## Configuration

In the normal course of things, configuration parameters might not be the most exciting part of a specification. However, to gain an understanding of these parameters is to gain a huge insight into what kind of beast we are dealing with in the beacon chain.

You'll notice that most of them are powers of two. There's no huge significance to this. Computer scientists think it's neat, and it ensures that things cleanly divide other things in general. Justin Drake believes that it helps to minimise bike-shedding.

Some of the configuaration parameters below are quite technical and perhaps obscure. I'll use the opportunity here to introduce some concepts, and more detailed explanation will follow when they are used later in the spec.

Note: The default mainnet configuration values are included here for spec-design purposes. The different configurations for mainnet, testnets, and YAML-based testing can be found in the configs/constant_presets directory. These configurations are updated for releases and may be out of sync during dev changes.

To facilitate easier initial interoperability testing and testnets, a much lighter-weight minimal configuration was defined. This runs more quickly, with much lower resource use, than the below mainnet configuration: it has fewer, smaller committees, less shuffling, 6s rather than 12s slots, 8-slot rather than 64-slot epochs, and so on. The final beacon chain will be deployed with the mainnet configuration parameters below.

### Misc

Name Value
ETH1_FOLLOW_DISTANCE 2**10 (= 1,024)

This is the minimum depth of block on the Ethereum 1 chain that can be considered by the Eth2 chain: it applies to the Genesis process and the processing of deposits by validators. The Eth1 chain depth is estimated by multiplying this value by the target average Eth1 block time, SECONDS_PER_ETH1_BLOCK.

The value of ETH1_FOLLOW_DISTANCE is very conservatively set to ensure that we never end up with some block that we've relied on subsequently being reorganised out of the Ethereum 1 chain.

MAX_COMMITTEES_PER_SLOT 2**6 (= 64)

Validators are organised into committees to do their work. At any one time, each validator is a member of exactly one beacon chain committee, and is called on to make an attestation exactly once per epoch. (An attestation is a vote for a beacon chain block that has been proposed for a slot.)

In the beacon chain, the 64 committees active in a slot effectively act as a single committee as far as the fork-choice rule is concerned. They all vote on the proposed block for the slot, and their votes/attestations are pooled. (In a similar way, all committees active during an epoch act effectively as a single committee as far as justification and finalisation are concerned.)

The number 64 is designed to map onto one committee per shard once Phase 1 is deployed, since these committees will also vote on shard crosslinks.

TARGET_COMMITTEE_SIZE 2**7 (= 128)

To achieve a desirable level of security, committees need to be larger than a certain size. This is to make it infeasible for an attacker to randomly end up with a majority in a committee even if they control a significant number of validators. This target is a kind of lower-bound on committee size. If there are not enough validators to make all committees have at least 128 members, then, as a first measure, the number of committees per slot is reduced to maintain this minimum. Only if there are fewer than SLOTS_PER_EPOCH * TARGET_COMMITTEE_SIZE = 4096 validators in total will the committee size be reduced below TARGET_COMMITTEE_SIZE. With so few validators, the system would be insecure in any case.

See the note below for how this value 128 was arrived at.

MAX_VALIDATORS_PER_COMMITTEE 2**11 (= 2,048)

This is just used for sizing some data structures, and is not particularly interesting. Reaching this limit would imply over 4 million active validators, staked with a total of 128 million Ether, which exceeds the total supply today.

MIN_PER_EPOCH_CHURN_LIMIT 2**2 (= 4)

Validators are allowed to exit the system and cease validating, and new validators may apply to join at any time. For interesting reasons, a design decision was made to apply a rate-limit to entries (activations) and exits. Basically, it is important in proof of stake protocols that the validator set not change too quickly.

In the normal case, a validator is able to exit fairly swiftly: it just needs to wait MAX_SEED_LOOKAHEAD (currently four) epochs. However, if there are large numbers of validators wishing to exit at the same time, a queue forms with a limited number of exits allowed per epoch. The minimum number of exits per epoch (the minimum "churn") is MIN_PER_EPOCH_CHURN_LIMIT, so that validators can always eventually exit. The actual allowed churn per epoch is calculated in conjunction with CHURN_LIMIT_QUOTIENT.

