In the cloud motif, the data processor is usually:

The party that assigns access rights is typically the data controller (cloud customer), not necessarily the data processor.

The cloud customer is typically the data controller, determining why and how data is processed, but generally not the one physically processing it in the cloud provider's infrastructure.

The cloud provider acts as the data processor because it performs operations (such as storage, computation, network routing) on personal data on behalf of the cloud customer. The customer, in turn, remains the data controller, determining the purposes and means of processing the data.

A Cloud Access Security Broker (CASB) is a security enforcement point between cloud consumers and cloud service providers, facilitating security policies, but not primarily the entity that processes the data itself.