Communication Complexity Clause Samples
The Communication Complexity clause defines the procedures and requirements for how parties must communicate with each other regarding matters covered by the agreement. It typically outlines the acceptable methods of communication, such as email or written notice, and may specify timelines for responses or acknowledgments. By establishing clear communication protocols, this clause helps prevent misunderstandings and ensures that all parties are informed and able to respond appropriately, thereby reducing the risk of disputes arising from miscommunication.
Communication Complexity. The most expensive steps in the protocol are the run of BA(κ) in Step 1 (which itself consists of κ parallel runs of BA(1)) and the distribution of the long blocks in Step 2. The costs for Step 1 are bounded as O(κ6 · n) since every run of BA(1) costs O(κ5 · n). The costs for Step 2 are bounded by O(l · n). Overall, we obtain a complexity of O(n · l + κ6 · n).
Communication Complexity. Our protocol incurs a communication com- plexity of O(n4(κ + r log(r))) bits, where κ is the size of a signature and r is the number of rounds. Using threshold signatures for the (conditional) graded broadcast primitive, we can save a linear factor n. It remains open to explore solutions with improved communication.
Communication Complexity. It is the total number of bits communicated by the honest parties in the protocol; (c)
Communication Complexity. Let |S| be a bound on the encoding of the union of the input sets, |r| a bound on the encoding of the round number and λH a bound on the output of a collision-resistant hash. Denote BCost(L) the cost of 5-graded gossip for an L-bit message. Let TCost(L) denote the communication cost of 5-graded f -threshold gossiping a set with encoding size L, and GCost(L) the communication cost of gradecasting an L-bit message. Theorem 4.12. Let n be an upper bound on the number of participating public keys with non-zero grades (i.e., those eligible to generate messages in the protocol), n′ an upper bound on the number of participating propose-round keys and p a lower bound on the probability that an iteration elects an honest leader. The communication complexity of the BA protocol, such that the probability of choosing an honest leader in each iteration is at least p, is at most n · TCost(|S|) + (1 + 1/p) · 2n · TCost(λH) + n′ · GCost(|S|) .
Communication Complexity. To avoid the use of n different instances of binary agreement, we adopt the ap- proach of external validity to the information theoretic setting. External validity [12] is a very successful framework in the authenticated setting for multi-valued agreement. To adopt to the information theoretic setting we define the notion of an asynchronous validity predicate, which is an information-theoretic asyn- chronous alternative to external validity functions. Such predicates act “like func- tions”, but the results are delivered asynchronously. That is, if a value is valid, all parties will eventually see that it is, but otherwise they might not receive any output from the predicate. We then construct an agreement with an asyn- chronous validity predicate. To adopt this approach of using an asynchronous validity predicate we first extend the use of information theoretic protocols [5] from partial synchrony to asynchrony. This requires using asynchronous verifi- able secret sharing to randomly choose leaders. Since we do not have a perfect leader election mechanism, we use the techniques of [3] that build a weaker no- tion of proposer election and adopt them to the information theoretic setting. Among other things, this requires re-formulating the gather primitive to support non-cryptographic primitives and changing the HotStuff variant to support an non-cryptographic asynchronous view change protocol.
Communication Complexity. We briefly explain the notions from commu- nication complexity we use. For formal definitions, background and more details, see the textbook [44]. For a function f and a distribution µ on its inputs, define Dµ(f ) as the minimum communication complexity of a protocol that correctly computes f with error 1/3 over the inputs from µ. Define D×(f ) = max Dµ(f ): µ is a product distribution . Define the unbounded error communication complexity U (f ) of f as the minimum communication complexity of a randomized private-coin8 protocol that correctly computes f with probability strictly larger than 1/2 on every input. The two works [64] and [63] showed that there are functions with small distribu- tional communication complexity under product distributions, and large unbounded error communication complexity. In [64] the separation is as strong as possible but it is not for an explicit function, and the separation in [63] is not as strong but the underlying function is explicit. 6Interestingly, their motivation for considering sign rank comes from image processing. 7The paper [49] considered a different type of combinatorial description from [14] and [18], and therefore considered a different formulation of the stretchability problem. However, it is possible to transform between these descriptions in polynomial time. 8In the public-coin model every boolean function has unbounded communication complexity at most two. The matrix A with d = 2 and n ≥ 3 in our example from § 2.2 corresponds to the following communication problem: ▇▇▇▇▇ gets a point p ∈ P , Bob gets a line ℓ ∈ L, and they wish to decide whether p ∈ ℓ or not. Let f : P × L → {0, 1} be the corresponding function and let m = log2(N ) . A trivial protocol would be that ▇▇▇▇▇ sends Bob the name of her point using m bits, Bob checks whether it is incident to the line and outputs accordingly. Theorem 7 implies the following consequences. Even if we consider protocols that use randomness and are allowed to err with probability less than but arbitrarily close to 1/2, then still one cannot do considerably better than the above trivial protocol. However, if the input (p, ℓ) P L is distributed according to a product distribution then there exists an O(1) protocol that errs with probability at most 1/3.
Communication Complexity n+1 rounds, 2n− 1 messages (2 per mem- ber Ui, i ƒ= {n− 1, n}; 1 message + 1 broadcast for Un−1, 1 broadcast for Un), O(log2 max(g) ) msg. size SECURITY: Passive (DDH/GDH assumption) PUBLIC DOMAIN PARAMETERS: g of prime order q such that G = (g) Every member ▇▇ chooses a random xi and executes the following: ROUND i; i ∈ [1, n − 1]:
Communication Complexity. In the first stage, (3t + 1) (t log n) words are being sent by quorum nodes to relayers and (n log n) words by relayers to parties. In the second stage, parties send (n log n) words to relayers, and relayers send (t log n) (3t + 1) words to quorum nodes. Finally, in stage 3, (3t + 1) 2t words are sent by quorum nodes to potentially undecided parties. Now, we prove Complete Correctness, splitting it into Agreement (every two honest nodes decide the same value), Termination (every honest node eventually decides), and Validity (if an honest party decides value v, then v = vin).
Communication Complexity. We define the communication complexity of an algorithm as the number of words sent by honest nodes, where each word contains a constant number of signatures and a constant number of bits. As pointed out by ▇▇▇▇▇▇▇▇▇▇ et al. [25], there doesn’t exist a partially synchronous algorithm that solves BA sending a bounded number of messages if messages sent before GST are counted. Therefore, when speaking about the communication complexity of partially synchronous algorithms, we refer only to messages sent after GST.
Communication Complexity. Let |S| be a bound on the encoding of the union of the input sets, |r| a bound on the encoding of the round number and λH a bound on the output of a collision-resistant hash. Denote BCost(L) the cost of 5-graded gossip for an L-bit message. Let TCost(L) denote the communication cost of 5-graded f -threshold gossiping a set with encoding size L, and GCost(L) the communication cost of gradecasting an L-bit message.