# The consensus algorithm¶

This document provides a description of Emmy*, the Tezos proof-of-stake consensus algorithm, as implemented in the Granada protocol.

## History¶

Before Emmy*, there was Emmy+ (introduced in this blog post), and before Emmy+, there was Emmy, a Nakamoto-style consensus first described in 2014, in the Tezos whitepaper:

our proof-of-stake mechanism is a mix of several ideas, including Slasher, chain-of-activity, and proof-of-burn.

The specificity of Emmy with respect to other proof-of-stake consensus algorithms, including some protocols introduced later such as Ouroboros and Snow White, is the combined use of priorities and endorsements in the so called minimal delay function. Thanks to these concepts, Emmy offers a better protection against selfish baking and a shorter time to finality. The time to finality can be measured in terms of the number of confirmations a user must have seen for a block to be considered as final. We recall that, being a Nakamoto-style consensus, Emmy provides probabilistic finality.

## Emmy*¶

Emmy* improves Emmy+ in that it brings smaller block times and faster times to finality.

### Terminology¶

A block in the blockchain consists of a header and a list of operations. The header has a shell part (common to all protocols) and a protocol-specific part. In Emmy*, the protocol-specific part of the header contains, most notably, a timestamp, a priority (a natural number), and the endorsements for the block at the previous level. Endorsements are operations that can be seen as votes for a given block. Each block is signed.

Before being endorsed, blocks are baked. Baking is the action of producing and signing a block. Corresponding to these two actions of baking and endorsing, at each level, two lists of slots are being created: a (conceptually) infinite list of baking slots and a list of ENDORSERS_PER_BLOCK endorsing slots (the value of ENDORSERS_PER_BLOCK is one of the parameters of the consensus protocol). The index of a baking slot is called a priority. Each slot is associated to a participant. A participant can appear several times in both lists. The selection of participants is at random, independently for each slot, and is stake based.

An endorsement for a block at level $$\ell$$ is valid if it is signed by a participant that has an endorsing slot at level $$\ell$$. The endorsing power of an endorsement is the number of slots the endorser owns at level $$\ell$$. The endorsing power of a block is the sum of the endorsing powers of the endorsements it contains.

### Minimal block delay function¶

At the heart of Emmy*, there is the minimal block delay function. This function serves to compute the minimal time between blocks depending on the current block’s priority p, and its endorsing power e. Namely, Emmy* defines the minimal block delay function as follows:

$\begin{split}delay^*(p, e) = \begin{cases} md & \text{ if } p = 0 \wedge w \geq \frac{3}{5} te\\ delay^+(p, e) & \text{ otherwise} \end{cases}\end{split}$

where

• $$md$$ stands for the minimal block delay,

• $$te$$ stands for the “total endorsing power”, and

• $$delay^+(p, e)$$ is the minimal block delay function in Emmy+, namely:

$delay^+(p, e) = bd + dp \cdot p + de \cdot max(0, ie - e)$

where

• $$bd$$ stands for the minimal time between the blocks,

• $$dp$$ stands for “delay per priority”,

• $$de$$ stands for “delay per (missed) endorsement”, and

• $$ie$$ stands for the “initial endorsing power”.

The delay function serves to determine if a produced block can be considered as valid.

### Block validity condition¶

A block with timestamp $$t'$$, priority $$p$$, and endorsing power $$e$$ is valid at level $$\ell$$ if:

• the endorsements in the block are valid for level $$\ell-1$$,

• it is signed by the baker that has baking slot $$p$$, and

• $$t' \geq t + delay^+(p,e)$$, where $$t$$ is the timestamp of the previous block.

We note that, by the definition of the delay function, the higher the priority and the smaller the endorsing power, the longer it takes before the block is considered as valid. However, if the block has priority 0 and contains endorsements with endorsing power at least $$ie$$, then there is no time penalty.

### Emmy* abstractly¶

We refer to someone trying to reach consensus by the generic notion of participant. Emmy* can be described in an abstract manner as follows:

• A participant continuously observes blocks and endorsements.

• A participant always adopts the fittest, that is, the longest (valid) chain it observes.

• A participant that has at least an endorsement slot at level $$\ell$$, emits an endorsement for the first block it observes at level $$\ell$$.

