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Cryptography: Review Day David Brumley dbrumley@cmu.edu Carnegie Mellon University

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Cryptonium Pipe Goals: Privacy, Integrity, and Authenticity 2 Alice Bob Public Channel Eve E D cc’ m keke m or error keke read/write access

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Privacy and Encryption 4

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Perfect Secrecy [Shannon1945] (Information Theoretic Secrecy) Defn Perfect Secrecy (informal): We’re no better off determining the plaintext when given the ciphertext. 5 AliceBob Eve 1.Eve observes everything but the c. Guesses m 1 2.Eve observes c. Guesses m 2 Goal: \Pr[m = m_1] = \Pr[m = m_2]

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The One Time Pad 6 Miller, 1882 and Vernam, 1917 \begin{align*} E(k,m) &= k \oplus m = c\\ D(k,c) &= k \oplus c = m\\ \end{align*} \[ \begin{split} D(k,E(k,m)) &= D(k, k \oplus m)\\ &= k \oplus (k \oplus m)\\ &= 0 \oplus m \\ &= m \end{split} \] m:0110110 k:1101000 c:1011110 k:1101000 m:0110110 M = C = K = {0,1} n

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Block Ciphers Modes of operations – CBC, CTR, etc. – What modes do for security, e.g., why ECB is bad, why randomize an IV for CBC, etc. Definitions – Is a block cipher a PRP or PRF Attacks 7

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Exhaustive Search for block cipher key Goal: given a few input output pairs (m i, c i = E(k, m i )) i=1,..,nfind key k. Attack: Brute force to find the key k. Homework: What is the probability that the key k found with one pair is correct? For two pairs? 8

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Meet in the middle attack Define 2E( (k 1,k 2 ), m) = E(k 1, E(k 2, m) ) key-len = 112 bits for 2DES Idea: key found when c’ = c’’: E(k i, m) = D(k j, c) m c' … … c … … c’’ m E(k 2,⋅)E(k 1,⋅) c 9

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Semantic Security Game 10 E 2. Pick b=0 3. k=KeyGen(l) 4. c = E(k,m b ) A 1. Picks m 0, m 1, |m 0 | = |m 1 | 5. Guess and output b’ m 0,m 1 c World 0 E 2. Pick b=1 3. k=KeyGen(l) 4. c = E(k,m b ) A 1. Picks m 0, m 1, |m 0 | = |m 1 | 5. Guess and output b’ m 0,m 1 c World 1 A doesn’t know which world he is in, but wants to figure it out. Semantic security is a behavioral model getting at any A behaving the same in either world when E is secure.

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Semantic security under CPA 11 Modes that return the same ciphertext (e.g., ECB, CTR) for the same plaintext are not semantically secure under a chosen plaintext attack (CPA) (many-time-key) if c b = c 0 output 0 else output 1 m 0, m 0 ∊ M C 0 ← E(k,m) m 0, m 1 ∊ M C b ← E(k,m b ) Challenger k ← K Adversary A

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Semantic security under CPA 12 Modes that return the same ciphertext (e.g., ECB, CTR) for the same plaintext are not semantically secure under a chosen plaintext attack (CPA) (many-time-key) if c b = c 0 output 0 else output 1 m 0, m 0 ∊ M C 0 ← E(k,m) m 0, m 1 ∊ M C b ← E(k,m b ) Challenger k ← K Adversary A Encryption modes must be randomized or use a nonce (or are vulnerable to CPA)

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Hashes and MACS 13

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Message Integrity Goal: integrity (not secrecy) Examples: – Protecting binaries on disk. – Protecting banner ads on web pages Security Principles: – Integrity means no one can forge a signature 14

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PRF Security Game (A behavioral model) 15 E 2. if(tbl[x] undefined) tbl[x] = rand() return y =tbl[x] A 1. Picks x 5. Guess and output b’ x y World 0 E y = PRF(x) A 1. Picks x 3. Outputs guess for b x y World 1 A doesn’t know which world he is in, but wants to figure it out. For b=0,1: W b := [ event that A(W b ) =1 ] Adv SS [A,E] := | Pr[ W 0 ] − Pr[ W 1 ] | ∈ [0,1] Always 1

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Secure PRF: An Alternate Interpretation 16 For b = 0,1 define experiment EXP(b) as: Def: PRF is a secure PRF if for all efficient A: Challenger F Adversary

