Main Result. Time-varying Communication Topology the communication graph. Then (9) yields min . V˙ ≤ −λ .BT BΣ |x¯|2 + |x¯| B T B δ √ uΣ m In this section we treat the case when the communication ǁ ǁ topology is time-varying, allowing each agent to lose/create ≤ −λmin .BT BΣ |x¯| |x¯| − BT B δu√m λmin(BT B) new communication links with other agents as the closed- loop system evolves. The problem in this case is that it’s Thus, all solutions of the closed-loop system enter the ball not possible to use V = 1 x¯T x¯ as a common Lyapunov . BT Σ √ B δu m function for the switched system, since the vector x¯ changes discontinuously whenever edges are added or deleted when x : |x¯| ≤ λmin (BT B) the communication topology changes. A different energy function is used and in particular, the function λmin(BT B) centered at x¯ = 0 of radius ǁB Bǁδu m in finite time. In the case of a logarithmic quantizer we have q = ql and |ql (x¯) − x¯| ≤ δl |x¯| and (9) yields W = max {x1, . . . , xN } − min {x1, . . . , xN } (16) which can act as a common Lyapunov function for the ∆ V˙ ≤ −λmin .BT BΣ |x¯| + BT B δl |x¯|2 , switched system. ∆ Let xmax = max {x1, . . . , xN } , xmin = so that V˙ ≤ − |x¯|

Main Result. In the following, an inner bound on the pairwise key capacity region of the source model with rate-limited public channel is

Main Result. ‌ We will use a notion from Daemen et al. [15], namely that of the multicollision limit function. ∈ Definition 1 (multicollision limit function). Let M, c, r N. Consider the experiment of throwing M balls uniformly at random in 2r bins, and let μ be the maximum number of balls in a single bin. We deﬁne the multicollision limit r,c function νM as the smallest natural number x that satisﬁes 2c Pr (μ > x) ≤ x . We derive the following result on the keyed duplex under leakage. Theorem 1. Let b, c, r, k, u, α, λ ∈ N, with c + r = b, k ≤ b, α ≤ b − k, and $ λ ≤ 2b. Let p ←− perm(b) be a random permutation, and K ←D−K− ({0, 1}k)u a L { { } × { } → { } } random array of keys. Let = L : 0, 1 b 0, 1 b 0, 1 λ be a class of leakage functions. For any distinguisher D quantiﬁed as in Sect. 5.1, KD AdvL-naLR(D) νfixN 2νM N 2νM νM (L + Ω)+ νfix −1 (L + Ω) r,c ≤ 2c−(R+1)λ + 2c−(R+1)λ + r,c + 2c r,c + 2 2c−Rλ + .M−L−qΣ + (M − L − q)(L + Ω) 2b−λ .M+NΣ + .NΣ 2b . Σ + qIV N + q(M − q) 2H∞(DK )+xxx{c,max{b−α,c}−k}−(R+qδ)λ 2H∞(DK )−qδλ + 2 2H∞(DK ) . In addition, except with probability at most the same bound, the ﬁnal output states have min-entropy at least b − λ. The proof is given in Sect. 5.4; we ﬁrst give an interpretation of the bound in Sect. 5.3.

