# Core Verification

## Problem statement

We assume that the light client knows a (base) header inithead it trusts (by social consensus or because the light client has decided to trust the header before). The goal is to check whether another header newhead can be trusted based on the data in inithead.

The correctness of the protocol is based on the assumption that inithead was generated by an instance of Tendermint consensus.

### Failure Model

For the purpose of the following definitions we assume that there exists a function validators that returns the corresponding validator set for the given hash.

The light client protocol is defined with respect to the following failure model:

Given a known bound TRUSTED_PERIOD, and a block b with header h generated at time Time (i.e. h.Time = Time), a set of validators that hold more than 2/3 of the voting power in validators(b.Header.NextValidatorsHash) is correct until time b.Header.Time + TRUSTED_PERIOD.

Assumption: “correct” is defined w.r.t. realtime (some Newtonian global notion of time, i.e., wall time), while Header.Time corresponds to the BFT time. In this note, we assume that clocks of correct processes are synchronized (for example using NTP), and therefore there is bounded clock drift (CLOCK_DRIFT) between local clocks and BFT time. More precisely, for every correct light client process and every header.Time (i.e. BFT Time, for a header correctly generated by the Tendermint consensus), the following inequality holds: Header.Time < now + CLOCK_DRIFT, where now corresponds to the system clock at the light client process.

Furthermore, we assume that TRUSTED_PERIOD is (several) order of magnitude bigger than CLOCK_DRIFT (TRUSTED_PERIOD >> CLOCK_DRIFT), as CLOCK_DRIFT (using NTP) is in the order of milliseconds and TRUSTED_PERIOD is in the order of weeks.

We expect a light client process defined in this document to be used in the context in which there is some larger period during which misbehaving validators can be detected and punished (we normally refer to it as UNBONDING_PERIOD due to the “bonding” mechanism in modern proof of stake systems). Furthermore, we assume that TRUSTED_PERIOD < UNBONDING_PERIOD and that they are normally of the same order of magnitude, for example TRUSTED_PERIOD = UNBONDING_PERIOD / 2.

The specification in this document considers an implementation of the light client under the Failure Model defined above. Mechanisms like fork accountability and evidence submission are defined in the context of UNBONDING_PERIOD and they incentivize validators to follow the protocol specification defined in this document. If they don’t, and we have 1/3 (or more) faulty validators, safety may be violated. Our approach then is to detect these cases (after the fact), and take suitable repair actions (automatic and social). This is discussed in document on Fork accountability.

The term “trusted” above indicates that the correctness of the protocol depends on this assumption. It is in the responsibility of the user that runs the light client to make sure that the risk of trusting a corrupted/forged inithead is negligible.

Remark: This failure model might change to a hybrid version that takes heights into account in the future.

### High Level Solution

Upon initialization, the light client is given a header inithead it trusts (by social consensus). When a light clients sees a new signed header snh, it has to decide whether to trust the new header. Trust can be obtained by (possibly) the combination of three methods.

1. Uninterrupted sequence of headers. Given a trusted header h and an untrusted header h1, the light client trusts a header h1 if it trusts all headers in between h and h1.

2. Trusted period. Given a trusted header h, an untrusted header h1 > h and TRUSTED_PERIOD during which the failure model holds, we can check whether at least one validator, that has been continuously correct from h.Time until now, has signed h1. If this is the case, we can trust h1.

3. Bisection. If a check according to 2. (trusted period) fails, the light client can try to obtain a header hp whose height lies between h and h1 in order to check whether h can be used to get trust for hp, and hp can be used to get trust for snh. If this is the case we can trust h1; if not, we continue recursively until either we found set of headers that can build (transitively) trust relation between h and h1, or we failed as two consecutive headers don’t verify against each other.

## Definitions

### Data structures

In the following, only the details of the data structures needed for this specification are given.

   type Header struct {
Height               int64
Time                 Time          // the chain time when the header (block) was generated

LastBlockID          BlockID       // prev block info
ValidatorsHash       []byte        // hash of the validators for the current block
NextValidatorsHash   []byte        // hash of the validators for the next block
}

Commit        Commit            // commit for the given header
}

type ValidatorSet struct {
Validators         []Validator
TotalVotingPower   int64
}

type Validator struct {
VotingPower   int64             // validator's voting power
}

type TrustedState {
ValidatorSet   ValidatorSet
}


### Functions

For the purpose of this light client specification, we assume that the Cosmos Full Node exposes the following functions over RPC:

    // returns signed header: Header with Commit, for the given height

// returns validator set for the given height
func Validators(height int64) (ValidatorSet, error)


Furthermore, we assume the following auxiliary functions:

    // returns true if the commit is for the header, ie. if it contains
// the correct hash of the header; otherwise false

// returns the set of validators from the given validator set that
// committed the block (that correctly signed the block)
// it assumes signature verification so it can be computationally expensive
func signers(commit Commit, validatorSet ValidatorSet) []Validator

// returns the voting power the validators in v1 have according to their voting power in set v2
// it does not assume signature verification
func votingPowerIn(v1 []Validator, v2 ValidatorSet) int64

// returns hash of the given validator set
func hash(v2 ValidatorSet) []byte


In the functions below we will be using trustThreshold as a parameter. For simplicity we assume that trustThreshold is a float between 1/3 and 2/3 and we will not be checking it in the pseudo-code.

