ok main thing is, is it ok to use header ES256 for jwt created using secp256k1 keys?

https://datatracker.ietf.org/doc/html/rfc8812 says, secp256k1 curve should only be used with ES256k header, but in authfusion even if we give k1 pair keys and then use sign and encode a JWT using EC, it will come as ES256 only, is that okay?

Another doubt is, those jwt (k1 curve keys + ES256) created in authfusion is only able to verify in jose4j with .setRelaxVerificationKeyValidation() //needed if the key is smaller than 256 bits.

Without it we get the error:

JWT processing failed. Additional details: [[17] Unable to process JOSE object (cause: org.jose4j.lang.InvalidKeyException: ES256/SHA256withECDSA expects a key using P-256 but was null):

Description inside setRelaxVerificationKeyValidation :

Bypass the strict checks on the verification key. This might be needed, for example, if the JWT issuer is using 1024-bit RSA keys or HMAC secrets that are too small (smaller than the size of the hash output)

Is it the correct way to validate jwt created using ec in authfusion?

No, FusionAuth doesn't support adding JWT headers to FusionAuth generated JWTs. I looked at the code and don't think it'd be a ton of work to add support; there's already some scaffolding in the fusionauth-jwt OSS project.

I highly encourage anyone with this problem to file a feature request here with more details about your needs: https://github.com/fusionauth/fusionauth-issues/issues

We consult that in our roadmap planning. We also offer professional services if you need us to build it on a schedule. Please send a request to our sales department if that is an option you'd like to pursue.

An alternative would be to build a service that would re-sign your JWTs from FusionAuth with the needed header changes. Not optimal, I understand, but another avenue that might get you what you need.