By Szabo A., Boucher K., Jones D.

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D. (independent and identically distributed) assumption, one can study asymptotic quantities such as key rates. 4. An asymptotic protocol Π is a sequence of pairs (Πk , τk ) where, for any k ∈ N, Πk is a protocol and τk ∈ N. The rate of Π is deﬁned by rate(Π) := lim k→∞ k . 5. , limk→∞ εk = 0) such that, for any k ∈ N, Π k ×k R×τk −→ . εk S See below for examples of asymptotic protocols. 6. Let Π = {(Πk , τk )}k∈N and Π ′ = {(Πk′ , τk′ )}k∈N be asymp¯ := Π ′ ◦ Π is then deﬁned by the protocol totic protocols.

2). Let E be an event with Pr[E] = 1 − ε such that Hmax (EX|Y ) = Hεmax (X|Y ). 5) Let Y be the range of the random variable Y . 7, there exists a function dF from U × Y to X such that, for any y ∈ Y, Pr E ∧ (dF (F (X), Y ) = X) Y = y ≤ |{x ∈ X : PEXY (x, y) > 0}| |U| = 2Hmax (EX|Y )−ℓ . Moreover, we have Pr dF (F (X), Y ) = X ≤ Pr E ∧ (dF (F (X), Y ) = X) + (1 − Pr[E]) ≤ max Pr E ∧ (dF (F (X), Y ) = X) Y = y + ε. 5) concludes the proof. 5 Privacy Ampliﬁcation Privacy ampliﬁcation is the art of shrinking a partially secure string S to a highly secret string S ′ by public discussion.

Finally, we mention a result showing that an arbitrarily large gap can separate the secrecy required for constructing the distribution from the amount of extractable secrecy. 42 U. Maurer et al. Information of Formation Instead of transforming weakly correlated and partially secure data into a secure key, one could also do the opposite [233]. 10. The information of formation (also called key cost) of a tripartite probability distribution PXY Z is deﬁned by Form Iform (PXY Z ) := rate(SK1 × AuthA→B =⇒ Source(PXY Z ))−1 .