# Plasma Cash verification cost

If there is a Plasma block every minute, then if one holds a coin for a year the size of the non-spend proof for a payment is

60 * 24 * 365 * 100 * 10 * 256 = 17 Gbyte

This assumes that an average payment transfers 10 coins and that the depth of the Merkle tree is 100

This needs to be multiplied by 2 because the receiver needs to send the change back. So it is 34 GByte total traffic

How feasible is the entire payment then? How long will it take for sender to upload and for receiver to download and verify 17 GB of data and then pay the change back and then for the sender to verify the proof for the change? Uploading 17 GB of data will mean days for many people that have asymmetric DSL.

There are two proposals to alleviate this, what do you think of them?

## Compression

Most of the coin validity proofs are exclusion proofs, and those can be compressed using SNARKs/STARKs. For e.g. if we use constant-sized proofs this means that a coin validity proof will only increase in size when a coin is spent.

Back-of-the-envelope calculation: if a coin is spent once per day, the proof size becomes 1.2 MB per coin.

## Checkpointing

We know that withdrawing and re-depositing a coin will “reset” the length of the coin validitiy proof; this construction allows you to do this at an amortized cost of 1 bit of storage (SSTORE) per coin.

Back-of-the-envelope calculation: if we do this every 3 months would make the proof size is upper-bounded at 0.4 GB per coin at the cost of a few cents per year.

## Combinations Thereof

We can combine these two techniques by doing a checkpoint when even the compressed proofs get too big. Analysis: with checkpoints, we can have the coin validity proof grow as O(f(n)) and the per-unit-time cost of owning a coin grow as O(g(n)) with the constraint that fg \in O(n) where n is the total number of plasma blocks (i.e. increases by 1 per minute).

Three points on this tradeoff space are

1. Bounded proof sizes, constant rent (e.g. 1 cent/year/coin)
2. Constant cost to own a coin, but proof size grows linearly (e.g. 1.7gb /year)
3. Both cost and proof size are O(\sqrt n)

By combining them we relax the constraint to fg \in O(s) where s is the number of spends of the coin (i.e. increases by 1 every time the coin is spent).

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You will pay several USD of ethereum costs when you withdraw/redeposit.

And then you have to wait each time you withdraw/redeposit. I do not think this is a solution people will use.

For SNARKS - what are the computational costs of SNARK calculation - I understand it makes things smaller but how does it influence computational costs? How much time does it take to compute/verify SNARKS?

You will pay several USD of ethereum costs when you withdraw/redeposit

Please reread what I wrote more carefully; I am not suggesting a withdrawal and redeposit. The cost with today’s gas fees should be one cent or less.

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For succint/scalable verification technology in general, this depends on the exact construction and there are trade-offs, eg STARKs have lower prover cost and longer proof sizes (hundreds of kilobytes vs hundreds of bytes). The trade-offs come into play because depending on exactly how you want to compress the proofs, maybe the verification must be doable on-chain, or maybe it is sufficient to do them client-side.

For the specific zk-SNARK construction I believe that the verification costs is low (dominated by two elliptic curve pairings) but the costs of the trusted setup as well as prover cost are quite high (@kfichter estimated hours per plasma block to produce a proof of the validity of an entire block). For asymptotic costs I have the following table in my notes (note: I don’t understand everything in this table!) for the cost of doing various things for an arithmetic circuit of with N wires, l of which are public:

Proof Size Prover Cost Verifier Cost Setup
discrete-log based O(\log N) O(N) exponentiations O(N) multiplications public-coin
STARKs O(\sqrt N) O(N) multiplications O(N) additions public-coin
pairing-based SNARKs O(1) O(N) exponentiations O(l) exponentiations + O(1) pairings structured setup

Thanks to Jens Groth for the table, but all mistakes are mine.

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Yeah the ~hours figure came from discussion with vitalik and then spending ~10 mins generating a height 4 merkle tree on my laptop.

@ldct Great table!! Thank you!

Do you know how does the number of wires N depend on the number of transactions ? Is it linear in the number of transactions?
If I have a coin which is one year old, then for the non-spend proof , will the number of wires be linear in time?

Also, what is public coin setup and structured setup?

@kfichter - I watched Eli Ben Sasson lecture