A process in a system comprising a server which is in communication with a client computer over a computer network, and a user's mobile device which is in communication with the client computer for conducting communications over the computer network, the process comprising using said client computer to perform the following process actions: facilitating an authentication by the server of the user who is attempting to access a network site associated with the server, wherein said facilitation comprises: receiving a representation of a secret value and a challenge number generated by the mobile device from the mobile device, wherein the representation is computed using a prescribed cryptographic hash function, wherein the mobile device generates the representation of the secret value whenever needed, and wherein the server knows the secret value; combining the representation of the secret value and a password which is entered into the client computer by the user and known to the server, to produce a combined representation; transmitting the challenge number generated by the mobile device, a user identification, and the combined representation to the server for use in determining if the password and secret value known to the server were used to generate the combined representation; receiving a notice from the server as to whether access to said network site has been granted or not, wherein access is granted only if the password and secret value known to the server were used to generate the combined representation; and informing the user as to whether access to said network site has been granted or not. facilitating an authentication by the server of the user who is attempting to access a network site associated with the server, wherein said facilitation comprises: receiving a representation of a secret value and a challenge number generated by the mobile device from the mobile device, wherein the representation is computed using a prescribed cryptographic hash function, wherein the mobile device generates the representation of the secret value whenever needed, and wherein the server knows the secret value; combining the representation of the secret value and a password which is entered into the client computer by the user and known to the server, to produce a combined representation; transmitting the challenge number generated by the mobile device, a user identification, and the combined representation to the server for use in determining if the password and secret value known to the server were used to generate the combined representation; receiving a notice from the server as to whether access to said network site has been granted or not, wherein access is granted only if the password and secret value known to the server were used to generate the combined representation; and informing the user as to whether access to said network site has been granted or not. receiving a representation of a secret value and a challenge number generated by the mobile device from the mobile device, wherein the representation is computed using a prescribed cryptographic hash function, wherein the mobile device generates the representation of the secret value whenever needed, and wherein the server knows the secret value; combining the representation of the secret value and a password which is entered into the client computer by the user and known to the server, to produce a combined representation; transmitting the challenge number generated by the mobile device, a user identification, and the combined representation to the server for use in determining if the password and secret value known to the server were used to generate the combined representation; receiving a notice from the server as to whether access to said network site has been granted or not, wherein access is granted only if the password and secret value known to the server were used to generate the combined representation; and informing the user as to whether access to said network site has been granted or not.