Proof Outline Sample Clauses

Proof Outline. Our lower bound is derived from the following two lemmas. The first lemma shows a random (g, e, d) constitutes an ideally-secure PKE protocol, even against adversaries that have access to the oracles (u, v), in addition to (g, e, d). The second lemma shows that the security of any proposed CKE protocol (CRSGen, Init, Comm, Derive), instantiated with a random O := (g, e, d), may be broken by an adversary making at most a polynomial number of queries to (O, u, v). The black-box separation will then follow. 2λ/2 Lemma 4.3 (O is secure against (O, u, v)). For any polynomial-query adversary A: Pr[AO,u,v(pk, c) = b] ≤ 1/2 + 1 , where (g, e, d, u, v) →$ Ψ, O := (g, e, d), b →$ {0, 1}, sk →$ {0, 1}λ, pk = g(sk), r →$ {0, 1}λ and c = e(pk, b; r). The following lemma shows how to break compact CKE constructions relative to the PKE oracles. The lemma shows that even for encrypting single-bit keys (i.e., |K| = 1), a CKE ciphertext cannot be sub-linear in n.
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