The problem describes an application with two Cloud Run services (Service A - frontend, Service B - backend) and requires setting up IAM with the principle of least privilege. Service A calls Service B.

Principle of Least Privilege: This principle dictates that each entity (in this case, a Cloud Run service) should only have the minimum permissions necessary to perform its function.

Separate Service Accounts for Separate Services: To adhere to the principle of least privilege, it's best practice to assign a unique service account to each distinct service or component. This ensures that a compromise of one service account does not grant excessive permissions across other services. Service A needs permissions to run itself and to invoke Service B. Service B only needs permissions to run itself.Extract Reference: "Assign a service account to a Cloud Run service. The service account acts as the identity for your service and determines what permissions your revisions have when executing requests. It is a best practice to grant each service account only the permissions that are required to run the specific service (principle of least privilege)." (Google Cloud documentation: https://cloud.google.com/run/docs/configuring/service-accounts)

Authentication for Cloud Run Services: When one Cloud Run service (caller) needs to invoke another Cloud Run service (callee), the caller must be authorized to do so. This is typically achieved by assigning the roles/run.invoker role on the callee service to the caller's service account.Extract Reference: "To allow a service to invoke another service, grant the roles/run.invoker role on the called service to the caller's service account." (Google Cloud documentation: https://cloud.google.com/run/docs/securing/service-to-service)

Let's evaluate the options:

A. Create a new service account with the permissions to run service A and service B. Require authentication for service B. Permit only the new service account to call the backend. This violates the principle of least privilege by giving a single service account permissions for both services. If that service account were compromised, both services would be affected.

B. Create two separate service accounts. Grant one service account the permissions to execute service A, and grant the other service account the permissions to execute service B. Require authentication for service B. Permit only the service account for service A to call the back-end. This aligns perfectly with least privilege. Service A gets its own identity, Service B gets its own identity. Service A's service account is then granted run.invoker permissions on Service B, allowing it to call the backend while Service B requires authentication. This is the recommended approach.

C. Use the Compute Engine default service account to run service A and service B. Require authentication for service B. Permit only the default service account to call the backend. The Compute Engine default service account often has broad permissions (e.g., editor role in its project). Using it violates the principle of least privilege and is generally discouraged for production applications due to the potential for excessive permissions.

D. Create three separate service accounts. Grant one service account the permissions to execute service A. Grant the second service account the permissions to run service B. Grant the third service account the permissions to communicate between both services A and B. Require authentication for service B. Call the back-end by authenticating with a service account key for the third service account. This introduces unnecessary complexity with a third service account just for communication. More critically, using a service account key for authentication is generally discouraged in Cloud Run environments where ADC (Application Default Credentials) can be used, as managing keys securely becomes an operational overhead and security risk. Cloud Run services automatically use their attached service accounts for authentication when making calls to other Google Cloud services, including other Cloud Run services.

Therefore, option B is the best solution, adhering to the principle of least privilege and Google Cloud best practices for Cloud Run service-to-service authentication.