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. Time-varying Communication Topology
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) .
Theorem 1: In the described setup, all rates in the closure of the convex hull of the set of all key rate triples (R12, R13, R23)
I( S21,S31;X2,X3|X1) +I(S12,S32;X1,X3|X2) +I(S13,S23;X1,X2|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 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) ≤ 2c−(R+1)λ + 2c−(R+1)λ + 2c + 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 У 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 У such that Pa /= Pb for all distinct a, b ∈ {1,..., r} is at least (2n)r 2nq .
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. Upper Bound on the Length of a Multiparty Secret Key
Main Result. We prove that stable Xxxxxxxx invariants of a curve over a number field are polynomial in the Belyi degree. We use our results to give algorithmic, geometric and Diophantine applications in the following two chapters. Let X be a smooth projective connected curve over Q of genus g. In [5] Xxxxx proved that there exists a finite morphism X → P1 ramified over at most three points. Let degB (X) denote the Belyi degree of X (introduced in Section 1.9). Since the topological fundamental group
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. Our bound uses a function that is defined in terms of a simple balls-into-bins problem.
