# Plasma Prime design proposal

#1

Here is Plasma Prime design proposal from BANKEX Foundation team. At the moment it’s at the draft stage. Waiting for discussions & сriticism. Thank you.

## Abstract

Below we are going to describe Plasma design based on plasma cashflow improved by RSA accumulators and range chunking. Range chunking simplifies block verification for block observer by compressing history.

\DeclareMathOperator{\included}{included} \DeclareMathOperator{\parent}{parent} \DeclareMathOperator{\child}{child} \DeclareMathOperator{\inclusionproof}{inclusionproof} \DeclareMathOperator{\exclusionproof}{exclusionproof} \DeclareMathOperator{\True}{True} \DeclareMathOperator{\False}{False}

## Segments

Similar to plasma cashflow this proposal is based on the UTXO model where each unspent output defines ownership of specific segment. Length of the segment is equal to a value of the asset that was deposit to plasma.
Each segment is mapped to a prime number generated by the prime hash function, usage of that prime number will be shown later.
Due to the limitation of prime numbers that we generate with reasonable efforts plasma has a finite set of segments that can belong to different owners during the plasma lifetime.
We can say that we may have up to 2^30 different segments in the demo implementation.

## Segments chunking

Suppose there exists a segment X. During the timeA some transaction that affects X was published, e.g. split or merge transactions.
During the time B transactions that affect segments different from A was published.
At the time moment C transaction that affects X, was published, at that moment we want to check the history of that transaction is valid.

To do that we need to generate proof that covers only transactions related only to range X, and not the whole transaction history of plasma.

#### Example

Lets split plasma segments space into the four chunks.

The image below illustrates scenario when two sequential deposits transactions (txA, txB) creates two segments:

• txA creates the segment (0 - 1.75) that matches to chunk (1 - 2)
• txB creates the segment (1.75 - 5) that matches to chunks (3 - 4) and (5 - 6)

Note: Deposit transactions are coming from the smart contract, each one located in a separate block, and doesn’t have reference to any previous block

Suppose after N blocks we got following layout of segments inside plasma:

To verify that C exit is valid, block observer only needs to check the history of the segment that matches to chunks (3 - 4) and (5 - 6). Proof of inclusion of range C to mentioned chunks will include transactions that affect the same two chunks.

## RSA proof of segment inclusion and non-inclusion

The similar prooving schema was proposed by @vbuterin here. We proposed schema with shorter proofs of inclusion (because we need to proof only inclusion of several numbers) and same proof of non-inclusion.

Let’s define boolean function \included for each aligned slice P_i and introduce following auxiliary variables \alpha and \beta, mapping each aligned slice to two unique prime numbers:

\alpha(P) = \included(P) \bigwedge_{Q \in \parent(P)}! \included(Q)
\beta(P) = \exists Q \in \child(P) \cup P: \alpha(Q)

Then determine following functions:

\inclusionproof(P) = \exists Q \in \parent(P) \cup P: \alpha(Q) \bigwedge_{R\in\parent(Q) \cup Q} \beta(R)
\exclusionproof(P) = !\beta(P) \bigwedge_{Q\in\parent(P) \cup P} !\alpha(Q)

It is obviously to check, that

\inclusionproof(P) \Rightarrow \inclusionproof(Q), \forall Q \in \child(P).
\exclusionproof(P) \Rightarrow !\inclusionproof(Q), \forall Q \in \child(P) \cup P.
\inclusionproof(P) \Rightarrow !\exclusionproof(Q), \forall Q \in \child(P) \cup P.

In plasma we use the following properties to get only necessary parts of plasma history.

## Block structure

We propose the following block structure:

Notes: TX Netto is actually transaction content without signatures. A transaction can include one or two signatures, two - in case of atomic swap and one in all case of a split and merge.

struct Block {
Transactions[] transactions
}

SumMerkleRoot sumMerkleRoot,
uint2048, RSAAccumulator,
RSAInclusionProof RSAChainProof
}

struct RSAInclusionProof {
uint2048 b,
uint256 r
}

struct Transaction {
TransactionContent content,
Signatures[] signatures
}

struct TransactionContent {
Input[] inputs,
Output[] outputs,
uint64 maxBlockIndex
}

struct Input
{
uint160 owner,
uint64 blockIndex,
uint32 txIndex,
uint8 outputIndex,
Segment amount
}

struct Output
{
uint160 owner,
Segment amount
}

struct Segment
{
uint256 begin
uint256 end
}



One transaction may have multiple txIndex values because it can be fragmented to different segments.

#### SumMerkleRoot

We use SumMerkleTree to quickly verify that there’s no overlap between different transaction segments in the block and efficient generation of inclusion proof.

We build the tree in the following way:

node.length = left.length + right.lengh
node.hash = Hash(left.length, left.hash, right.lengh, right.hash)


The leaf of this tree corresponds to the transaction hash of the transaction or null hash.

