  In this paper an enhanced fuzzy vault scheme is proposed which we refer to as
fuzzy-fuzzy vault scheme. The proposed scheme builds on the classical fuzzy
vault by adding the concept of uncertainty and imprecision to the classical
scheme. To lock a secret key K in the classical fuzzy vault the locking and
unlocking elements are crisp or real elements and consequently the locking and
unlocking operations are strict imperative. In the fuzzy-fuzzy vault scheme,
Alice locks the secret key K using a set of fuzzy elements that belong to
multi-fuzzy set A~ obtained from a universe public set of fuzzy elements in a
multi-fuzzy set F~_q and projecting them on polynomial p. The elements in
multi-fuzzy sets F~_q and A~ are fuzzy using m membership functions MF_i,
i=1,2,...,m. Alice selects a set k of fuzzy elements fuzzy with a specific
membership function MF_K from A~ to lock the vault. To hide the genuine locking
points Alice generates a set of fuzzy chaff points that some of them do not lie
on polynomial p while the other fuzzy chaff points may lie on polynomial p but
fuzzy with different membership functions other than the membership function
MF_K used to lock the vault. To unlock the fuzzy-fuzzy vault and retrieve the
secret key K , Bob should have a set of unlocking fuzzy elements belonging to
multi-fuzzy set B~ which substantially overlap with A~ is required. Then Bob
selects t'_(TF_ki) fuzzy elements from B~ which are close to the t_(TF_k) fuzzy
elements from A~ used by Alice to lock the vault. We show that adding
uncertainty and imprecision by introducing fuzzy theory will enhance the
security threshold of the fuzzy vault.
