1. A method for verifying a digital signature, the method comprising: accessing a digital signature associated with an entity; accessing an implicit certificate issued by a certificate authority and associated with the entity, the implicit certificate includes a first elliptic curve point P representing a public key reconstruction value of the entity; accessing a second elliptic curve point C 1 representing a pre-computed multiple of a public key C of the certificate authority; and using the first elliptic curve point P and the second elliptic curve point C 1 to verify the digital signature by data processing apparatus.