Length of the leaf is the length of the corresponding segment in the transaction.

Note: Each transaction in the block can produce more than one leaf of this tree in case transaction contains more than one segment inside.

Similar structure is proposed here. Our approach differs, because we store only one tx hash or zero hash inside each leaf.

## Deposits, withdrawals, and fragmentation

Segment based model may have a problem of excessive fragmentation since withdrawing leave a hole between segments. To solve that plasma operator we can:

• Make deposits to the holes
• Take extra fee for the withdrawal that makes significant fragmentation

Note: Exit possible with part of UTXO (one of the segments)

## Exit game

### Force inclusion of the transaction

This idea is proposed in Plasma Cashflow spec. A transaction may be force included into a block by following game:

1. Somebody can present the unsigned transaction (corresponding TransactionContent structure) and the bond
2. Anybody can commit signatures (with auto refund the gas)
3. Anybody can challenge the transaction by presenting spend of inputs (including withdrawals) or non-inclusion proof for inputs
4. If the transaction is challenged, the challenger takes the bond. Else if all signatures collected, the transaction is included into a special block with the same exit priority as the oldest input of the transaction. In both cases (signatures collected or not) the transaction is considered to be removed from all other blocks

### Standard exit game

1. Somebody starts the exit process from segment [x_1, x_2] and attach his output to this process and set the bond.
2. (1) can be challenged by non-inclusion proof or spend (including withdrawals) immediately and get the bond
3. Anybody can start a special challenge request.
4. If the game is challenged by challenge requests, the bond is split to all

### Challange request

1. Anybody present output \in [x_1, x_2] older than the output of tx exiting
2. Anybody can challenge him presenting the spend. The spend must not be newer than output exiting in standard exit game.
3. Anybody can replace the challenge presented an older challenge
4. Challenger (1) can continue challenge game selecting utxo \in [x_1, x_2] from tx in remained challenge
5. Same as (2) or (3)
6. The first challenger in (2-3) gets the gas refund. The second challenger in (5) get the bond. Or challenge request is considered to be completed.

See the schema with better resolution.

### Merkle proof structure

We use two kinds of sum Merkle proof:

• Standard sum Merkle proof to proof mapping from segment to transaction
• Full sum Merkle proof to prove inclusion of all segments corresponding the transaction

We use standard Merkle proof inside the transaction:

1 bit for tx netto proof (must be zero for outputs, inputs, max_blockid and must be one for signatures)
5 bits for Merkle proof for outputs, inputs, and maxBlockId
3 bits for Merkle proof for signatures.


### Exit game methods


function withdrawal(Input point,
RSAInclusionProof proof) external payable returns (bool);

function withdrawalChallangeSpend(Input point,
Transaction tx, FullSumMerkleProof txProof, uint8 spendIndex,
RSAInclusionProof spendInclusionProof) external returns (bool);

function withdrawalChallangeBlock(
Input point, SumMerkleProof txProof,MerkleProof inputProof,
uint64 maxBlockIndex, MerkleProof maxBlockIndexProof
) external returns (bool);



## RSA proof of chain validity

Plasma owner can potentially fork RSA accumulator chain. Then the spend may be included into any subsegment of blocks, but not included into the segment to which the subsegment belongs.

Let’s put into each block header special prime number and the proof that this number is included into [A_{n-1}, A_{n}].

If the operator forks the chain, he does not present the proof, because we have no division and modulo operations in exponential rate in RSA.

# Bibliography

Plasma Prime spec?
Plasma World Map - the hitchhiker’s guide to the plasma
#2

I have a question.
Is this D's exit?

And in example, if txD and txC were withheld by operator, and the operator makes an invalid exit with segment B. Exit game work fine?

I think that the owner of B try to challenge invalid exit by txB, but operator can break challenge procedure by txC or txD. If owners of txC or txD are not operator, they wanna exit with txC, txD. Can they exit before operator’s invalid exit?

#3

Self-solved.
This challenge period is half of exit period, so users can re-challenge. right?

#4

For example, we have the following chain affecting the same coins:

A -> B -> C -> D -> E

С and D are withheld. D is a defect. The operator tries to withdraw E.

The operator may be challenged by the following way:
Challenger commits B and requests to prove chain continuity. Operator challenges him by withheld C. As we can see in Challenge request part, challenger have second try and commit output of C. The operator cannot challenge C by defect D and finally his exit E is rejected.

#5

Thank you for replying! I got it.

I have another question.
How do we deposit various token type?
If userA deposit 1.5 ETH and userB deposit 0.5 WBTC in next.
It will cause the problem like fragmentation?

#6

My expectation for this is that you split the Merkle tree in sub-trees, where each sub-tree is a coin. So if you have ETH and WBTC, you have the left subtree having coins which are only ETH, and the right subtree with WBTC. Go down N levels for N Coins.

