An authentication method authenticates between subscribers of a communications system using an asymmetric elliptic curve encryption algorithm. The method involves providing a first and at least one second subscriber having a first or second secret key known only to the respective subscriber and a public key; authenticating an inquiry transmitted by the first subscriber with respect to the validity of the first certificate contained therein and associated with the first subscriber; calculating the response of the second subscriber associated with the inquiry; randomized encryption of the calculated response and a second certificate associated with the second subscriber using the public key; decryption and authentication of the response transmitted by the second subscriber with respect to the validity of the second certificate contained therein.