# # Sortition Algorithm

## # Sortition Seed

Sortition algorithm requires a seed that is random and publicly verifiable. In each height, the proposer has to generate a new seed based on previous seed and the adversary is not able to manipulate the seed generation.

To achieve this, we use BLS signature scheme to generate the seed. Since BLS signature is unique and deterministic, the adversary cannot generate more than one valid signature per height.

During the block proposal stage, the proposer should take the previous seed and sign it with its public key. The result is the seed for the next block.

$seed_{n+1}=generate(sk_{proposer}, hash(seed_{n}))$

Everyone knows the proposer's public key, therefore they can easily validate the seed for the next block. If the seed is not valid, the proposed block will be rejected.

$verify(pub_{proposer}, hash(seed_{n}), seed_{n-1})==True$

- Note: Seed is 48 bytes.
- Note: Seed for genesis block is 0.

## # Verifiable Random Function

Verifiable Random Function is a pseudo-random function that the owner of key $s$ at any point of $x$
can evaluate $v=f_s(x)$ and also provides $proof_x$ efficiently proving that v is correct. We call
such a mathematical object a verifiable(pseudo-)random function, VRF for brevity ^{[1]}.

In Zarb we are using BLS signature scheme as source of VRF. Since BLS signature are unique, in the random oracle model, the hash of a BLS signature can be used as a secure VRF.

VRF takes the publicly-known seed for each block and generates an index and proof.

$<index, proof>=VRF(sk_{validator}, seed_{n})$

The result of VRF is deterministic and Zero Knowledge Provable.

## # How VRF Works?

In each height, validators outside the committee run VRF to generate a verifiable random number between 0 and 1. The seed for VRF is known to all validators and it is of part previous block header. A validator based on its stake and its luck can enter the committee for the next height. The formula is quite simple:

$R = index {\times} S_{p}$

*which*

- $index$ is VRF output between 0~1
- $S_{p}$ is sum of the stake in validator pool

Validator pool contains validators that are not in the committee. We can easily calculate the stake of pool at any given height:

$S_{p} = S_{t} - S_{c}$

*which*

- $S_{t}$ is the total stake of the network
- $S_{c}$ is the sum of the stake of committee members

If the result $R$ is less than the validator's stake, then this validator is eligible to enter the set for the next height. The validator needs to send a Sortition transaction with the $proof$ of VRF. Other validators can easily verify the sortition:

$verify(pub_{validator}, proof, seed_n) < S_v$

*which*

- $pub_{validator}$ os the public key of the validator
- $S_v$ is the stake of validator at height $n$

There is no need to send $index$ alongside $proof$ because the result should be less than validator's stake and validator's stake is known at height $n$ for all the validators in the network.

If a validator can generate a valid sortition proof, it can broadcast a sortition transaction. Sortition transactions are valid for 7 blocks. When a valid sortition transaction is committed into a block, the new validator from the pool can enter the committee and the oldest validator in the committee will leave the committee to make it balanced.