**Attack**

Let’s assume that an attacker controls some proportion a of validator deposits in the VMC, and some proportion b of mining power in the main chain. Because the current `getEligibleProposer`

method is subject to blockhash grinding the attacker can make himself the eligible proposer on a shard (actually, several shards, depending on how fast the attacker can grind) with proportion a + (1 - a)*b.

If we set a = b (i.e. the attacker controls the same proportion of validator deposits and mining power) and solve for a + (1 - a)*a = 0.5 (i.e. solve for the attacker having controlling power) we get a = 0.292. That is, an attacker controlling just 30% of the network can do 51% attacks on shards.

**Defenses**

One defense strategy is to use a “perfectly fair” validator sampling mechanism with no repetitions, e.g. see here. Another strategy is to improve the random number generator to something like RANDAO or Dfinity-style BLS random beacons.