This obviously reduces the amount of value per coin that your tree can hold, but with a deep-enough tree this shouldn’t matter. The disadvantage of this is that you need to know beforehand all types of coins that will be in the contract so that you split the tree accordingly.

Alternatively, you could have 1 tree for each coin, make a new merkle tree and from the merkle roots of each tree’s, and commit the new merkle tree’s root.

Coin A MT -> RootA
Coin B MT -> RootB

Committed merkle root -> MT of RootA and RootB

(Wrote this very fast, hope it makes sense)

#7

thanks! It makes sense for me.
Actually, I was thinking about change token type by each chunk because we wanna deposit multiple token types. But it will make worse fragmentation.
Yes, using merkle branch for token type is better, even the depth of merkle tree become little bit deep. I’ll thinking about this more.

#8

I’m sorry for asking many times

I have a question in another case.

A -> B -> C

In this case, the operator can include an invalid transaction between B and C.
If user withdraws C, the operator can challenge by invalid tx. If user withdraws B, the operator can challenge by C. And of course also operator can’t withdraw invalid tx. Is this right?

I think, in Plasma Cash, the user sends 2 transactions in exit, so operator can’t challenge by this invalid tx.

#9

# Plasma cashflow with S[NT]ARK exclusion proofs

Here is Plasma design proposal update from BANKEX Foundation team.

## Abstract

Plasma cashflow brings the idea of plasma slice space, the critical problem of that design is that the size of exclusion proofs is growing excessively during the plasma lifetime.

In that document, we describe plasma design that uses S[NT]ARK based exclusion proof to bypass the computation complexity of RSA accumulator.

## Slices

Similar to plasma cashflow this proposal is based on the UTXO model where each unspent output defines ownership of a specific slice. Length of the slice is equal to a value of the asset that was deposit to plasma.

## Block structure

We propose the following block structure:

Notes: TX Netto is actually transaction content without signatures. A transaction can include one or two signatures, two - in case of atomic swap and one in all case of a split and merge.

struct Block {
Transactions[] transactions
}

SumMerkleRoot sumMerkleRoot
}

struct Transaction {
TransactionContent content,
Signatures[] signatures
}

struct TransactionContent {
Input[] inputs,
Output[] outputs,
uint64 maxBlockIndex
}

struct Input
{
uint160 owner,
uint64 blockIndex,
uint32 txIndex,
uint8 outputIndex,
Slice amount
}

struct Output
{
uint160 owner,
Slice amount
}

struct Slice
{
uint256 begin
uint256 end
}



One transaction may have multiple txIndex values because it can be fragmented to different slices.

### SumMerkleRoot

We use SumMerkleTree to quickly verify that there’s no overlap between different transaction slices in the block and efficient generation of inclusion proof.

We build the tree in the following way:

node.length = left.length + right.lengh
node.hash = Hash(left.length, left.hash, right.lengh, right.hash)


The leaf of this tree corresponds to the transaction hash of the transaction or null hash.

Length of the leaf is the length of the corresponding slice in the transaction.

Note: Each transaction in the block can produce more than one leaf of this tree in case transaction contains more than one slice inside.

Similar structure is proposed here. Our approach differs because we store only one tx hash or zero hash inside each leaf.

### Merkle proof

We are planning to use Pedersen hash function. The computation of Merkle proofs onchain is expensive by the gas. That’s why we put all proofs into the zk-SNARK. SNARK computation is expensive too, but we can use the Truebit protocol to make it cheaper.

## Deposits, withdrawals, and fragmentation

The slice-based model may have a problem of excessive fragmentation since withdrawing leave a void between slices. To solve that plasma operator we can:

• Make deposits to the voids
• Split deposits to the several voids
• Take extra fee for the withdrawal that makes significant fragmentation

Note: Exit is possible with part of UTXO (one of the slices in the transaction)

## Exit game

### Priority increasing game (it is like @kfichter’s special exit). Not multisig case.

1. Alice publishes the transaction on mainnet plasma contract.
2. Bob can challenge the submission by presenting spend of inputs of the transaction with another hash
3. If the submission is unchallenged, we consider the transaction included into the block of youngest input of the transaction

#### Priority increasing game. Multisig case.

1. Alice publishes the unsigned transaction on mainnet plasma contract. Alice must be an owner of any of input or outputs of the transaction. If Alice is an owner of any of inputs, Alice must present the signature.
2. Bob can present remain signatures.
3. Anybody can challenge the submission by presenting spend of inputs of the transaction with another hash
4. If the submission is unchallenged, we consider the transaction included into the block of youngest input of the transaction. If the submission is unchallenged, but the signatures are not collected, we consider all transactions with the same hash from the plasma blockchain to be removed.

### Exit game

1. Exiter publish the exit output
2. Anybody can challenge the output for spend or non-inclusion
3. Anybody can present an output with an earlier priority. If this output is unchallenged by spend, exit (1) is challenged.