Main Result. ⊕ The main theorem of Xxxxxxx’s mirror theory, simply dubbed “mirror theorem”, is the following. It corresponds to “Theorem Pi Pj for any ξmax ” of Patarin [40, Theorem 6]. Theorem 2 (mirror theorem). Let ξ ≥ 2. Let E be a system of equations over the unknowns P that is (i) circle-free, (ii) ξ-block-maximal, and (iii) non- degenerate. Then, as long as (ξ − 1)2 · r ≤ 2n/67, the number of solutions for P such that Pa ƒ= Pb for all distinct a, b ∈ {1,..., r} is at least (2n)r 2nq . The quantity measured in above theorem (the number of solutions...) is called hr in [40]. Hr is subsequently deﬁned as 2nqhr. The parameter H has slightly diﬀerent meanings in [39, 41, 42], namely the number of oracles whose outputs could solve the system of equations. In the end, these deﬁnitions yielded the naming of the H-coeﬃcient technique of Theorem 1. For the mirror theorem, we have opted to stick to the convention of [40] as its deﬁnition is pure in the sense that it is independent of the actual oracles in use. − − − · ≤ In Appendix A, we give a proof sketch of Xxxxxxx 2, referring to [40] for the details. In the proof sketch, it becomes apparent that the side condition (ξ 1)2 r 2n/67 can be improved (even up to 2n/16) quite easily. Patarin ﬁrst derived the side condition symbolically and only then derived the speciﬁc constants. Knowing the constants in advance, we reverted the reasoning. How- ever, to remain consistent with the theorem statement of [40], we deliberately opted to leave the 67 in; the improvement is nevertheless only constant. The term (ξ 1)2 is present to cover worst-case systems of equations; it can be improved to (ξ 1) in certain cases [44]. Fortunately, in most cases ξ is a small number and the loss is relatively insigniﬁcant.

Main Result. The main theorem of Xxxxxxx’s mirror theory, tailored to the case where we have a partition of the unknowns into two disjoint sets, is given below. We follow [20], with the side condition on 2n/64 from [22]. Theorem 3 (mirror theorem). Let {1,..., r} = f1 ∪ f2 be a partition of the indices. Let S be a system of equations over the unknowns У that is (i) circle-free, (ii) ξ-block-maximal, and (iii) non-degenerate. Then, as long as ξ2 · max{|f1|, |f2|} ≤ 2n/64, the number of solutions for У such that Pi i, j ∈ fA (l = 1, 2) is at least Pj for all NonEq(f1, f2; S) , 2nq where NonEq(f1, f2; S ) denotes the number of solutions to У that satisfy Pi Pj ∈ f S for all i, j A (l = 1, 2) as well as the inequalities imposed by (but the equalities themselves released). A lower bound on the technical quantity NonEq(f1, f2; S ) can be derived as follows. Every equation Pϕ(a) ⊕ Pϕ(b) = λ = 0 in S imposes Pϕ(a) =/ Pϕ(b). As ϕ(a) ∈ f1 and ϕ(b) ∈ f2 are in distinct index sets, this inequality Pϕ(a) Pϕ(b)

Main Result. In this section we present and discuss the statement of our main result of this chapter; a theorem about the existence of a dimension gap under some different assumptions to the analogous theorem in [KPW]. Before we state the result, we introduce some additional notation: for a finite word w ∈ Σ∗ we denote the periodic point in Σ obtained by repeating the finite word w by (w)∞ (note that since Σ is the full shift space this is well defined for any w ∈ Σ∗). We denote the projection of this periodic point (which is periodic for T ) by zw = Π((w)∞). S For simplicity, in what follows we’ll assume that if Tj > 0 then I1 = (0, a) for some a < 1 and if Tj < 0 then I1 = (b, 1) for some b > 0.

Main Result. ‌ Our bound uses a function that is defined in terms of a simple balls-into-bins problem. r,c r,c

Main Result. Upper Bound on the Length of a Multiparty Secret Key |K| In this section, we present a new methodology for proving converse results for the multiparty SK agreement problem. Our main result is an upper bound on the length log of a SK generated by multiple parties, using interactive public communication. Consider a (nontrivial) partition π = {π1, ..., πl} of the set M. Heuristically, if the underlying distribution of the observations PXMZ is such that XM are conditionally independent across the partition π given Z, the length of a SK that can be generated is 0. Our approach is to bound the length of a generated SK in terms of “how far” is the distribution PXMZ from another distribution Qπ that renders XM conditionally independent across the partition π given Z – the closeness of the two distributions is measured by βs PXMZ, QXMZ . XMZ . π Σ Speciﬁcally, for a partition π with |π| ≥ 2 parts, let Q(π) be the set of all distributions Qπ XMZ that factorize as follows: (x ,..., x |z) = Q1 m |π| Qπ XM|Z π Xπi |Z i=1 (xπi |z). (7) Our main result is given below. Σ| ≤ −

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