• A participant produces a block as soon as it is allowed to, that is, as soon as it can produce a valid block (see the validity condition above).

### Emmy* concretely¶

In Tezos, a participant is a delegate that has at least one roll, and is active. For simplicity we just refer to participants as delegates (and omit the “active” and “with rolls” attributes). A delegate plays two roles:

• that of a baker, that is, it creates blocks, or

• that of an endorser, that is, it contributes in agreeing on a block by endorsing that block.

To these roles correspond the two types of actions mentioned above, baking and endorsing. As mentioned above, the baking and endorsing rights of a delegate are given by its baking, respectively endorsing slots, whose selection is described here. The mechanism behind baking slots is meant to ensure that if the delegate whose turn is to bake is for some reason unable to bake, the next delegate in the list can step up and bake the block.

There are two more notions which are defined abstractly at the level of the shell and concretized in Emmy*, the fitness, and the protocol-specific header:

• the fitness of a block is 1 plus the fitness of the previous block;

• the protocol-specific header of a block has the following fields:

• signature: a digital signature of the shell and protocol headers (excluding the signature itself).

• priority: the position in the priority list of delegates at which the block was baked.

• seed_nonce_hash: a commitment to a random number, used to generate entropy on the chain. Present in only one out of BLOCKS_PER_COMMITMENT (see Constants).

• proof_of_work_nonce: a nonce used to pass a low-difficulty proof-of-work for the block, as a spam prevention measure.

The consensus algorithm is implemented in Tezos in five components: the shell, the economic protocol, and the three daemons: the baker, the endorser, and the accuser.

There are mainly two rules that the shell uses when receiving a new valid block:

• The shell changes the head of the chain to this new block only if it has a higher fitness than the current head.

• The shell does not accept a branch whose fork point is in a cycle more than PRESERVED_CYCLES in the past. More precisely, if n is the current cycle, the last allowed fork point is the first level of cycle n-PRESERVED_CYCLES.

The parameter PRESERVED_CYCLES therefore plays a central role in Tezos: any block before the last allowed fork level is immutable.

Finally, the economic protocol provides the rules for when block and endorsements are valid, as explained above, and defines the economic incentives of delegates. Finally, the three daemons are responsible for injecting blocks, endorsements, and respectively accusations (see below) on behalf of delegates.

### Economic Incentives¶

In Emmy*, participation in consensus is rewarded and bad behavior is punished.

#### Rewards¶

To incentivize participation in the consensus algorithm, delegates are rewarded for baking and endorsing. The reward for baking a block with priority $$p$$ and endorsing power $$e$$ is given by the formula $$baking\_reward(p,e)$$. The rewards for endorsing a block with priority $$p$$ and having the corresponding endorsement included in the block is given by the formula $$endorsing\_reward(p,e)$$, where $$e$$ is the endorsement’s endorsing power. These reward formulas are as follows:

$\begin{split}baking\_reward(p,e) = \begin{cases} \frac{e}{te}\cdot \frac{level\_rewards\_prio\_zero}{2} & \mbox{ if } p = 0\\ \frac{e}{te} \cdot level\_rewards\_prio\_nonzero & \mbox{ otherwise } \end{cases}\end{split}$
$\begin{split}endorsing\_reward(p,e) = \begin{cases} baking\_reward(0, e) & \mbox{ if } p = 0\\ \frac{2}{3} \cdot baking\_reward(0, e) & \mbox{ otherwise } \end{cases}\end{split}$

where

• $$level\_rewards\_prio\_zero$$ and $$level\_rewards\_prio\_nonzero$$ are constants.

The motivation behind this choice of design is given in the Carthage blog post.

Besides the reward for baking, the baker receives all the fees paid for the transactions included in the baked block.

Rewards and fees are not distributed immediately, instead they are frozen for a period of PRESERVED_CYCLES.

#### Slashing¶

If a delegate deviates from the consensus rules by baking or endorsing two different blocks at the same level, we say that a delegate double signs. As a counter-measure against double signing a security deposit is frozen from the delegate’s account. Precisely, each delegate key has an associated security deposit account. When a delegate bakes or endorses a block the security deposit is automatically moved to the deposit account where it is frozen for PRESERVED_CYCLES cycles, after which it is automatically moved back to the baker’s main account.

