It seems you're gonna have to generate the scripts for the linked servers as you can see on How to Transfer Linked Servers from one server to another automatically?, therefore, if you follow the Restore the master Database doc before executing the generated scripts on the new instance, the linked servers recreated will be encrypted with the same service master key from the old server.
According to what the How to Create a Service Master Key in SQL Server article says,
The service master key is a symmetric key stored in the master
database. As such it can be found using the sys.symetric_keys catalog
view in the that very database.
So, I'd say the linked servers won't stop working since the service master key would be restored along with the
master database. To make sure that's the way it works you could easily test the procedure:
- Backup the master database on the old instance;
- Generate the linked server scripts;
- Restore the master database on the new instance;
- Execute the scripts for the linked servers on the new instance;
- Test if the linked servers are still working.
Also, it's recommended to Back Up the Service Master Key.