The same applies to new validator activations, once a validator has been marked as eligible for activation.

CHURN_LIMIT_QUOTIENT 2**16 (= 65,536)

This is used in conjunction with MIN_PER_EPOCH_CHURN_LIMIT to calculate the actual number of validator exits and activations allowed per epoch. The number of exits allowed is max(MIN_PER_EPOCH_CHURN_LIMIT, n // CHURN_LIMIT_QUOTIENT), where n is the number of active validators. The same applies to activations.

SHUFFLE_ROUND_COUNT 90

The beacon chain implements a rather interesting way of shuffling validators in order to select committees, called the "swap-or-not shuffle". This shuffle proceeds in rounds, and the degree of shuffling is determined by the number of rounds: SHUFFLE_ROUND_COUNT. The time taken to shuffle is linear in the number of rounds, so for light-weight, non-mainnet configurations, the number of rounds can be reduced.

The value 90 was introduced in Vitalik's initial commit without explanation. The original paper describing the shuffling technique seems to suggest that a cryptographically safe number of rounds is $6\log{N}$. With 90 rounds, then, we should be good for shuffling 3.3 million validators, which is close to the maximum number possible (given the Ether supply).

The main advantage of using this shuffling method is that light clients and others that are interested in only a small number of the committees at any time can compute only the committees they need without having to shuffle the entire set of active validators. This can be a big saving on computational resources. See compute_shuffled_index.

MIN_GENESIS_ACTIVE_VALIDATOR_COUNT 2**14 (= 16,384)

MIN_GENESIS_ACTIVE_VALIDATOR_COUNT is the minimum number of full validator stakes that must have been deposited before the beacon chain can start producing blocks. The number is chosen to ensure a degree of security. It allows for four 128 member committees per slot, rather than the 64 eventually desired to support Phase 1. But fewer validators means higher rewards per validator, so it is designed to attract early participants to get things bootstrapped.

MIN_GENESIS_ACTIVE_VALIDATOR_COUNT used to be much higher (65,536 = 2 million Ether staked), but was reduced when MIN_GENESIS_TIME, below, was added.

MIN_GENESIS_TIME 1578009600 (Jan 3, 2020)

MIN_GENESIS_TIME is the earliest date that the beacon chain can start, and will be adjusted as we near deployment. It's currently set to January the 3rd, 2020 as that is the 11th anniversary of the Bitcoin genesis. It had been suggested that it would be cool to start up on that date, but it was never remotely practical.

Having a MIN_GENESIS_TIME allows us to start the chain with fewer validators than was previously thought necessary. The previous plan was to start the chain as soon as there were MIN_GENESIS_ACTIVE_VALIDATOR_COUNT validators staked. But there were concerns that with a lowish initial validator count, a single entity could form the majority of them and then act to prevent other validators from entering (a "gatekeeper attack"). A minimum genesis time allows time for all intending depositors to make their deposits before they could be excluded by a gatekeeper attack.

HYSTERESIS_QUOTIENT 4
HYSTERESIS_DOWNWARD_MULTIPLIER 1
HYSTERESIS_UPWARD_MULTIPLIER 5

These parameters relate to the way that effective balance is changed (see EFFECTIVE_BALANCE_INCREMENT below). As described there, effective balance for a validator follows changes to the actual balance in a step-wise way, with hysteresis. This is to ensure that it does not change very often.

The original hysteresis design had an unintended effect that might have encouraged stakers to over-deposit or make multiple deposits in order to maintain a balance above 32 Ether at all times. This is because, if a validator's balance were to drop below 32 Ether soon after depositing, however briefly, the effective balance would immediately drop to 31 Ether, and would take a long time to recover. This would result in a 3% reduction in rewards for a period.

