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Peras as an abstract protocol
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‎docs/abstract-protocol.md‎

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# Ouroboros Peras as an abstract protocol
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## Preliminaries
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### Adversary
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Conservatively, we assume a single coordinated adversary.
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### Time
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Time is divided into slots, and we assume that all parties have access to a perfectly synchronized clock indicating the current slot.
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### Network
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We assume that all parties can diffuse arbitrary messages via an abstract network functionality.
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The network is $\Delta$-semi-synchronous, which means:
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- If an honest party diffuses a messages in slot $s$, all other honest parties receive that message no later than in slot $s + \Delta$.
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- The adversary gets to decide the delay for each honest party individually, subject to the above constraint.
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### Stake
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Each party has some amount of stake $\sigma$.
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The adversary is assumed to control parties having less than 50% of all stake.
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### Leader schedule
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A party with stake $\sigma$ is elected in any given slot with probability $\phi(\sigma) = 1 - {(1-\mathrm{asc})}^\sigma$ (independent events for different slots).
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### Transactions
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We assume that all honest parties have access to some source of transactions.
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## Praos
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### Fundamental properties
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#### Common Prefix
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Let $C_1,C_2$ be the chains adopted by two parties at slot $s_1 \le s_2$.
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Let $C_1'$ be the the chain obtained by removing the last `kcp` blocks from $C_1$.
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Then $C_1'$ is a prefix of $C_2$.
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#### Chain Growth
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Consider a chain $C$ selected by an honest party.
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Then any window of slots of size $T_{cp}$ ending before the tip of $C$ contains at least `kcp` blocks.
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#### Chain Quality
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Consider a chain $C$ selected by an honest party.
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Then any window of slots of size $T_{cq}$ ending before the tip of $C$ contains at least one honest block.
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### Protocol description
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#### State
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- The currently selected (best) chain $C_{\mathrm{pref}}$, initialized at the empty chain rooted at Genesis.
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- The set of chains $\mathcal{C}$ received so far, initialized at the singleton set containing the empty chain rooted at Genesis.
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#### Logic
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At the beginning of every slot, every honest party does the following:
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##### Fetching chains
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1. Add all new valid chains to $\mathcal{C}$.
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2. Remove all chains from $\mathcal{C}$ that intersect with $C_{\mathrm{pref}}$ more than `kcp` blocks in the past.
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3. Set $C_{\mathrm{pref}}$ to the longest chain in $\mathcal{C}$.
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##### Forging
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1. Check if the party is elected in the current slot.
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2. If so, extend $C_{\mathrm{pref}}$ with a new block with as many transactions as possible.
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3. Diffuse $C_{\mathrm{pref}}$.
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## Peras
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### Committees, votes and certificates
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Time is divided into consecutive *rounds* for Peras, each of size `perasRoundSlots` in terms of slots.
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Every round has a *weighted committee*, ie a set of parties with associated weights (summing to one).
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The core requirement is that the adversary always has less than 50% of the weight.
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Every party that is part of the committee of a round can cast a vote for a particular block.
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A certificate for a round is a (representation of a) collection of votes with at least 75% weight, all voting for the same block.
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By a quorum intersection argument, there can be at most one certificate per round.
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### Fundamental properties
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Like for Praos, but now with weight instead of the number of blocks:
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#### Common Prefix
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Let $C_1,C_2$ be the chains adopted by two parties at slot $s_1 \le s_2$.
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Let $C_1'$ be the the chain obtained by removing blocks from $C_1$ until we reach a block that was buried under weight `kcp`.
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Then $C_1'$ is a prefix of $C_2$.
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#### Chain Growth
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Consider a chain $C$ selected by an honest party.
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Then any window of slots of size $T_{cp}$ ending before the tip of $C$ contains blocks having at least weight `kcp`.
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#### Chain Quality
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Consider a chain $C$ selected by an honest party.
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Then any window of slots of size $T_{cq}$ ending before the tip of $C$ contains at least one honest block.
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#### State
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- The currently selected (best) chain $C_{\mathrm{pref}}$, initialized at the empty chain rooted at Genesis.
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- The set of chains $\mathcal{C}$ received so far, initialized at the singleton set containing the empty chain rooted at Genesis.
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- The set of votes $\mathcal{V}$ received so far
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- The set of certificates $\mathrm{Certs}$ received so far, as well as the respective arrival/emergence slots.
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For convenience:
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- The most recent certificate $\mathrm{cert}'$.
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- The most recent certificate $\mathrm{cert}^*$ on $C_{\mathrm{pref}}$.
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#### Logic
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At the beginning of every slot $s$, every honest party does the following:
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##### Fetching
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1. Add all new valid chains to $\mathcal{C}$, and all new valid votes to $\mathcal{V}$.
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2. Add all certificates in $\mathcal{C}$ to $\mathrm{Certs}$.
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3. Turn any quorum in $\mathcal{V}$ into a certificate, and add it to $\mathrm{Certs}$.
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2. Remove all chains from $\mathcal{C}$ that intersect with $C_{\mathrm{pref}}$ more than `kcp` weight in the past.
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3. Set $C_{\mathrm{pref}}$ to the weightiest chain in $\mathcal{C}$.
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##### Forging
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1. Check if the party is elected in the current slot, otherwise, abort.
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2. If so, extend $C_{\mathrm{pref}}$ with a new block
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- with as many transactions as possible, and
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- with $\mathrm{cert}'$ if
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1. there is no round-$`(r-2)`$ certificate in $\mathrm{Certs}$, and
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2. $r - \mathrm{round}(\mathrm{cert}') \le \mathrm{perasCertMaxRounds}$, and
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3. $\mathrm{round}(\mathrm{cert}') > \mathrm{round}(\mathrm{cert}^*)$,
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where $r$ is the current round.
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3. Diffuse $C_{\mathrm{pref}}$.
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##### Voting
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1. Check if $s$ is the first slot of a round $r$, otherwise, abort.
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2. Check if the party is part of the committee of round $r$, otherwise, abort.
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2. Let $B$ be the block to potentially vote for, given by the most recent block on $C_{\mathrm{pref}}$ that is at least $\mathrm{perasBlockMinSlots}$ older than $s$.
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3. Check if either (VR-1A) and (VR-1B), or (VR-2A) and (VR-2B) are satisfied, otherwise abort.
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- (VR-1A): $\mathrm{round}(\mathrm{cert}') = r - 1$, and $\mathrm{cert}'$ was received within the first $X$ slots of round $r-1$.
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- (VR-1B): $B$ extends the block boosted by $\mathrm{cert}'$.
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- (VR-2A): $r \ge \mathrm{round}(\mathrm{cert}') + \mathrm{perasIgnoranceRounds}$.
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- (VR-2B): $r = \mathrm{round}(\mathrm{cert}^*) + c \cdot \mathrm{perasCooldownRounds}$ for some $c>0$.
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3. Diffuse a vote for $B$.

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