An authentication system of a terminal on a public switched telephone network provides a security node associated with a local exchange and a network terminal. For one-way authentication, the terminal responds to a call initiation by sending a unique authentication code comprising a number and a secret key encrypted according to a first algorithm, the secret key being specific to the terminal. The security node constructs the expected authentication code from the number, using the first algorithm and a second key which is a function of a terminal identification number, and compares the expected code with the received code. In two-way authentication, the security node responds to the call initiation by sending a transaction number to the terminal encrypted according to a second algorithm. The terminal generates the authentication code as a function of the first algorithm, the secret key and the transaction number. The authentication code is sent back to the security node. An expected code is compared with the received one in the same way. In both cases, a match between expected and received authentication codes constitutes authentication of the terminal allowing the user access to the network.