The authentication of GNSS navigation signals is an active research area in recent years due to the concerns of interference and spoofing. Galileo, the European GNSS, is implementing the Open Service Navigation Message Authentication (OSNMA) service to ensure the integrity and authenticity of the GNSS navigation message. The OSNMA service is based on the Timed Efficient Stream Loss-tolerant Authentication (TESLA) protocol, which consists of two main parts: the transmission of Message Authentication Code (MAC), and the delay transmission of the TESLA key. The main OSNMA functionalities include public key verification through Merkle tree, digital signature verification by Elliptic Curve Digital Signature Algorithm (ECDSA), TESLA key verification using a hash function, and MAC verification by HMAC-SHA-256. Such authentication service could mitigate the risks under spoofing and jamming attacks, thereby providing more reliable and secure navigation services. The paper describes the development of an algorithm to receive and decrypt Galileo OSNMA messages and verification of the algorithm by collecting and processing Galileo signals. A simulation of the OSNMA data is developed based on specification of the Galileo I/NAV messages. Test vectors are generated to simulate and verify the cryptographic functions. In addition, actual Galileo signals has been collected and data messages decoded for the verification of TESLA key and MAC. The paper will present the development and experiment results with emphasis on the assessment of the additional resource to accomplish the cryptographic functions. In practice, the delay in the authentication is an issue if the GNSS receiver wants to authenticate the navigation message in real time. One way to deal with this issue is to employ navigation messages from other satellites in the reception of whole Digital Signature Message (DSM). A scheme is then devised to realize this operation and the benefits/impacts are analyzed. It is anticipated that the algorithm under study will lead to an effective implementation in the processing of the Galileo OSNMA data messages.