Theorem 4. Every conventional linear code of length n with minimum Ham- ming distance d can be converted to a code of length 2n with minimum 0-1 distance d by replacing every bit in the original codewords by a pair of bits, namely by replacing 0 by 01 and 1 by 10. We omit the proof since it is obvious. The code obtained from the linear code with the method of Theorem 4.3.5 is called a 0-1 code, and its codewords are called 0-1 codewords. Note that the number of 1’s is equal to that of 0’s in any 0-1 codeword. Xxxxx and Xxx can agree on a linear code. For any source state, a corresponding codeword is obtained according to the encoding rule of the linear code. Subsequently, the codeword can be changed into a 0-1 codeword. Then both Xxxxx and Xxx are able to construct the authenticator for the source state by taking together some bits from her or his initial string, where the positions of the bits are determined by the indices of 1-entries of the 0-1 codeword. More precisely, the authentication scheme can be described as follows. Let Xxxxx be the sender and Bob the receiver.
Theorem 4. Let ts, ti such that 2ti + ts < n, ti ≤ ts < n . There exists an T2 = O(1) round deterministic protocol GCS∗ab that satisfies the following in the context of the graded consensus primitive (See Definition 2) in a ∆-synchronous network if the round distance between protocol initiation times of any two honest parties is at most 1. – Authenticated ts-validity. – Authenticated ts-termination. – Sabotaged ti-security. Furthermore, given ΠAW has at most O(λn2) bit complexity and constant round complexity, so too does GC∗Sab.
Theorem 4. Suppose that an [n, k, d] linear code is employed in Scheme 4.
Theorem 4. Assume that δ > 0 and γ − r +δ ≥ 0 or δ = 0 and γ − r > σ2 . Define f by (4.11), b by Proposition 4.1 and Proposition 4.2. Then the initial value of stock loan with automatic termination clause is f (S 0).
Theorem 4. (On the connection between the family (𝑆𝑡)𝑡≥0 and the semigroup with the generator 𝐿) Suppose that 𝑔 𝑋, 𝐵 𝑋𝐻 , 𝐶 𝑋, and for every 𝑥 𝐻 we have 𝑔(𝑥) 𝑔0 const > 0 and 𝐶(𝑥) 0. As 𝐶 𝑋, there exists a sequence (Xx) 𝐷, converging to 𝐶 uniformly; let us additionally claim that this sequence can be selected in such a way that Xx(𝑥) 0 for all 𝑗 N and all 𝑥 𝐻. Then the following holds:
Theorem 4. 2.1. Define fn, f∞ : Θ → R by (4.2) and (4.3), where Θ ⊂ (Hs)k for s ≥ 1 is given by (4.4), respectively. Under Assumptions 3.1-3.3 any sequence of minimizers µ(n) of fn are, with probability one, weakly compact and any weak limit µ(∞) is a minimizer of f∞. Furthermore if µ(nm) ~ µ(∞) in Hs then µ(nm) → µ(∞) in L2.
Theorem 4. AERID can be solved with high probability in an n-node d-regular expander network with up to |B| = o(n/ log n) Byzantine nodes in O˜(n) rounds. Once again, we point out that nodes do not need to to know if they are part of the broadcasting or receiving subsets to solve the AERID primitive. And in fact, in the presented AERID protocol, all nodes ”attempt” to be in both subsets, and those who fail are not aware of it. A major disadvantage of the presented AERID protocol lies in its congestion issues, even when we consider only the Θ(n log n) good tokens that contain any given node’s ID. Indeed, all honest nodes generate Θ(n log n) random walks (or communication paths) but are only incident to O(1) edges. Therefore, the congestion of the good tokens on these edges is Θ(n log n). This leads to a significant Θ(n log n) slowdown caused by congestion when this protocol is used in our common coin primitive, in Section 4.
Theorem 4. For any given k ≥ 1, the sequence {(xkl, ukl, θkl)} of solutions to (Pk′l) 0 k≥1 converges to (xk, uk, θk) ∈ W1,∞ × L∞ × Θ. (ii) The sequence {(xk, uk, θk)} converges to a solution (x , u , θ0) of (P).
Theorem 4. If a ceded loss function f∗ ∈ Fπ satisfies f∗(x) and If∗(x) are both non-decreasing functions in x ≥ 0, and f∗ + uR = f∗(uI + uR + P0), (6) then f∗ is an optimal ceded loss function in Fπ, which maximizes the joint survival probability and satisfies (5). Proof: For any f ∈ Fπ, it holds that Jf = Pr{If (X) ≤ uI + Pf, f (X) ≤ uR + Pf}
Theorem 4. If n = 3t + δ, then there is an almost-surely terminating ABA protocol with expected running time O( t ).
