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It's pretty clear how signing a message with a Bitcoin key can be useful in proving ownership of funds.

However, it's far from clear how encrypting a message with a Bitcoin key can do anything useful. This feature is available from Tools/Encrypt/decrypt message, and appears to use ECIES or some variant of it:

https://en.wikipedia.org/wiki/Integrated_Encryption_Scheme

It seems that at least one developer is doubtful about the security of such schemes:

https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2014-August/006581.html

Security aside, what practical reason would anyone have to encrypt a message with a Bitcoin key?

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