I know that RSACryptoServiceProvider can encrypt with the public key, then it can be decrypted with the private key.
Is it possible to encrypt with the private key and decrypt with the public key using the RSACryptoServiceProvider ?
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No. That’s not how any public/private key encryption works. You can only encrypt with the public key, and only decrypt with the private key.
If you want to apply the private key to a message, maybe you’re looking for a signature, rather than encryption? This is a different cryptographic scheme that can also use RSA keys.