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Let's say that I want to send the hash of a key to an entity and I want the entity to be sure that the hash could have come only from me.

It is possible to create a message with the hash of the key and then send it signed to the other entity?

Will this cost any gas?

Will a transaction be generated and mined in a block in the blockchain?

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You can sign whatever you want off-chain with your private key and deliver it to someone else however you want (e.g. HTTP or email). This is how "payment channels" work without incurring transaction fees.

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