Can Json web token sign with public key?

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I saw that JWT can sign a JSON object with private key, and verified by a public key.

I am thinking that would it be possible to use JWT to sign the API request payload using JWT. However, instead of using the private key, I want to use the public key to sign the API request payload and verify it on server side with my private key. Would it be possible? As I see the library implemented JWT seems only sign with private key and verify by public key.

Isn't it impossible to sign with public key?

Thanks

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