Lemma 6 Clause Samples
Lemma 6. Suppose ta ≤ ts and ta + 2ts < n, and assume protocol Πts,ta rity guarantees of Corollary 4. Then, protocol Πts,ta ts,ta HMPC [1] ▇▇▇▇▇ ▇▇▇▇▇▇▇, ▇▇▇▇▇ ▇▇▇▇▇, and ▇▇▇▇▇▇ ▇. ▇▇▇▇▇▇▇. An almost-surely terminating polynomial protocol forasynchronous byzantine agreement with optimal resilience. In Rida ▇. ▇▇▇▇▇ and ▇▇▇▇ ▇▇▇▇-▇▇▇▇▇▇, editors, 27th ACM PODC, pages 405–414. ACM, August 2008. [2] ▇▇▇▇▇ ▇▇▇▇▇▇▇, ▇▇▇▇▇▇ ▇▇▇▇▇▇, ▇▇▇▇▇▇ ▇▇▇▇▇, ▇▇▇▇ ▇▇▇, and Maofan Yin. Sync HotStuff: Simple and practical synchronous state machine replication. Cryptology ePrint Archive, Report 2019/270, 2019. https: //▇▇▇▇▇▇.▇▇▇▇.▇▇▇/▇▇▇▇/▇▇▇. [3] ▇▇▇▇▇▇▇▇▇▇ ▇▇▇▇▇▇, ▇▇▇▇ ▇▇▇ ▇▇▇▇▇▇▇▇▇, ▇▇▇▇▇▇▇ ▇▇▇▇, and ▇▇▇▇▇▇▇▇ ▇▇▇▇. Two round information- theoretic MPC with malicious security. In ▇▇▇▇▇ ▇▇▇▇▇ and ▇▇▇▇▇▇▇ ▇▇▇▇▇▇, editors, EUROCRYPT 2019, Part II, volume 11477 of LNCS, pages 532–561. Springer, Heidelberg, May 2019. [4] ▇▇▇▇▇ ▇▇▇-▇▇▇▇ and ▇▇▇▇▇▇ ▇▇▇▇▇▇. Non-cryptographic fault-tolerant computing in constant number of rounds of interaction. In ▇▇▇▇▇ ▇▇▇▇▇▇▇▇, editor, 8th ACM PODC, pages 201–209. ACM, August 1989. [5] ▇▇▇▇▇▇ ▇▇▇▇▇▇, ▇▇▇▇▇▇ ▇▇▇▇▇▇, and ▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇. The round complexity of secure protocols (extended abstract). In 22nd ACM STOC, pages 503–513. ACM Press, May 1990. [6] ▇▇▇▇▇ ▇▇▇▇▇▇▇, Viet ▇▇▇▇ ▇▇▇▇▇, and ▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇. Foundations of garbled circuits. In Proceedings of the 2012 ACM conference on Computer and communications security, pages 784–796, 2012. [7] Michael Ben-Or, ▇▇▇▇▇ ▇▇▇▇▇▇▇▇▇▇, and ▇▇▇ ▇▇▇▇▇▇▇▇▇. Completeness theorems for non-cryptographic fault- tolerant distributed computation (extended abstract). In 20th ACM STOC, pages 1–10. ACM Press, May 1988. [8] Michael Ben-Or, ▇▇▇▇ ▇▇▇▇▇▇, and ▇▇▇ ▇▇▇▇▇. Asynchronous secure computations with optimal resilience (extended abstract). In ▇▇▇ ▇▇▇▇▇▇▇▇ and ▇▇▇ ▇▇▇▇▇, editors, 13th ACM PODC, pages 183–192. ACM, August 1994. [9] ▇▇▇▇▇ ▇▇▇▇, ▇▇▇▇▇▇▇▇ ▇▇▇▇, and ▇▇▇▇▇▇ Loss. Synchronous consensus with optimal asynchronous fallback guarantees. In ▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇ and ▇▇▇▇ ▇▇▇▇▇, editors, TCC 2019, Part I, volume 11891 of LNCS, pages 131–150. Springer, Heidelberg, December 2019. [10] ▇▇▇▇▇ ▇▇▇▇, ▇▇▇▇▇▇▇▇ ▇▇▇▇, and ▇▇▇▇▇▇ Loss. Network-agnostic state machine replication. Cryptology ePrint Archive, Report 2020/142, 2020. ▇▇▇▇▇://▇▇▇▇▇▇.▇▇▇▇.▇▇▇/2020/142. [11] ▇▇▇▇▇ ▇▇▇▇, ▇▇▇▇-▇▇ ▇▇▇ Zhang, and Julian Loss. Always have a backup plan: Fully secure synchronous MPC with asynchronous fallback. In ▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇▇▇ and ▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇▇▇,...
Lemma 6. If all honest parties in C1 input b to ABA(1) then the output of ABA(1) is b. Lemma 7. ΠsprABA satisfies validity if t ≤ (1 − 6ε)n/3 parties are corrupted.
Lemma 6. Suppose (1) C > 3, and (2) no digest z is broadcast via Alg. 4 by t + 1 correct processes. Then, every correct process delivers CRB in O(1) message delays.
Lemma 6. The subsheaf &Xr,k,l ⊆ &Xr,k,l [1/S] on Xr,k,5 is integrally closed. Similarly, the subsheaf &IGn,r,k,l ⊆ &IGn,r,k,l [1/S] on IGn,r,k,5 is integrally closed.
Lemma 6. Suppose δ and δ′ are symbolic runs of a register automaton such that strace(δ)= strace(δ′). Then δ = δ′.
Lemma 6. If L is a line bundle on a split torus Gr,an , then L is trivial.
Lemma 6. Assume there exists an SRDS scheme based on the LOSSW multi-signature scheme, where Setup(1κ, 1n) generates ppms = (G, GT , p, g, e), as per Definition 6.2, with n/ log |GT | < 1. Let 0 < α < 1 be a constant and let s(n) = α · n. Then, there exist SNARGs for average-case (s(n), GT )-Subset-Product (as defined in Definition 6.4).
Lemma 6. 3.1. Let f ∈ Hr(R) with r > 1 . Then there exists f˜ ∈ L∞(R), uniformly continuous with f = f˜ a.e.
Lemma 6. For any k ∈ Z≥0 and l ≤ l′ ∈ 1/pkZ≥0∪{∞}, I = [l, l′], set pkI := [lpk, l′pk]. Then the rescaling isomorphism restricts to an isomorphism rk : Mk,I → M0,pk I. More generally, for any d ≤ k there is an isomorphism rd : Mk,I → Mk−d,pd I, Sk ←' Sk−d. Proof. Rescaling by definition sends Sk ←' T and thus for l′ < ∞ identifies the subspaces M (|p| ≤ |S | = 0, |S | ≤ |p| = 0) −∼→ M (|p| ≤ |T 5pk | = 0, |T 5 p | ≤ |p| = 0).