Answer: yes! Star7ng point of public- key cryptography: Merkle (1974), Diffie- Hellman (1976), RSA (1977) More recently: ID- based enc. (BF 2001), Func7onal enc. (BSW 2011)
8 End of Segment Dan Boneh
9 Online Cryptography Course Dan Boneh Basic key exchange Merkle Puzzles
10 Key exchange without an online TTP? Goal: Alice and Bob want shared key, unknown to eavesdropper For now: security against eavesdropping only (no tampering) Alice Bob eavesdropper?? Can this be done using generic symmetric crypto? Dan Boneh
11 Merkle Puzzles (1974) Answer: yes, but very inefficient Main tool: puzzles Problems that can be solved with some effort Example: E(k,m) a symmetric cipher with k {0,1} 128 puzzle(p) = E(P, message ) where P = 0 96 ll b 1 b 32 Goal: find P by trying all 2 32 possibili7es
12 Merkle puzzles Alice: prepare 2 32 puzzles For i=1,, 2 32 choose random P i {0,1} 32 and x i, k i {0,1} 128 set puzzle i E( 0 96 ll P i, Puzzle # x i ll k i ) Send puzzle 1,, puzzle 2 32 to Bob Bob: choose a random puzzle j and solve it. Obtain ( x j, k j ). Send x j to Alice Alice: lookup puzzle with number x j. Use k j as shared secret Dan Boneh
13 In a figure Alice puzzle 1,, puzzle n x j Bob k j k j Alice s work: O(n) (prepare n puzzles) Bob s work: O(n) (solve one puzzle) Eavesdropper s work: O( n 2 ) (e.g me)
14 Impossibility Result Can we achieve a befer gap using a general symmetric cipher? Answer: unknown But: roughly speaking, quadra7c gap is best possible if we treat cipher as a black box oracle [IR 89, BM 09]
15 End of Segment Dan Boneh
16 Online Cryptography Course Dan Boneh Basic key exchange The Diffie- Hellman protocol
17 Key exchange without an online TTP? Goal: Alice and Bob want shared secret, unknown to eavesdropper For now: security against eavesdropping only (no tampering) Alice Bob eavesdropper?? Can this be done with an exponen7al gap? Dan Boneh
18 The Diffie- Hellman protocol (informally) Fix a large prime p (e.g. 600 digits) Fix an integer g in {1,, p} Alice choose random a in {1,,p- 1} Bob choose random b in {1,,p- 1} B a (mod p) = (g b ) a = k AB = g ab (mod p) = (g a ) b = A b (mod p) Dan Boneh
19 Security (much more on this later) Eavesdropper sees: p, g, A=g a (mod p), and B=g b (mod p) Can she compute g ab (mod p)?? More generally: define DH g (g a, g b ) = g ab (mod p) How hard is the DH func7on mod p?
20 How hard is the DH func7on mod p? Suppose prime p is n bits long. Best known algorithm (GNFS): run 7me exp( ) cipher key size modulus size 80 bits 1024 bits 128 bits 3072 bits 256 bits (AES) bits Ellip7c Curve size 160 bits 256 bits 512 bits As a result: slow transi7on away from (mod p) to ellip7c curves
21 Ellip7c curve Diffie- Hellman Dan Boneh
22 Insecure against man- in- the- middle As described, the protocol is insecure against acjve afacks Alice MiTM Bob
23 Another look at DH Facebook g a g b g c g d Alice a Bob b Charlie c David d K AC =g ac K AC =g ac
24 An open problem Facebook g a g b g c g d Alice a Bob b Charlie c David d K ABCD K ABCD K ABCD K ABCD
25 End of Segment Dan Boneh
26 Online Cryptography Course Dan Boneh Basic key exchange Public- key encryp7on
27 Establishing a shared secret Goal: Alice and Bob want shared secret, unknown to eavesdropper For now: security against eavesdropping only (no tampering) Alice Bob This segment: a different approach eavesdropper??
28 Public key encryp7on Alice Bob E D
29 Public key encryp7on Def: a public- key encryp7on system is a triple of algs. (G, E, D) G(): randomized alg. outputs a key pair (pk, sk) E(pk, m): randomized alg. that takes m M and outputs c C D(sk,c): det. alg. that takes c C and outputs m M or Consistency: (pk, sk) output by G : m M: D(sk, E(pk, m) ) = m
30 Seman7c Security For b=0,1 define experiments EXP(0) and EXP(1) as: b Chal. (pk,sk) G() pk m 0, m 1 M : m 0 = m 1 c E(pk, m b ) Adv. A b {0,1} EXP(b) Def: E =(G,E,D) is sem. secure (a.k.a IND- CPA) if for all efficient A: Adv SS [A,E] = Pr[EXP(0)=1] Pr[EXP(1)=1] < negligible
31 Establishing a shared secret Alice (pk, sk) G() Alice, pk Bob choose random x {0,1} 128
32 Security (eavesdropping) Adversary sees pk, E(pk, x) and wants x M Seman7c security adversary cannot dis7nguish { pk, E(pk, x), x } from { pk, E(pk, x), rand M } can derive session key from x. Note: protocol is vulnerable to man- in- the- middle
33 Insecure against man in the middle As described, the protocol is insecure against acjve afacks Alice (pk, sk) G() MiTM (pk, sk ) G() Bob Alice, pk Bob, E(pk, x) Bob, E(pk, x) choose random x {0,1} 128
34 Public key encryp7on: construc7ons Construc7ons generally rely on hard problems from number theory and algebra Next module: Brief detour to catch up on the relevant background Dan Boneh
35 Further readings Merkle Puzzles are Op7mal, B. Barak, M. Mahmoody- Ghidary, Crypto 09 On formal models of key exchange (sec7ons 7-9) V. Shoup, 1999
36 End of Segment Dan Boneh
18733: Applied Cryptography Anupam Datta (CMU) Basic key exchange. Dan Boneh
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