nomos-specs/carnot/committee_sizes.py

"""
This algorithm given the number_of_nodes in the network,  fraction of Byzantine nodes in the network, network_adversary_threshold,
fraction of Byzantine modes in a committee, adversaries_threshold_per_committee, and the probability of failure_threshold  which can be tolerated computes  the maximum number_of_committees and  committee_size.

The algorithm computes  the current_probability for the  number_of_nodes=committee_size*number_of_committees+remainder nodes where  the  committee_size and committee_size+1 number of nodes are assigned, respectively, to the  number_of_committees-remainder and remainder number of  committees. 

Initially, all number_of_nodes are in one committee, and in subsequent iterations, the number_of_committees is increased by two until the current_probability <= failure_threshold. When the latter condition is violated  then the algorithm stops and outputs the number_of_committees, committee_size, remainder and current_probability.  

A more detailed description of the algorithm,  and of its mathematical aspects, is provided in the "Carnot paper" available at https://www.notion.so/Nomos-Specification-419bfb7a939648e9b3894a90d188c3be?pvs=4   
"""
import math
from scipy.stats import binom


CARNOT_ADVERSARY_THRESHOLD_PER_COMMITTEE: float = 1/3
CARNOT_NETWORK_ADVERSARY_THRESHOLD: float = 1 / 4


def compute_optimal_number_of_committees_and_committee_size(
        number_of_nodes: int,
        failure_threshold: float,
        adversaries_threshold_per_committee: float,
        network_adversary_threshold: float
):
    assert failure_threshold > 0
    
    # number_of_nodes is the number of nodes in the network 
    # failure_threshold is the prob. of failure which can be tolerated
    # adversaries_threshold_per_committee is the fraction of Byzantine modes in a committee
    # network_adversary_threshold is the fraction of Byzantine nodes in the network
    
    number_of_committees = 1
    committee_size = number_of_nodes
    remainder = 0
    current_probability = 0.0
    odd_committee = 0
    while current_probability < failure_threshold:
        previous_number_of_committees = number_of_committees
        previous_committee_size = committee_size
        previous_remainder = remainder
        previous_probability = current_probability
        odd_committee = odd_committee + 1
        number_of_committees = 2 * odd_committee + 1
        committee_size = number_of_nodes // number_of_committees
        remainder = number_of_nodes % number_of_committees

        committee_size_probability = binom.cdf(
            math.floor(adversaries_threshold_per_committee * committee_size),
            committee_size,
            network_adversary_threshold
        )
        if 0 < remainder:
            committee_size_plus_one_probability = binom.cdf(
                math.floor(adversaries_threshold_per_committee * (committee_size + 1)),
                committee_size + 1,
                network_adversary_threshold
            )
            current_probability = (
                    1 - committee_size_probability ** (number_of_committees - remainder)
                    * committee_size_plus_one_probability ** remainder
            )
        else:
            current_probability = 1 - committee_size_probability ** number_of_committees
            
    # return the number_of_committees, committee_size, remainder and current_probability
    # computed at the previous iteration. 

    return previous_number_of_committees, previous_committee_size, previous_remainder, previous_probability