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Secure MAC Game Security goal: A cannot produce a valid tag on a message – Even if the message is gibberish 17 Challenger 1. k = KeyGen(l) 3. Compute i in 0...q: t i = S(m i, k) 5. b = V(m,t,k) Adversary A 2. Picks m 1,..., m q 4. picks m not in m 1,...,m q Generates t m 1,...,m q t 1,...,t q m,t b = {yes,no} existential forgery if b=“yes”

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Birthday Paradox Rule of Thumb Given N possibilities, and random samples x 1,..., x j, PR[x i = x j ] ≈ 50% when j = N 1/2 18

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Generic attack on hash functions Let H: M {0,1} n be a hash function ( |M| >> 2 n ) Generic alg. to find a collision in time O(2 n/2 ) hashes Algorithm: 1.Choose 2 n/2 random messages in M: m 1, …, m 2 n/2 (distinct w.h.p ) 2.For i = 1, …, 2 n/2 compute t i = H(m i ) ∈{0,1} n 3.Look for a collision (t i = t j ). If not found, got back to step 1. How well will this work? 19

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Brute Force Online Brute Force Attack: input: hp = hash(password) to crack for each i in dictionary file if(h(i) == hp) output success; Time Space Tradeoff Attack: precompute: h(i) for each i in dict file in hash tbl input: hp = hash(password) check if hp is in hash tbl 20 “rainbow tables”

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Salts Enrollment: 1.compute hp=h(password + salt) 2.store salt || hp Verification: 1.Look up salt in password file 2.Check h(input||salt) == hp What is this good for security, given that the salt is public? 21 Salt doesn’t increase security against online attack, but does make tables much bigger.

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Authenticated Encryption 22

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Motivating Question: Which is Best? E(k E, m||tag) S(k I, m) m Encryption Key = K E ; MAC key = k I Option 1: SSL (MAC-then-encrypt) mtagm S(k I, c)E(k E, m) m Option 2: IPsec (Encrypt-then-MAC) mmtag S(k I, m)E(k E, m) m Option 3: SSH (Encrypt-and-MAC) mmtag 23

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An authenticated encryption system (E,D) is a cipher where As usual: E: K × M × N ⟶ C but D: K × C × N ⟶ M ∪{ ⊥ } Security: the system must provide – Semantic security under CPA attack, and – ciphertext integrity. The attacker cannot create a new ciphertext that decrypts properly. reject ciphertext as invalid 24

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CCA Game Definition 25 Let ENC = (E,D) over (K,M,C). For b = {0,1}, define EXP(0) and EXP(1) b Chal. k K Adv. b’ {0,1} m i,0, m i,1 M : |m i,0 | = |m i,1 | c i E(k, m i,b ) for i=1,…,q: (1) CPA query: c i C : c i ∉ {c 1, …, c i-1 } m i D(k, c i ) (2) CCA query: Ex: could query a changed c i

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Public Key Cryptography 26

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Eve observes: g, g a, g b Goal: compute a (or b) (i.e., calculate the discrete log) or compute g ab 27 3. g a mod p 4. g b mod p 1. Pick a from [0,p-1)2. Pick b from [0,p-1) 5. Compute (g a ) b mod p as secret key 6. Compute (g b ) a mod p as secret key Alice Bob Eve

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MITM Adversary As described, Diffie-Hellman is insecure against active Man In The Middle (MITM) attacks AliceBobMITM g a mod pg m mod p g b mod p g m mod p g ma mod p g mb mod p 28

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Public Key Encryption Def: a public-key encryption system is a triple of algorithms (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): determisitic alg. that takes c∈C and outputs m ∈ M or ⊥ Consistency: ∀(pk, sk) output by G : ∀m∈M: D(sk, E(pk, m) ) = m Note: Without randomization, an attacker can determine E(pk,m 1 ) = E(pk,m 2 ) when m 1 =m 2 29

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Semantic Security For b=0,1 define experiments EXP(b) (i.e., EXP(0) and EXP(1)): Def: Enc = (G,E,D) is sem. secure (a.k.a IND-CPA) if for all efficient A: Adv SS [A, Enc ] = |Pr[EXP(0)=1] – Pr[EXP(1)=1] | < negligible Chal. b Adv. A (pk,sk) G() m 0, m 1 M : |m 0 | = |m 1 | c E(pk, m b ) b’ {0,1} EXP(b) pk No query encryptions of messages. Why? 30

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Easy and Hard Problems Factoring Discrete Log Exponentiation 31

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32 Questions?

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END

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34 Thought

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