Is it possible to issue a token that can be sent only using a different protocol, say for example ring signature?


Yes, a token is simply a contract that manages user balances, so you can program it to use any system you can implement within the gas limit. However:

  • If you change the interfaces used to call it, for example by requiring different signature parameters, it will be incompatible with existing wallets that follow the ERC 20 standard.
  • Currently, the standard system of transactions with a per-account nonce and an ECC signature is the only way you can actually send a transaction, so you will have to wrap the signatures and parameters used in your scheme in a transaction using the traditional scheme. There is a proposal to change this under discussion, currently as EIP 208.

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