Password based key exchange protocols on elliptic curves which conceal the public parameters

We here describe a new Password-based Authenticated Key Exchange (PAKE) protocol based on elliptic curve cryptography. We prove it secure in the Bellare-Pointcheval-Rogaway (BPR) model. A significant novelty in our work is that the elliptic curve public parameters remain private. This is important in the context of ID contactless devices as, in this case, there will exist most probably a way to link these parameters with the nationality of the ID document owners.