A sensor network has one or more base stations that talk to a large set of sensors, where the sensor life depends a small battery that consumes most power during communication. Before this network can be applied, many security problems must be solved, yet traditional security protocols usually need a lot of communication overhead. Recently, the μTesla protocol was proposed to address the authentication problem which ensures that no adversaries can impersonate the legal sender to control the sensors. The protocol adds a 24-bytes MAC (Message Authentication Code) to each 30-bytes message, where the MAC is based on the symmetric technique but can achieve the asymmetric property. The protocol is applied to a general case that a sensor can either compute the MAC to authenticate the message or just skip the MAC to see the message content. In this paper, we propose a C-μTesla protocol for a specific case where a sensor wants to authenticate all messages. We reduce the overhead from 30 bytes to 4 bytes,by asking the sender to encrypt all messages without creating any MACs, so that a sensor authenticate messages through decryption. And each message adds a 4-bytes Cyclic Redundancy Check (CRC) before encryption, so that a modified encrypted message will not pass the CRC after decryption. Moreover, the C-μTesla protocol introduces a new confidentiality property that allows no adversaries to hear information of the sender through putting a fabricated sensor.