This problem was addressed by making the hysteresis configurable via these parameters. Specifically, these settings mean:

1. if a validators' balance falls 0.25 Ether below its effective balance, then its effective balance is reduced by 1 Ether
2. if a validator's balance rises 1.25 Ether above its effective balance, then its effective balance is increased by 1 Ether

These calculations are done in process_final_updates().

• For the safety of committees, TARGET_COMMITTEE_SIZE exceeds the recommended minimum committee size of 111; with sufficient active validators (at least SLOTS_PER_EPOCH * TARGET_COMMITTEE_SIZE), the shuffling algorithm ensures committee sizes of at least TARGET_COMMITTEE_SIZE. (Unbiasable randomness with a Verifiable Delay Function (VDF) will improve committee robustness and lower the safe minimum committee size.)

Given a proportion of the validators controlled by an attacker, what is the probability that the attacker ends up controlling a 2/3 majority in a randomly selected committee drawn from the full set of validators? This is what Vitalik looks at in the presentation, and where the 111 number comes from (a $\smash{2^{-40}}$ chance, one-in-a-trillion, of an attacker with 1/3 of the validators gaining by chance a 2/3 majority in any one committee).

Another issue is that the randomness that we are using (a RANDAO) is not unbiasable. If an attacker happens to control a number of block proposers at the end of an epoch, they can decide to reveal or not to reveal their blocks, gaining one bit of influence per validator on the next random number. This might allow an attacker to gain more control in the next round and so on. In this way, an attacker can gain some influence over committee selection. Having a good lower-bound on committee size (TARGET_COMMITTEE_SIZE) helps to defend against this. Alternatively, we could use an unbiasable source of randomness such as a verifiable delay function (VDF). Use of a VDF is not currently planned for Eth2, but may be implemented in future.

### Gwei values

Name Value
MIN_DEPOSIT_AMOUNT Gwei(2**0 * 10**9) (= 1,000,000,000)

MIN_DEPOSIT_AMOUNT is not actually used anywhere within the Phase 0 Beacon Chain Specification document. Where it is used is within the deposit contract to be deployed to the Ethereum 1 chain. Any amount less than this value sent to the deposit contract is reverted.

MAX_EFFECTIVE_BALANCE Gwei(2**5 * 10**9) (= 32,000,000,000)

There is a concept of "effective balance" for validators: whatever a validator's actual total stake (balance), its voting power is weighted by its effective balance, even if it has much more at stake. Effective balance is also the amount on which all rewards, penalties, and slashings are calculated—it's used a lot in the protocol

The MAX_EFFECTIVE_BALANCE is the highest effective balance that a validator can have: 32 Ether. Any balance above this is ignored. Note that this means that staking rewards don't compound in the usual case (unless our balance somehow falls below 32 Ether at some point).

There is a discussion in the Design Rationale of why 32 Ether was chosen as the staking amount. In short, we want enough validators to keep the chain both alive and secure under attack, but not so many that the message overhead on the network becomes too high.

EJECTION_BALANCE Gwei(2**4 * 10**9) (= 16,000,000,000)

If a validator's balance falls below 16 Ether then it is exited from the system (kicked out of the active validator set). This is most likely to happen as a result of the "quadratic leak" which gradually reduces the balances of inactive validators in order to maintain the liveness of the beacon chain.

EFFECTIVE_BALANCE_INCREMENT Gwei(2**0 * 10**9) (= 1,000,000,000)

Throughout the protocol, a quantity called "effective balance" is used instead of the validators' actual balances. Effective balance tracks the actual balance, with two differences: (1) effective balance is capped at MAX_EFFECTIVE_BALANCE no matter how high the actual balance of a validator is, and (2) effective balance is much more granular - it changes only in steps of EFFECTIVE_BALANCE_INCREMENT rather than Gwei.

This discretisation of balance is designed to reduce the amount of hashing required when making state updates. As we shall see, validators' actual balances are stored as a contiguous list in BeaconState. This is easy to update. Effective balances are stored in the individual validator records and are more costly to update (more hashing required). So we try to update effective balances relatively infrequently.

