A system has λ≥2 servers. At least each of a set of authentication servers stores a key-share ski of secret key sk, shared between q of the λ servers, of a key-pair (pk, sk). An access control server sends an authentication value to a subset of the authentication servers. The authentication value was formed using a predetermined function of a first ciphertext for a user ID and a second ciphertext produced by encrypting a password attempt under public key pk using a homomorphic encryption algorithm. The authentication value decrypts to a predetermined value if the password attempt equals the user password for that user ID. Each authentication server in the subset produces a decryption share dependent on the authentication value using the key-share ski. The access control server uses decryption shares to determine if the authentication value decrypts to the predetermined value, if so permitting access to a resource.