1. Dan Boneh Stream ciphers Stream ciphers are semantically secure Online Cryptography Course Dan Boneh Goal: secure PRG ⇒ semantically secure stream cipher

2. Dan Boneh Stream ciphers are semantically secure Thm: G:K {0,1} n is a secure PRG ⇒ stream cipher E derived from G is sem. sec. ∀ sem. sec. adversary A, ∃ a PRG adversary B s.t. Adv SS [A,E] ≤ 2 ∙ Adv PRG [B,G]

3. Dan Boneh Proof: Let A be a sem. sec. adversary. For b=0,1: W b := [ event that b’=1 ]. Adv SS [A,E] = | Pr[ W 0 ] − Pr[ W 1 ] | Chal. b Adv. A kKkK m 0, m 1  M : |m 0 | = |m 1 | c  m b ⊕ G(k) b’  {0,1} r  {0,1} n

4. Dan Boneh Proof: Let A be a sem. sec. adversary. For b=0,1: W b := [ event that b’=1 ]. Adv SS [A,E] = | Pr[ W 0 ] − Pr[ W 1 ] | For b=0,1: R b := [ event that b’=1 ] Chal. b Adv. A kKkK m 0, m 1  M : |m 0 | = |m 1 | c  m b ⊕ r b’  {0,1} r  {0,1} n

5. Dan Boneh Proof: Let A be a sem. sec. adversary. Claim 1: | Pr[R 0 ] – Pr[R 1 ] | = Claim 2: ∃ B: | Pr[W b ] – Pr[R b ] | = ⇒ Adv SS [A,E] = | Pr[W 0 ] – Pr[W 1 ] | ≤ 2 ∙ Adv PRG [B,G] 01 Pr[W 0 ]Pr[W 1 ]Pr[R b ]

6. Dan Boneh Proof of claim 2: ∃ B: | Pr[W 0 ] – Pr[R 0 ] | = Adv PRG [B,G] Algorithm B: Adv PRG [B,G] = PRG adv. B (us) Adv. A (given) c  m 0 ⊕ y y ∈ {0,1} n m 0, m 1 b’ ∈ {0,1}

7. Dan Boneh End of Segment