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. In the following, an inner bound on the pairwise key capacity region of the source model with rate-limited public channel is given. First, we define: r12 = [I(S12; X2 |S23S32) − I(S12; X3, S13 |S23, S32)]+, r21 = [I(S21; X1 |S13S31) − I(S21; X3, S23 |S13, S31 )]+, r13 = [I(S13; X3 |S23S32) − I(S13; X2, S12 |S23, S32)]+, r31 = [I(S31; X1 |S12S21) − I(S31; X2, S32 |S12, S21 )]+, r23 = [I(S23; X3 |S13S31) − I(S23; X1, S21 |S13, S31)]+, r32 = [I(S32; X2 |S12S21) − I(S32; X1, S31 |S12, S21 )]+, I12 = I(S12; S21 |X3, S13, S23) , I13 = I(S13; S31 |X2, S12, S32) , I23 = I(S23; S32 |X1, S21, S31) , I1 = I(S21; S31 |X1) , I2 = I(S12; S32 |X2) , I3 = I(S13; S23 |X3) .
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. 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| ≤ −
Main Result. — We will consider a situation when at time k a correct process value yi(k — j) is not available and some other value y(1)(k — j) is supplied to Kalman filter in place of it. Suppose that later on, at time step k + h, the correct value of yi(k j) is received by the filter. The correction of the filter state and restoring the optimality is possible using a set of pre-computed gains. First, fusing the ‘fake’ data y(1)(k — j) results in the following data step at time k xˆk|m¯ (k)∪(i,j) = xˆk|m¯ (k) + K(m(k), i, j) y(1)(k — j) — Cxˆk—j|m¯ (k) (176) i Now, assume that the correct sample arrives h time-steps later. Then, after updating the state as in 175 by all freshly arrived samples that have not been replaced by fakes in the past, the correction of the incorrectly fused sample y(1)(k — j) is done as xˆk+h|m¯ (k+h) := xˆk+h|m¯ (k+h) + K (m(t + h) ∪ {(i, j + h)} , i, j + h) y (k — j) — y(1)(k — j) (177) Note that the two Kalman gains used in 176 and 177 are generally different. Note that the correction requires the ‘fake’ measurement to be stored until the correct value is available. In the case when the filter was optimal up to time k and, if no further sample was missed, the optimality is restored at time k + h after the correction 177 was performed. The non-optimality between times k and k + h is due to the fact that the estimator was mislead by pretending that it was given correct data at time k. As a result, the Kalman filter wrongly considers the state covariance matrix smaller than the real one and therefore, its responsiveness to prediction errors is reduced. Nevertheless, due to the assumption that the correct sample arrives within a finite time horizon and the subsequent optimality restoration, the impact on the overall estimation quality is limited. This approach greatly reduces the number of gains to be stored, depending on the strategy used. For instance, we can handle optimally samples arriving with delay up to nd1, while others, whose delay is between nd1 and nd max, are handled by replacement/correction technique. The savings in the number of gains to be stored is due to the reduction of possible values of the set m(k). As a special case we can consider the one (also used in [56] for time-varying filters) which replaces (and later corrects) all delayed samples. Then, the number of gains to be stored equals to the product of the number of sensors and the maximum delay, corresponding to m(k) = {(i, j)}. As for the value to be used in place of th...
Main Result. (4) For any specific compact C ⊂ R, there exist βi2, βi2 ∈ IL such that if (ηi(0), ζi(0)) ∈ Si(µi(0), µi(0)) with µi(0) ≤ µi(0) belonging to C and κi ∈ [κi, κi] with κi ≤ κi belonging to C, then there exist µi(t) and µi(t) satisfying − βi2(κi − µi(0), t) + κi ≤ µi(t) ≤ µi(t) ≤ βi2(µi(0) − κi, t) + κi (16) such that (η (t), ζ (t)) ∈ S (µ (t), µ (t)) (17) The main result of this paper is given by Xxxxxxx 1.
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. 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. Our bound uses a function that is defined in terms of a simple balls-into-bins problem. r,c r,c
