# Where did the number of 32 Eth to become a validator come from?

What was the Math behind coming to the number of 32 Eth as a threshold to become a validator? Which parameters were considered to come to this number? Did the parameters of Eth distribution in the community, and total members in the community play a role? What role does the Sybil attack or 51% attack play in determining the number 32 ETH?

The 32 ETH stake size is a trade-off between network overhead, number of validators, and time to finality.

As for the long answer, I will try to sum up the explanations from Ben Edgington and Vitalik that I encourage you to read for a more thorough understanding.

The math behind 32 ETH is the following:

if n is the number of validators in a network, f is the time to finalize a block, and ω is the overhead in messages per second, then we have: ω ≥ 2n/f.

To put this in concrete terms, the hard limit on the number of validators is the total Ether supply divided by the stake size. If we take a stake of 32 ETH and assume a total supply of 2^27 ETH this means n=2^22. With that, we want to reach finality in two epochs (=768 seconds), which leads to about 11,000 messages per second which is possible thanks to BLS aggregate signature.

I've summed up the "official" explanation. In my opinion, there is also the fact that when the choice was made, 32 ETH didn't seem as high as it is now. So a rule of thumb was made to ensure decentralization and communication going well.

If you are interested, linked to this subject is a proposition of Vitalik to usher the protocol to single slot finality, thus changing drastically the activity of validators.