VerifySingle. The function VerifySingle attempts to validate given untrusted header and the corresponding validator sets based on a given trusted state. It ensures that the trusted state is still within its trusted period, and that the untrusted header is within assumed clockDrift bound of the passed time now. Note that this function is not making external (RPC) calls to the full node; the whole logic is based on the local (given) state. This function is supposed to be used by the IBC handlers.

func VerifySingle(untrustedSh SignedHeader,
untrustedVs ValidatorSet,
untrustedNextVs ValidatorSet,
trustedState TrustedState,
trustThreshold float,
trustingPeriod Duration,
clockDrift Duration,
now Time) (TrustedState, error) {

if untrustedSh.Header.Time > now + clockDrift {
}

}

// we assume that time it takes to execute verifySingle function
// is several order of magnitudes smaller than trustingPeriod
error = verifySingle(
trustedState,
untrustedSh,
untrustedVs,
untrustedNextVs,
trustThreshold)

if error != nil return (state, error)

// the untrusted header is now trusted
newTrustedState = TrustedState(untrustedSh, untrustedNextVs)
return (newTrustedState, nil)
}

// return true if header is within its light client trusted period; otherwise returns false
trustingPeriod Duration,
now Time) bool {

return header.Time + trustedPeriod > now
}


Note that in case VerifySingle returns without an error (untrusted header is successfully verified) then we have a guarantee that the transition of the trust from trustedState to newTrustedState happened during the trusted period of trustedState.SignedHeader.Header.

TODO: Explain what happens in case VerifySingle returns with an error.

verifySingle. The function verifySingle verifies a single untrusted header against a given trusted state. It includes all validations and signature verification. It is not publicly exposed since it does not check for header expiry (time constraints) and hence it’s possible to use it incorrectly.

func verifySingle(trustedState TrustedState,
untrustedVs ValidatorSet,
untrustedNextVs ValidatorSet,
trustThreshold float) error {

untrustedCommit = untrustedSh.Commit

trustedVs = trustedState.ValidatorSet

// validate the untrusted header against its commit, vals, and next_vals
if error != nil return error

}
} else {
error = verifyCommitTrusting(trustedVs, untrustedCommit, untrustedVs, trustThreshold)
if error != nil return error
}

// verify the untrusted commit
return verifyCommitFull(untrustedVs, untrustedCommit)
}

// returns nil if header and validator sets are consistent; otherwise returns error
if hash(vs) != header.ValidatorsHash return ErrInvalidValidatorSet
if hash(nextVs) != header.NextValidatorsHash return ErrInvalidNextValidatorSet
return nil
}

// returns nil if at least single correst signer signed the commit; otherwise returns error
func verifyCommitTrusting(trustedVs ValidatorSet,
commit Commit,
untrustedVs ValidatorSet,
trustLevel float) error {

totalPower := trustedVs.TotalVotingPower
signedPower := votingPowerIn(signers(commit, untrustedVs), trustedVs)

// check that the signers account for more than max(1/3, trustLevel) of the voting power
// this ensures that there is at least single correct validator in the set of signers
if signedPower < max(1/3, trustLevel) * totalPower return ErrInsufficientVotingPower
return nil
}

// returns nil if commit is signed by more than 2/3 of voting power of the given validator set
// return error otherwise
func verifyCommitFull(vs ValidatorSet, commit Commit) error {
totalPower := vs.TotalVotingPower;
signedPower := votingPowerIn(signers(commit, vs), vs)

// check the signers account for +2/3 of the voting power
if signedPower * 3 <= totalPower * 2 return ErrInvalidCommit
return nil
}


VerifyHeaderAtHeight. The function VerifyHeaderAtHeight captures high level logic, i.e., application call to the light client module to download and verify header for some height.

func VerifyHeaderAtHeight(untrustedHeight int64,
trustedState TrustedState,
trustThreshold float,
trustingPeriod Duration,
clockDrift Duration) (TrustedState, error)) {

now := System.Time()
}

newTrustedState, err := VerifyBisection(untrustedHeight,
trustedState,
trustThreshold,
trustingPeriod,
clockDrift,
now)

if err != nil return (trustedState, err)

now = System.Time()
}

return (newTrustedState, err)
}


Note that in case VerifyHeaderAtHeight returns without an error (untrusted header is successfully verified) then we have a guarantee that the transition of the trust from trustedState to newTrustedState happened during the trusted period of trustedState.SignedHeader.Header.