The values of the security deposits are BLOCK_SECURITY_DEPOSIT per block created and ENDORSEMENT_SECURITY_DEPOSIT per endorsement slot.

The evidence for double signing at a given level can be collected by any accuser and included as an accusation operation in a block for a period of PRESERVED_CYCLES. The inclusion of the accusation leads to forfeiting the entirety of the security deposits and fees obtained during the cycle when the double signing was made. Half of this amount is burned, and half goes to the baker who included the accusation.

In the current protocol, accusations for the same incident can be made several times after the fact. This means that the deposits and fees for the entire cycle are forfeited, including any deposit made, or fees earned, after the incident. Pragmatically, any baker who either double bakes or endorses in a given cycle should immediately stop both baking and endorsing for the rest of that cycle.

### Consensus protocol parameters¶

In this section we map the above notation to their corresponding parameter values. Note that these parameters are part of the larger set of protocol constants.

Mapping

Notation

Parameter name

Parameter value

$$md$$

MINIMAL_BLOCK_DELAY

30 seconds

$$bd$$

TIME_BETWEEN_BLOCKS[0]

60 seconds

$$dp$$

TIME_BETWEEN_BLOCKS[1]

40 seconds

$$de$$

DELAY_PER_MISSING_ENDORSEMENT

4 seconds

$$ie$$

INITIAL_ENDORSERS

192

$$te$$

ENDORSERS_PER_BLOCK

256

$$\frac{level\_rewards\_prio\_zero}{te \cdot 2}$$

BAKING_REWARD_PER_ENDORSEMENT[0]

0.078125 ꜩ

$$\frac{level\_rewards\_prio\_nonzero}{te}$$

BAKING_REWARD_PER_ENDORSEMENT[1]

0.011719 ꜩ

$$endorsing\_reward(0,1)$$

ENDORSEMENT_REWARD[0]

0.078125 ꜩ

$$endorsing\_reward(p,1)$$ for $$p \geq 1$$

ENDORSEMENT_REWARD[1]

0.052083 ꜩ

BLOCK_SECURITY_DEPOSIT

640 ꜩ

ENDORSEMENT_SECURITY_DEPOSIT

2.5 ꜩ

Since blocks are at least TIME_BETWEEN_BLOCKS[0], that is 30 seconds apart, and since a cycle has BLOCKS_PER_CYCLE, that is 8192 blocks, a cycle lasts at least 2 days, 20 hours, and 16 minutes, and PRESERVED_CYCLES cycles, that is 5 cycles, last at least 14 days, 5 hours, and 20 minutes.

Given that MINIMAL_BLOCK_DELAY is 30 seconds, the minimal block delay function says that:

• if the block is baked at priority 0 and it contains at least 60% of the endorsements (namely, at least 153 endorsements) then the minimal delay is 30 seconds;

• otherwise, the higher the priority and the fewer endorsements a block carries with respect to the 192 endorsements threshold, the longer it takes before it can be considered valid, where the delay of 60 seconds is incremented by 40 seconds with each missed priority and with 4 seconds with each missed endorsement.

The value for BAKING_REWARD_PER_ENDORSEMENT[0] is chosen such that the inflation from block rewards and endorsement rewards, which is given by ENDORSERS_PER_BLOCK * (ENDORSEMENT_REWARD[0] + BAKING_REWARD_PER_ENDORSEMENT[0]) is 80 ꜩ which in turn preserves the 5.51% annual inflation.

Since deposits are locked for a period of PRESERVED_CYCLES, one can compute that at any given time, about ((BLOCK_SECURITY_DEPOSIT + ENDORSEMENT_SECURITY_DEPOSIT * ENDORSERS_PER_BLOCK) * (PRESERVED_CYCLES + 1) * BLOCKS_PER_CYCLE) tokens of all staked tokens should be held as security deposits. For instance, if the amount of staked tokens is 720,000,000 ꜩ, then roughly 8.74% of this amount is stored in security deposits. This percentage also gives an indication of the minimal amount of tokens a delegate should own in order to not miss out on creating a block or an endorsement. Please refer to this section of the documentation for a discussion on (over-)delegation.