To make the answer more complete, we can mention the creation of non-self-signed certificates with the ecdsa-sha384 signature, because this is slightly different. The trick is to use the "-md sha384" option with the "openssl ca" command.
Example Including Client Certificate
Create a key for a certification authority:
openssl ecparam -out ca.key -name secp384r1 -genkey
Create your own signed certificate for ca:
openssl req -x509 -new -key ca.key -out ca-ca.pem -outform pem -sha384
Create a key for the client:
openssl ecparam -out host1.key -name secp384r1 -genkey
Create a certificate request for a client key:
openssl req -new -nodes -key host1.key -outform pem -out host1.req -sha384
Create a certificate for the client in response to the request:
openssl ca -keyfile ca.key -cert ca-ca.pem -in host1.req -out ca-host1-cert.pem -md sha384 -outdir .
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