Effective balance is changed according to a process with hysteresis to avoid situations where it changes frequently. See HYSTERESIS_QUOTIENT above.

### Initial values

Name Value
GENESIS_FORK_VERSION Version('0x00000000')

Forks/upgrades are expected, if only when we move to Phase 1. This is the fork version the beacon chain starts with at its "Genesis" event: the point at which the chain first starts producing blocks.

BLS_WITHDRAWAL_PREFIX Bytes1('0x00')

Not actually used in this core beacon chain spec, but used in the deposit contract spec.

Validators need to register two public/private key pairs. The signing key is used constantly for signing attestations and blocks. The withdrawal key will be used in future after a validator has exited to allow the validator's Ether balance to be transferred to an Eth2 account. The withdrawal credentials are stored in the validator's record so that, in future, the owner of the validator can lay claim to the original stake and accrued rewards. The withdrawal credentials is the 32 byte SHA256 hash of the validators withdrawal public key, with the first byte set to BLS_WITHDRAWAL_PREFIX as a version number, in case of future changes.

### Time parameters

Name Value Unit Duration
GENESIS_DELAY 172800 seconds 2 days

The GENESIS_DELAY is a grace period to allow nodes and node operators time to prepare for the Genesis event. The Genesis event cannot occur before MIN_GENESIS_TIME. If there are not MIN_GENESIS_ACTIVE_VALIDATOR_COUNT registered validators sufficiently in advance of MIN_GENESIS_TIME, then Genesis will occur GENESIS_DELAY seconds after enough validators have been registered.

The Genesis event (beacon chain start) was originally designed to take place at midnight UTC, even for testnets, which was not always convenient. This has now been changed. Once we're past MIN_GENESIS_TIME - GENESIS_DELAY, Genesis could end up being at any time of the day, depending on when the last depost needed comes in.

SECONDS_PER_SLOT 12 seconds 12 seconds

This used to be 6 seconds, but is now 12, and has previously had other values. The main limiting factors in shortening this is the time necessary for block proposals to propagate among committees, and for validators to communicate and aggregate their votes for the block.

This slot length has to account for shard blocks as well in later phases. There was some discussion around having the beacon chain and shards on differing cadences, but the latest Phase 1 design tightly couples the beacon chain with the shards. Shard blocks under the new proposal are much larger, which led to the lengthening of the slot to 12 seconds.

There is a general intention to shorten this in future, perhaps to [8 seconds](https://github.com/ethereum/eth2.0-specs/issues/1890#issue-638024803, if it proves possible to do this in practice.

SECONDS_PER_ETH1_BLOCK 14 seconds 14 seconds

The assumed block interval on the Eth1 chain, used when calculating how long we will wait before trusting that an Eth1 block will not be reorganised.

The average Eth1 block time since January 2020 has actually been nearer 13 seconds, but never mind. The net effect is that we will be going a little deeper back in the Eth1 chain than ETH1_FOLLOW_DISTANCE would suggest, which ought to be safer.

MIN_ATTESTATION_INCLUSION_DELAY 2**0 (= 1) slots 12 seconds

A design goal of Eth2 is not to heavily disadvantage validators that are running on lower-spec systems, or, conversely, to reduce any advantage gained by running on high-spec systems.

One aspect of performance is network bandwidth. When a validator becomes the block proposer, it needs to gather attestations from the rest of its committee. On a low-bandwidth link, this takes longer, and could result in the proposer not being able to include as many past attestations as other better-connected validators might, thus receiving lower rewards.

MIN_ATTESTATION_INCLUSION_DELAY was an attempt to "level the playing field" by setting a minimum number of slots before an attestation can be included in a beacon block. It was originally set at 4, with a 6 second slot time, allowing 24 seconds for attestations to propagate around the network.

It was later set to one—attestations are included as early as possible—and, now we are crosslinking shards every slot, this is the only value that makes sense. So it exists today as a kind of relic of the earlier design.

