Is it permissible to use OP_CHECKMULTISIG with 0 signatures or 0 keys? I.e. are 0-of-X or 0-of-0 outputs allowed per the protocol? Are they standard?

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The minimum is 0. 0-of-0 multisigs are consensus valid, as are 0-of-n. Bare multisigs must have at least one key and signature. However it can be wrapped in a P2SH or P2WSH and will still be standard.

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