Proof of Prop Sample Clauses

Proof of Prop. 2 Note that the first term in the expression above is maximized ∼ N ∼ N Recall that s (0, Q). Choose x (0, P ) to be a when α = 1. In this case we have that Gaussian random variable independent of s and let E[xs] = . √ Σ √ 1 (1 + ∆)(P + Q + 1 + 2ρ PQ) ρ PQ. Select u = x + αs and the lower bound follows by R = log P + Q + 1 + ∆ + 2ρ√PQ

Proof of Prop. 3 We evaluate the upper bound in Theorem 2 for the choice is not known to the eavesdropper.

Proof of Prop. 3 We evaluate the upper bound in Theorem 2 for the choice ze = zr + zδ, where zδ ∼ N (0, ∆) is independent of zr. next available symbol of tni and samples the channel input symbol from the distribution px|s,u. • At the end of the transmission it looks for the bin index of sn in Cs and declares this to be κs. • The overall secret-key is (κ1, . . . , κM , κs).

Proof of Prop. 2 √ ∼ N ∼ N Recall that s (0, Q). Choose x (0, P ) to be a Gaussian random variable independent of s and let E[xs] = ρ PQ. Select u = x + αs and the lower bound follows by evaluating r R = I(u; y ) − I(u; y ) the secret key κ. Thus we can bound the rate R = 1 H(κ) as below:

Proof of Prop. 1a): Due to Definition 1, for any rate point (r1, r2) ∈ Rcom(ǫ), there is a pair of transmit covariance matrices (Ψ1, Ψ2) such that com U stat(r1, r2, Ψ1, Ψ2) ≥ 1 − ǫ. (3.65) This implies that U stat(ri, Ψ1, Ψ2) ≥ 1 − ǫ, i = 1, 2. Hence, (r1, r2) ∈ Rind (ǫ1, ǫ2). i stat □ inst com Proof of Prop. 1b): For any rate point (r1, r2) ∈ Rcom(ǫ), we have U inst(r1, r2) = i Pr{B} ≥ 1 − ǫ. From (3.50) and (3.51), we get that U inst(r1, r2, t) ≥ 1 − ǫ = 1 − ǫi, inst since Pr{Ci} ≥ 0 and Pr{Di} ≥ 0. Hence, it follows that (r1, r2) ∈ Rind (ǫ1, ǫ2). □ This statement is quite intuitive; the definition of common outage says that the entire IC is in outage when at least one of the links is in outage. Individual outage is less restrictive since it allows one link to be in outage while the other is not.

Proof of Prop. 2 ∼ The lower bound expression follows from Theorem 1 by choosing x N (0, P) to be a Gaussian random variable independent of s and by choosing u = x + αs. In this case,

Proof of Prop. 3 We evaluate the upper bound in Theorem 2 for the choice ze = zr + zδ , where zδ ∼ N (0,∆) is independent of zr. I(s, x; yr|ye) = h(yr|ye)− h(yr|ye, x, s) = h(yr|ye)− h(zr|ze state with common knowledge,” in Proc. Int. Symp. In- form. Theory, Toronto, Canada, Jul. 2008, pp. 935– 939. [9] ——, “Coding and common reconstruction,” IEEE Trans. Inform. Theory, vol. 55, pp. 4995–5010, Nov. 2009.