The current slot time of 12 seconds (see above) is assumed to allow sufficient time for attestations to propagate and be aggregated sufficently within one slot. If this proves not to be the case, then it may be lengthened later.

SLOTS_PER_EPOCH 2**5 (= 32) slots 6.4 minutes

When slots were six seconds, there were 64 slots per epoch. So the time between epoch boundaries is unchanged compared with the original design.

As a reminder, epoch transitions are where the heavy beacon chain state-transition calculation occurs, so we don't want them too close together. On the other hand, they are also the targets for finalisation, so we don't want them too far apart.

MIN_SEED_LOOKAHEAD 2**0 (= 1) epochs 6.4 minutes

A random seed is used to select all the committees and proposers for an epoch. Every epoch, the beacon chain accumulates randomness from proposers via the RANDAO and stores it. The seed for the current epoch is based on the RANDAO output from the epoch MIN_SEED_LOOKUP + 1 ago. With MIN_SEED_LOOKAHEAD set to one, the effect is that we can know the seed for the current epoch and the next epoch, but not beyond (since the next-but-one epoch depends on randomness from the current epoch that hasn't been accumulated yet).

This mechanism is designed to allow sufficient time for committee members to find each other on the peer-to-peer network, and in Phase 1 to sync up any data they will need. But preventing committee makeup being known too far ahead limits the opportunity for coordinated collusion between validators.

MAX_SEED_LOOKAHEAD 2**2 (= 4) epochs 25.6 minutes

The above notwithstanding, if an attacker has a large proportion of the stake, or is, for example, able to DoS block proposers for a while, then it might be possible for the the attacker to predict the output of the RANDAO further ahead than MIN_SEED_LOOKAHEAD would normally allow. In which case the attacker might be able to manipulate the make up of committtees advantageously by performing judicious exits and activations of their validators.

To prevent this, we assume a maximum feasible lookahead that an attacker might achieve (that is, this parameter) and delay all activations and exits by this amount. With MAX_SEED_LOOKAHEAD set to 4, if only 10% of validators are online and honest, then the chance that an attacker can succeed in forecasting the seed beyond MAX_SEED_LOOK_AHEAD - MIN_SEED_LOOKAHEAD = 3 epochs is $\smash{0.9^{3\times32}}$, which is about 1 in 25,000.

MIN_EPOCHS_TO_INACTIVITY_PENALTY 2**2 (= 4) epochs 25.6 minutes

The inactivity penalty is discussed below. This parameter sets the time until it kicks in: if the last finalised epoch is longer ago than this, then the beacon chain starts operating in "leak" mode. In this mode, participating validators no longer get rewarded, and validators that are not participating get penalised.

EPOCHS_PER_ETH1_VOTING_PERIOD 2**5 (= 32) epochs ~3.4 hours

In order to safely onboard new validators, the beacon chain needs to take a view on what the Eth1 chain looks like. This is done by collecting votes from beacon block proposers - they are expected to consult an available Eth1 client in order to construct their vote.

EPOCHS_PER_ETH1_VOTING_PERIOD * SLOTS_PER_EPOCH is the total number of votes for Eth1 blocks that are collected. As soon as half of this number of votes are for the same Eth1 block, that block is adopted by the beacon chain and deposit processing can continue.

Rules for how validators select the right block to vote for are set out in the validator guide. ETH1_FOLLOW_DISTANCE is the minimum depth of block that can be considered: this is very conservatively set to ensure that we never end up with some block that we've relied on subsequently being reorganised out of the Ethereum 1 chain. For a detailed analysis of these parameters, see this ethresear.ch post

SLOTS_PER_HISTORICAL_ROOT 2**13 (= 8,192) slots ~27 hours

There have been several redesigns of the way the beacon chain stores its past history. The current design is a double batched accumulator. The block root and state root for every slot are stored in the state for SLOTS_PER_HISTORICAL_ROOT slots. When that list is full, both lists are merkleised into a single Merkle root, which is added to the ever-growing state.historical_roots list.