In case VerifyHeaderAtHeight returns with an error, then either (i) the full node we are talking to is faulty or (ii) the trusted header has expired (it is outside its trusted period). In case (i) the full node is faulty so light client should disconnect and reinitialize with new peer. In the case (ii) as the trusted header has expired, we need to reinitialize light client with a new trusted header (that is within its trusted period), but we don’t necessarily need to disconnect from the full node we are talking to (as we haven’t observed full node misbehavior in this case).

VerifyBisection. The function VerifyBisection implements recursive logic for checking if it is possible building trust relationship between trustedState and untrusted header at the given height over finite set of (downloaded and verified) headers.

func VerifyBisection(untrustedHeight int64,
trustedState TrustedState,
trustThreshold float,
trustingPeriod Duration,
clockDrift Duration,
now Time) (TrustedState, error) {

untrustedSh, error := Commit(untrustedHeight)
if error != nil return (trustedState, ErrRequestFailed)

// note that we pass now during the recursive calls. This is fine as
// for a smaller heights, and therefore should happen before.
if untrustedHeader.Time > now + clockDrift {
}

untrustedVs, error := Validators(untrustedHeight)
if error != nil return (trustedState, ErrRequestFailed)

untrustedNextVs, error := Validators(untrustedHeight + 1)
if error != nil return (trustedState, ErrRequestFailed)

error = verifySingle(
trustedState,
untrustedSh,
untrustedVs,
untrustedNextVs,
trustThreshold)

if fatalError(error) return (trustedState, error)

if error == nil {
// the untrusted header is now trusted.
newTrustedState = TrustedState(untrustedSh, untrustedNextVs)
return (newTrustedState, nil)
}

// at this point in time we need to do bisection
pivotHeight := ceil((trustedHeader.Height + untrustedHeight) / 2)

error, newTrustedState = VerifyBisection(pivotHeight,
trustedState,
trustThreshold,
trustingPeriod,
clockDrift,
now)
if error != nil return (newTrustedState, error)

return VerifyBisection(untrustedHeight,
newTrustedState,
trustThreshold,
trustingPeriod,
clockDrift,
now)
}

func fatalError(err) bool {
err == ErrNonIncreasingHeight OR
err == ErrNonIncreasingTime OR
err == ErrInvalidValidatorSet OR
err == ErrInvalidNextValidatorSet OR
err == ErrInvalidCommitValue OR
err == ErrInvalidCommit
}


### The case untrustedHeader.Height < trustedHeader.Height

In the use case where someone tells the light client that application data that is relevant for it can be read in the block of height k and the light client trusts a more recent header, we can use the hashes to verify headers “down the chain.” That is, we iterate down the heights and check the hashes in each step.

Remark. For the case were the light client trusts two headers i and j with i < k < j, we should discuss/experiment whether the forward or the backward method is more effective.

func VerifyHeaderBackwards(trustedHeader Header,
trustingPeriod Duration,
clockDrift Duration) error {

now := System.Time()
}

untrustedSh, error := Commit(i)
if error != nil return ErrRequestFailed

}

}

}

now := System.Time()
}

return nil
}


Assumption: In the following, we assume that untrusted_h.Header.height > trusted_h.Header.height. We will quickly discuss the other case in the next section.

We consider the following set-up:

• the light client communicates with one full node
• the light client locally stores all the headers that has passed basic verification and that are within light client trust period. In the pseudo code below we write Store.Add(header) for this. If a header failed to verify, then the full node we are talking to is faulty and we should disconnect from it and reinitialize with new peer.
• If CanTrust returns error, then the light client has seen a forged header or the trusted header has expired (it is outside its trusted period).
• In case of forged header, the full node is faulty so light client should disconnect and reinitialize with new peer. If the trusted header has expired, we need to reinitialize light client with new trusted header (that is within its trusted period), but we don’t necessarily need to disconnect from the full node we are talking to (as we haven’t observed full node misbehavior in this case).

## Correctness of the Light Client Protocols

### Definitions

• TRUSTED_PERIOD: trusted period
• for realtime t, the predicate correct(v,t) is true if the validator v follows the protocol until time t (we will see about recovery later).
• Validator fields. We will write a validator as a tuple (v,p) such that
• v is the identifier (i.e., validator address; we assume identifiers are unique in each validator set)
• p is its voting power
• For each header h, we write trust(h) = true if the light client trusts h.

### Failure Model

If a block b with a header h is generated at time Time (i.e. h.Time = Time), then a set of validators that hold more than 2/3 of the voting power in validators(h.NextValidatorsHash) is correct until time h.Time + TRUSTED_PERIOD.

