Optimal Strategic Mining Against Cryptographic Self-Selection in Proof-of-Stake

Cryptographic Self-Selection is a subroutine used to select a leader for modern proof-of-stake consensus protocols, such as Algorand. In cryptographic self-selection, each round $r$ has a seed $Q_r$. In round $r$, each account owner is asked to digitally sign $Q_r$, hash their digital signature to produce a credential, and then broadcast this credential to the entire network. A publicly-known function scores each credential in a manner so that the distribution of the lowest scoring credential is identical to the distribution of stake owned by each account. The user who broadcasts the lowest-scoring credential is the leader for round $r$, and their credential becomes the seed $Q_{r+1}$. Such protocols leave open the possibility of a selfish-mining style attack: a user who owns multiple accounts that each produce low-scoring credentials in round $r$ can selectively choose which ones to broadcast in order to influence the seed for round $r+1$. Indeed, the user can pre-compute their credentials for round $r+1$ for each potential seed, and broadcast only the credential (among those with a low enough score to be the leader) that produces the most favorable seed. We consider an adversary who wishes to maximize the expected fraction of rounds in which an account they own is the leader. We show such an adversary always benefits from deviating from the intended protocol, regardless of the fraction of the stake controlled. We characterize the optimal strategy; first by proving the existence of optimal positive recurrent strategies whenever the adversary owns last than $38\%$ of the stake. Then, we provide a Markov Decision Process formulation to compute the optimal strategy.