Lemma 2. .16.1. Let k = [k1, k2, k3, k4] be a key agreement scheme. Let α, β, γ, δ, δ′, η, η′, φ be functions, such that K1(a, b) = δ(k1(α(a), β(b))) K2(b, c) = η(k2(β(b), γ(c))) K3(a, e) = φ(k3(α(a), η′(e))) K4(d, c) = φ(k4(δ′(d), γ(c))). are well-defined. If δ′(δ(x)) = x and η′(η(x)) = x for all x in the appropriate domains, then K = [K1, K2, K3, K4] is a key agreement scheme.
Lemma 2. 2.6. Let A ≥ 1 and ε ∈ (0, 1 ] be fixed. Let X ≥ 1, 1 ≤ Q ≤ ∆ and ∆ = Xθ with 1 + 2ε ≤ θ ≤ 1. Then we have that X1/6+ε x<n≤x+q∆ Σ Σ ∫ 2X Σ q≤Q χ(q) X (Λ(n)χ(n) − δχ) dx ≪ Q3∆2X logA X , where we define δχ = 1 if χ = χ0 and δχ = 0 otherwise. Proof. The proof can be adapted from the proof given in [21, Section 4], as described in the Appendix.
Lemma 2. 3.12 (The Singular Series). Let h be a non-zero even integer and Q0 be defined as in (2.3.5). Let A > 3 be fixed and let (log X)19+ε ≤ H ≤ X log−A X. Then, for all but at most O(HQ−1/3) values of 0 < |h| ≤ H we φ2(q) = S(h) + O(Q0 log H). have that Σ µ2(q)cq(−h) q≤Q0 −1/3 −q
Lemma 2. 3.29 (Xxx´asz-Xxxxxxxxxx Inequality). Let T ≥ 1, q ≥ 2. Let T ⊂ [−T, T ] be a well-spaced set, and S = T × {χ mod q}. With the same assumptions as Lemma 2.3.27, we have that (tΣ,χ)∈S F (it, χ) 2 φ(q)X + (qT )1/2 (log(2qT )) Σ| | ≪ |S| ≤ q X<n 2X (n,q)=1 |an| . Proof. This is [20, Lemma 7.4].
Lemma 2. 9. If ξ is cone-mixing with angle θ > arctan(α + β), then ξ∗ is cone-mixing with angle 1 π.
Lemma 2. 2.9 Let 1 ≤ m = m(n) < n and p = p(n) be such that p = m/n. Let P be an event in the probability space of the random sets [n]p. If [n]p is in P w.o.p., then [n]m is in P ∩ [n] w.o.p. m C > 0, there exists a cohnstant Cr >i 0, where Cr → ∞has C → ∞, such tihat Proof Let Q be the complement of P . We shall show that, for any constant the following holds. If P O(n—C′ ). [n]p is in Q = O(n—C), then P [n]m is in Q ∩ [n] = Xxxxxx’x inequality (see [6, p. 35] and [23, p. 17]) states that Ph[n]m is in Q ∩ [n] i = O(√m) · Ph[n] is in Qi. (2.12) m p Since, by hypothesis, P [n]p is in Q = O(n—C) holds for any constant C > 0, inequality (2.12) implies that
Lemma 2. 2. The set D defined in (2.27) is normal. Furthermore, the constraint set is compact, bounded, and connected. The program- ming problem in (2.25) corresponds exactly to the monotonic optimization problem. Therefore, we can apply the outer polyblock approximation algorithm described in [3] to solve all three problems, the weighted sum-rate maximization in (2.17), the proportional fair problem in (2.18), and the max-min problem in (2.19).
Lemma 2. There is a value of λ such that the contract is complete if and only if the arbiter is biased in favor of honest parties.
Lemma 2. 4. If H is a normal subgroup of G of order prime to char(k), then one has + QuotG..V − V >HΣ/.V − V ≥HΣΣ = QuotG/H ..V ≥H − V >HΣ Proof. Since ∧ QuotG,H.Th.V /V ≥HΣΣΣ. QuotG = QuotG/H QuotG,H, it is sufficient to show that QuotG,H((V − V >H)/(V − V ≥H)) is isomorphic to (V ≥H − V >H)+ ∧ QuotG,H(Th(V /V ≥H)) as a G/H-space. Since the order of H is prime to char(k), there is an isomorphism V = V ≥H ⊕ (V /V ≥H ). Using this isomorphism, we get an isomorphism + .V − V >HΣ/.V − V ≥HΣ = .V ≥H − V >HΣ ∧ Th.V /V ≥HΣ. Since the action of H on V ≥H is trivial, we get + QuotG,H..V − V >HΣ/.V − V ≥HΣΣ = .V ≥H − V >HΣ Combining Lemmas 2.2 and 2.4, we get the following result. ∧ QuotG,H.Th.V /V ≥HΣΣ.
Lemma 2. 1.2. The morphism φ : X (p−nϵ) → X (p−(n−1)ϵ) has a canonical finite flat formal model φ : X(p−nϵ) → X(p−(n−1)ϵ). It reduces to the relative Frobenius map mod p1−є. We may interpret φ as a formal model XΓ0(pn)(ϵ)a → XΓ0(pn−1)(ϵ)a of the forgetful map.