Formally, [ \sum_{(v,p) \in validators(h.NextValidatorsHash) \wedge correct(v,h.Time + TRUSTED_PERIOD)} p > 2/3 \sum_{(v,p) \in validators(h.NextValidatorsHash)} p ]

The light client communicates with a full node and learns new headers. The goal is to locally decide whether to trust a header. Our implementation needs to ensure the following two properties:

• Light Client Completeness: If a header h was correctly generated by an instance of Tendermint consensus (and its age is less than the trusted period), then the light client should eventually set trust(h) to true.

• Light Client Accuracy: If a header h was not generated by an instance of Tendermint consensus, then the light client should never set trust(h) to true.

Remark: If in the course of the computation, the light client obtains certainty that some headers were forged by adversaries (that is were not generated by an instance of Tendermint consensus), it may submit (a subset of) the headers it has seen as evidence of misbehavior.

Remark: In Completeness we use “eventually”, while in practice trust(h) should be set to true before h.Time + TRUSTED_PERIOD. If not, the header cannot be trusted because it is too old.

Remark: If a header h is marked with trust(h), but it is too old at some point in time we denote with now (h.Time + TRUSTED_PERIOD < now), then the light client should set trust(h) to false again at time now.

Assumption: Initially, the light client has a header inithead that it trusts, that is, inithead was correctly generated by the Tendermint consensus.

To reason about the correctness, we may prove the following invariant.

Verification Condition: light Client Invariant. For each light client l and each header h: if l has set trust(h) = true, then validators that are correct until time h.Time + TRUSTED_PERIOD have more than two thirds of the voting power in validators(h.NextValidatorsHash).

Formally, [ \sum_{(v,p) \in validators(h.NextValidatorsHash) \wedge correct(v,h.Time + TRUSTED_PERIOD)} p > 2/3 \sum_{(v,p) \in validators(h.NextValidatorsHash)} p ]

Remark. To prove the invariant, we will have to prove that the light client only trusts headers that were correctly generated by Tendermint consensus. Then the formula above follows from the failure model.

## Details

Observation 1. If h.Time + TRUSTED_PERIOD > now, we trust the validator set validators(h.NextValidatorsHash).

When we say we trust validators(h.NextValidatorsHash) we do not trust that each individual validator in validators(h.NextValidatorsHash) is correct, but we only trust the fact that less than 1/3 of them are faulty (more precisely, the faulty ones have less than 1/3 of the total voting power).

VerifySingle correctness arguments

Light Client Accuracy:

• Assume by contradiction that untrustedHeader was not generated correctly and the light client sets trust to true because verifySingle returns without error.
• trustedState is trusted and sufficiently new
• by the Failure Model, less than 1/3 of the voting power held by faulty validators => at least one correct validator v has signed untrustedHeader.
• as v is correct up to now, it followed the Tendermint consensus protocol at least up to signing untrustedHeader => untrustedHeader was correctly generated. We arrive at the required contradiction.

Light Client Completeness:

• The check is successful if sufficiently many validators of trustedState are still validators in the height untrustedHeader.Height and signed untrustedHeader.
• If untrustedHeader.Height = trustedHeader.Height + 1, and both headers were generated correctly, the test passes.

Verification Condition: We may need an invariant stating that if untrustedSignedHeader.Header.Height = trustedHeader.Height + 1 then signers(untrustedSignedHeader.Commit) \subseteq validators(trustedHeader.NextValidatorsHash).

Remark: The variable trustThreshold can be used if the user believes that relying on one correct validator is not sufficient. However, in case of (frequent) changes in the validator set, the higher the trustThreshold is chosen, the more unlikely it becomes that verifySingle returns with an error for non-adjacent headers.

• VerifyBisection correctness arguments (sketch)*

Light Client Accuracy:

• Assume by contradiction that the header at untrustedHeight obtained from the full node was not generated correctly and the light client sets trust to true because VerifyBisection returns without an error.
• VerifyBisection returns without error only if all calls to verifySingle in the recursion return without error (return nil).
• Thus we have a sequence of headers that all satisfied the verifySingle
This is only ensured if upon Commit(pivot) the light client is always provided with a correctly generated header.
With VerifyBisection, a faulty full node could stall a light client by creating a long sequence of headers that are queried one-by-one by the light client and look OK, before the light client eventually detects a problem. There are several ways to address this:
• Each call to Commit could be issued to a different full node
• Instead of querying header by header, the light client tells a full node which header it trusts, and the height of the header it needs. The full node responds with the header along with a proof consisting of intermediate headers that the light client can use to verify. Roughly, VerifyBisection would then be executed at the full node.
• We may set a timeout how long VerifyBisection may take.