Best way to perform authentication between microservices?

I need to think about micro-service architecture at work.

We will need some security (only micro-services or "official micro-services consumers" can communicate).

What did you think about this architecture ?

Consumer or micro service need to get token for communicate with other micro-service.

Auth microservice with receive credentials (shared rsa private key ?) and generate a token, sending it to the consumer or micro-service.

Each micro-service will verify themselves validity of token before returning a response.

Am i wrong of it's good solution ? Have you better implementation ? (i'm really new on micro-service architecture)

All our micro service can send the same rsa private key to the auth micro service ?

Thanks!