MIN_VALIDATOR_WITHDRAWABILITY_DELAY 2**8 (= 256) epochs ~27 hours

Once a validator has made it through the exit queue it can stop participating. However, its funds remain locked for the duration of MIN_VALIDATOR_WITHDRAWABILITY_DELAY. In Phase 0 this is to allow some time for any slashable behaviour to be detected and reported so that the validator can still be penalised (in which case the validator's withdrawable time is pushed EPOCHS_PER_SLASHINGS_VECTOR into the future). In Phase 1 this delay will also allow for shard rewards to be credited and for proof of custody challenges to be mounted.

Note that in Phases 0 and 1 there is no mechanism to withdraw a validator's balance in any case. But being in a "withdrawable" state means that a validator has now fully exited from the protocol.

SHARD_COMMITTEE_PERIOD Epoch(2**8) (= 256) epochs ~27 hours

This really anticipates Phase 1. The idea is that it's bad for the stability of longer-lived shard committees if validators can appear and disappear very rapidly. Therefore, a validator cannot initiate a voluntary exit until SHARD_COMMITTEE_PERIOD epochs after it is activated. Note that it could still be ejected by slashing before this time.

### State list lengths

The following parameters set the sizes of some lists in the beacon chain state. Some lists have natural sizes, others such as the validator registry need an explicit maximum size to help with SSZ serialisation.

Name Value Unit Duration
EPOCHS_PER_HISTORICAL_VECTOR 2**16 (= 65,536) epochs ~0.8 years

This is the number of epochs of previous RANDAO mixes that are stored (one per epoch). Having access to past randao mixes allows historical shufflings to be recalculated.

[TODO: Does this make sense? How to recompute shufflings without the validator list? I expect the duration has something to do with the weak subjectivity period, but need to check.]

EPOCHS_PER_SLASHINGS_VECTOR 2**13 (= 8,192) epochs ~36 days

In the epoch in which a misbehaving validator is slashed, its effective balance is added to an accumulator in the state. In this way, the state.slashings list tracks the total effective balance of all validators slashed during the last EPOCHS_PER_SLASHINGS_VECTOR epochs.

At a time EPOCHS_PER_SLASHINGS_VECTOR // 2 after being slashed, a further penalty is applied to the slashed validator, based on the total amount of value slashed during the 4096 epochs before and the 4096 epochs after it was originally slashed.

The idea of this is to disproportionately punish coordinated attacks, in which many validators break the slashing conditions at the same time, while only lightly penalising validators that get slashed by making a mistake. Early designs for Eth2 would always slash a validator's entire deposit.

HISTORICAL_ROOTS_LIMIT 2**24 (= 16,777,216) historical roots ~52,262 years

Every SLOTS_PER_HISTORICAL_ROOT slots, the list of block roots and the list of state roots are merkleised and added to state.historical_roots list. This is sized so that it is possible to store these roots for the entire past history of the chain. Although this list is effectively unbounded, it grows at less than 10 KB per year.

Storing past roots like this allows historical Merkle proofs to be constructed if required.

VALIDATOR_REGISTRY_LIMIT 2**40 (= 1,099,511,627,776) validators

Every time the Eth1 deposit contract processes a deposit from a new validator (as identified by its public key), a new entry is appended to the state.validators list.

In the current design, validators are never removed from this list, even after exiting from being a validator. This is largely because there is nowhere yet to send a validator's remaining deposit and staking rewards, so they continue to need to be tracked in the beacon chain.

The maximum length of this list is VALIDATOR_REGISTRY_LIMIT, which is one trillion, so we ought to be OK for a while, especially given that the minimum deposit amount is 1 Ether.

### Rewards and penalties

Name Value
BASE_REWARD_FACTOR 2**6 (= 64)

This is the big knob to turn to change the issuance rate of Eth2. Almost all validator rewards are calculated in terms of a "base reward" which is calculated as,

effective_balance * BASE_REWARD_FACTOR // integer_squareroot(total_balance) // BASE_REWARDS_PER_EPOCH

where effective_balance is the individual validator's current effective balance and total_balance is the sum of the effective balances of all active validators.

