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fix: make dollar signs go away
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huitseeker committed Apr 19, 2022
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Expand Up @@ -29,9 +29,9 @@ The [Tusk consensus](https://github.com/MystenLabs/narwhal/tree/main/consensus)

A Narwhal instance sets up a message-passing system comprised of a set of $3f+1$ units of stake divided amongst a set of nodes, and assumes a computationally bounded adversary that controls the network and can corrupt parties holding up to f units of stake. The validators collaborate in forming a leaderless graph of batches of transactions - which the literature (in the context of DAG-based consensus) designates as _blocks_ and that we label as _collections_ - to emphasize we're in a context where the mempool data is used by an unspecified consensus algorithm.

The graph's *vertices* consist of certified collections. Each valid collection signed by its validator-author must contain a round number and must itself be signed by a quorum ($2f+1$) of validator stake. We call those $2f+1$ signatures a _certificate of availability_. Furthermore, that collection must contain hash pointers to a quorum of valid certificates (that is, certificates from validators with $2f+1$ units of stake) from the previous round (see Danezis & al. Fig 2), which constitute the *edges* of the graph.
The graph's *vertices* consist of certified collections. Each valid collection signed by its validator-author must contain a round number and must itself be signed by a quorum (2f+1) of validator stake. We call those 2f+1 signatures a _certificate of availability_. Furthermore, that collection must contain hash pointers to a quorum of valid certificates (that is, certificates from validators with 2f + 1 units of stake) from the previous round (see Danezis & al. Fig 2), which constitute the *edges* of the graph.

Each collection is formed in the following way: each validator _reliably broadcasts_ a collection for each round. Subject to specified validity conditions, if validators with $2f+ 1$ stake receive a collection, they acknowledge it with a signature each. Signatures from $2f + 1$ validators by stake form a certificate of availability that is then shared and potentially included in collections at round $r + 1$.
Each collection is formed in the following way: each validator _reliably broadcasts_ a collection for each round. Subject to specified validity conditions, if validators with 2f + 1 stake receive a collection, they acknowledge it with a signature each. Signatures from 2f + 1 validators by stake form a certificate of availability that is then shared and potentially included in collections at round r + 1.

The following figure represents five rounds of construction of such a DAG (1 to 5), with authorities A, B, C and D participating. For simplicity, each validator holds 1 unit of stake. The collections transitively acknowledged by A's latest round in A5 are represented in full lines in the graph.

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