Veil Wallet pool example
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testinprd.base.eth has a balance in the veil wallet pool. 

They need to pay deploy.base.eth 0.01 ETH. 

deploy.base.eth then chooses to withdraw 0.005 out of the pool into his own wallet.

Private transfers
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0.01ETH from user A to B

The recipient, amount and even the sender is private. 

On the blockchain all you see is that there was an update to the pool contract by the Veil relayer.

The inputs + outputs are encrypted before being pushed into the smart contract.

Arbitrary Amount Withdrawal
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User B withdraws some of the funds

deploy.base.eth wants to take out half the funds, they submit a withdrawal request. 

Remaining balance in the pool is private.