Thus, the total validator rewards per epoch (the Eth2 issuance rate) could in principle be tuned by increasing or decreasing BASE_REWARD_FACTOR.

WHISTLEBLOWER_REWARD_QUOTIENT 2**9 (= 512)

One reward amount that is not tied to the base reward is the whistleblower reward. This is a reward for providing a proof that a proposer or attestor has violated a slashing condition. The whistleblower reward is set at $\smash{\frac{1}{512}}$ of the effective balance of the slashed validator.

PROPOSER_REWARD_QUOTIENT 2**3 (= 8)

The whistleblower reward can optionally be divided between the reporter and the proposer that includes the report in a block in the ratio 7:1. That is, the proposer receives a proportion 1/PROPOSER_REWARD_QUOTIENT of the reward.

In the Phase 0 spec, however, the whistleblower reward always gets awarded in its entirety to the block proposer including the report, ignoring this parameter. In general, it's quite hard to avoid whistleblowing reports being stolen by block proposers in any case. zkProofs might help one day.

INACTIVITY_PENALTY_QUOTIENT 2**24 (= 16,777,216)

If the beacon chain hasn't finalised an epoch for longer than MIN_EPOCHS_TO_INACTIVITY_PENALTY epochs, then it enters "leak" mode. In this mode, any validator that does not vote (or votes for an incorrect target) is penalised an amount each epoch of effective_balance * finality_delay // INACTIVITY_PENALTY_QUOTIENT. The effect of this is the quadratic leak described below.

MIN_SLASHING_PENALTY_QUOTIENT 2**5 (= 32)

When a validator is first convicted of a slashable offence, an initial penalty is applied. This is calculated as,

validator.effective_balance // MIN_SLASHING_PENALTY_QUOTIENT

Thus, the initial slashing penalty is between 0.5 Ether and 1 Ether depending on the validator's effective balance (which is between 16 and 32 Ether; note that effective balance is denominated in Gwei).

A further slashing penalty is applied later based on the total amount of balance slashed during a period of EPOCHS_PER_SLASHINGS_VECTOR.

• The INACTIVITY_PENALTY_QUOTIENT equals INVERSE_SQRT_E_DROP_TIME**2 where INVERSE_SQRT_E_DROP_TIME := 2**12 epochs (about 18 days) is the time it takes the inactivity penalty to reduce the balance of non-participating validators to about 1/sqrt(e) ~= 60.6%. Indeed, the balance retained by offline validators after n epochs is about (1 - 1/INACTIVITY_PENALTY_QUOTIENT)**(n**2/2); so after INVERSE_SQRT_E_DROP_TIME epochs, it is roughly (1 - 1/INACTIVITY_PENALTY_QUOTIENT)**(INACTIVITY_PENALTY_QUOTIENT/2) ~= 1/sqrt(e).

The idea for the inactivity leak (aka the quadratic leak) was proposed in the original Casper FFG paper. The problem it addresses is that, if a large fraction of the validator set were to go offline at the same time, it would not be possible to continue finalising blocks, since a 2/3 majority of the whole validator set is required for finalisation.

In order to recover, the inactivity leak gradually reduces the stakes of validators who are not making attestations until, eventually, the remaining participating validators control 2/3 of the remaining stake. They can then begin to finalise checkpoints once again.

In the calculation here, we are solving the (discrete form of the) differential equation $\smash{\frac{dB}{dt}=-\frac{Bt}{\alpha}}$, where $B$ is the balance and $\alpha$ is the value of INACTIVITY_PENALTY_QUOTIENT, and the amount leaked at each step increases in proportion to the time $t$ since finality. The solution to this differential equation is $\smash{B(t)=B_0e^{-t^2/2\alpha}}$. From this it can be confirmed that INVERSE_SQRT_E_DROP_TIME, the time taken to reduce starting balances by $\smash{e^{\frac{1}{2}}}$, is $t=\sqrt{\alpha}$. The second half of the spec paragraph is just a calculus-avoiding way of expressing the same thing.

