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Message Integrity

Message Integrity. CS255 Winter ‘06. Verify tag: V (k, m, tag) = `yes’. ?. Message Integrity. Goal: provide message integrity. No confidentiality. ex: Protecting public binaries on disk. Protecting ads. Requires secret key k unknown to attacker.

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Message Integrity

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  1. Message Integrity CS255 Winter ‘06

  2. Verify tag: V(k, m, tag) = `yes’ ? Message Integrity • Goal: provide message integrity. No confidentiality. • ex: Protecting public binaries on disk. Protecting ads. • Requires secret key k unknown to attacker. • Def: MAC I = (S,V) defined over (K,M,T) is a pair of algorithms: - S(k,m) outputs tT , - V(k,m,t) outputs `yes’ or `no’ k k Message m tag Alice Bob Generate tag: tag  S(k, m)

  3. Secure MACs • Attacker’s power: chosen message attack. • for m1,m2,…,mq attacker is given ti S(k,mi) • Attacker’s goal: existential forgery. • produce some new valid message/tag pair (m,t). (m,t)  { (m1,t1) , … , (mq,tq) } • Note: attacker cannot even produce a valid tag for a nonsensical message.

  4. ti S(k,mi) (m,t) mi M b=1 when V(k,m,t) = `yes’ and (m,t)  { (m1,t1) , … , (mq,tq) } b=0 otherwise Secure MACs • For a MAC I=(S,V) and adv. A we define a MAC game: • Def: I=(S,V) is a secure MAC if for all “efficient” A: MAC Adv[A,I] = Pr[Chal. outputs 1] is “negligible.” Chal. Adv. kK b

  5. Any secure PRF is a secure MAC • Let F be a PRF over (K,X,Y). Define a MAC IF = (S,V): • S(k,m) = F(k,m) • V(k,m,t): output `yes’ if t = F(k,m) and `no’ otherwise. • Theorem: If F is a secure PRF and 1/|Y| is negligible then IF is a secure MAC. In particular, for any MAC adversary A attacking IF there exists a PRF adversary B attacking F s.t.: MAC Adv[A, IF]  PRF Adv[B, F] + 1/|Y|  IF is secure as long as |Y| is large, say |Y| = 280 .

  6. Proof Sketch • Intuition: • Adversary A issues chosen message queries m1,m2, … • Gets back F(k,m1) , F(k,m2) , … • Must guess F(k,m) for m  {m1, m2, … } • But F is a PRF, so prob A guesses F(k,m) is 1/|X| • Truncating MACs: • Suppose MAC is a PRF outputting n-bit tags (|Y| = 2n). • It is OK to truncate the MAC output to w<n bits. … as long as 1/2w is still negligible (say w64)

  7. Examples • AES: a MAC for 16-byte messages. • Main question: how to convert Small-MAC into a Big-MAC ? • Two main constructions: • CBC-MAC (banking – ANSI X9.9, X9.19, FIPS 186-3) • HMAC (Internet protocols: SSL, IPsec, SSH, …) • Both convert a small-PRF into a big-PRF.

  8. Raw CBC Construction 1: (E) CBC-MAC m[0] m[1] m[3] m[4]     F(k,) F(k,) F(k,) F(k,) F(k1,) tag Let F be PRF over (K,X,X) Define new PRF FCBC over (K2 , XL , X )

  9. CBC-MAC: Analysis • CBC-MAC Theorem: For any L>0, If F is a secure PRF over (K,X,X) then FCBC is a secure PRF over (K, XL, X). In particular, for a q-query PRF adv. A attacking FCBCthere exists a PRF adversary B s.t.: PRF Adv[A, FCBC]  PRF Adv[B, F] + 2 q2 Lo(1) / |X| • Note: CBC-MAC is secure as long as q << |X|1/2

  10. Why the last encryption step? • Suppose we define a MAC IRAW = (S,V) where S(k,m) = RawCBC(k,m) • Fact: IRAW is easily broken using a chosen msg attack. • Adversary works as follows: • Pick an arbitrary one-block message mM • Request tag for m. Get t = F(k,m) • Output t as MAC forgery for the message (m, tm) • Indeed: RawCBC(k, (m, tm) ) = F(k, t(tm) ) = t • Unimportant note: RawCBC is secure for prefix-free inputs.

  11. CBC-MAC Padding • What is length of m is not multiple of block-size? • Bad idea: pad m with 0’s • Vulnerable to chosen message attack: ask for tag on m and obtain tag on m||0 • ISO: pad with “100000”. Add new block if needed. • The “1” indicates beginning of pad. • CMAC: different padding. Never adds an extra block.

  12. P(k,3) P(k,1) P(k,0) P(k,2) Construction 2: PMAC • CBC-MAC is sequential. PMAC – Parallel MAC. m[0] m[1] m[3] m[4]     F(k,) F(k,) F(k,) F(k,)  F(k1,) tag

  13. PMAC: Analysis • PMAC Theorem: For any L>0, If F is a secure PRF over (K,X,X) then FPMAC is a secure PRF over (K, XL, X). In particular, for a q-query PRF adv. A attacking FPMACthere exists a PRF adversary B s.t.: PRF Adv[A, FPMAC]  PRF Adv[B, F] + 2 q2 L2 / |X| • Note: PMAC is secure as long as qL << |X|1/2 • Note: PMAC is incremental. Homework.

  14. Construction 3: HMAC (Hash-MAC) • Most widely used MAC on the Internet. • … but, we first we discuss hash function.

  15. Collision Resistant Hashing

  16. Collision Resistance • Let H: M T be a hash function. • A collision for H is a pair m0 , m1  M such that: H(m0) = H(m1) and m0  m1 • Def: A function H is collision resistant if for all (uniform) “efficient” algs. A: CR Adv[A,H] = Pr[ A outputs collision for H] is “negligible” • Used to have lots of examples: MD5, SHA1, … • Currently, only: SHA-256, SHA-512, Whirpool 44.5MB/sec, 11.4, 12.1 216MB/s 68

  17. MACs from Collision Resistance • Let I = (S,V) be a MAC for small messages over (K,M,T). Let H: Mbig M • Define: Ibig = (Sbig , Vbig ) over (K, Mbig, T) as: Sbig(k,m) = S(k,H(m)) ; Vbig(k,m,t) = V(k,H(m),t) • Theorem: If I is a secure MAC and H is collision resistant then Ibig is a secure MAC. • So: S(k,m) = AES(k, SHA-256(m)) is a secure MAC.

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