With these parameters, it would take about 21.4 days for a validator to leak half its deposit and then be ejected for falling below the EJECTION_BALANCE.

This inactivity penalty mechanism is designed to protect the chain long-term in the face of catastrophic events (sometimes referred to as the ability to surve World War III). The result might be that the beacon chain could permanently split into two independent chains either side of a network partition, but this is assumed to be a reasonable outcome for any problem that can't be fixed in a few weeks. In this sense, the beacon chain technically prioritises availability over consistency. (You can't have both.)

### Max operations per block

Name Value
MAX_PROPOSER_SLASHINGS 2**4 (= 16)
MAX_ATTESTER_SLASHINGS 2**1 (= 2)
MAX_ATTESTATIONS 2**7 (= 128)
MAX_DEPOSITS 2**4 (= 16)
MAX_VOLUNTARY_EXITS 2**4 (= 16)

These parameters are used to size lists in the beacon block bodies for the purposes of SSZ serialisation, as well as constraining the maximum size of beacon blocks so that they can propagate efficiently, and avoid DoS attacks.

With these settings, the maximum size of a beacon block (before compression) is 123,016 bytes. By far the largest object is the AttesterSlashing, at up to 33,216 bytes. However, a single attester slashing can be used to slash many misbehaving validators at the same time (assuming that in an attack, many validators would make the same conflicting vote).

With some assumptions on average behaviour and compressibility, this leads to an average block size of around 36 KBytes, compressing down to 22 KBytes, in the worst case (with the maximum number of validators, and the maximum average number of possible slashings).

Some calculations to support the above can be found in the next section and on this spreadsheet.

### Domain types

Name Value
DOMAIN_BEACON_PROPOSER DomainType('0x00000000')
DOMAIN_BEACON_ATTESTER DomainType('0x01000000')
DOMAIN_RANDAO DomainType('0x02000000')
DOMAIN_DEPOSIT DomainType('0x03000000')
DOMAIN_VOLUNTARY_EXIT DomainType('0x04000000')
DOMAIN_SELECTION_PROOF DomainType('0x05000000')
DOMAIN_AGGREGATE_AND_PROOF DomainType('0x06000000')

These domain types are used in two ways: for signatures and for seeds.

As a cryptographic nicety, each of the protocol's five signature types is augmented with the appropriate Domain before being signed:

• Signed block proposals incorporate DOMAIN_BEACON_PROPOSER
• Signed attestations incorporate DOMAIN_BEACON_ATTESTER
• RANDAO reveals are BLS signatures, and use DOMAIN_RANDAO
• Deposit data mesages from Ethereum 1 incorporate DOMAIN_DEPOSIT
• Validator voluntary exit messages incorporate DOMAIN_VOLUNTARY_EXIT

In each case, except for Eth1 deposits, the fork version is also incorporated before signing. Deposits are valid across forks, but other messages are not. Note that this would allow validators to participate, if they wish, in two independent forks of the beacon chain without fear of being slashed.

In addition, the first two domains are also used to separate the seeds for random number generation. The original motivation was to avoid occasional collisions between Phase 0 committees and Phase 1 persistent committees. So, when computing the beacon block proposer, DOMAIN_BEACON_PROPOSER is hashed into the seed, and when computing committees, DOMAIN_BEACON_ATTESTER is hashed into the seed.

The last two domains were introduced to implement attestation subnet validations for denial of service resistance. They are not part of the consensus-critical state-transition. In short, each slot, validators are selected to aggregate attestations from their committee. The selection is done based on the validator's signature over the slot number, mixing in DOMAIN_SELECTION_PROOF. Then the validator signs the whole aggregated attestation using DOMAIN_AGGREGATE_AND_PROOF. See the Honest Validator spec for more on this.