# Uniqueness

The PUF uniqueness property is formulated as:

The definition could be interpreted in an ambiguous way, since it seems to be overlapped, or at least very similar, to unclonability statements (Eqs. 10.2a and 10.2b). However this is not true, because the requirement to be unique for a *в* is the nonexistence of another *в'* that specifically belongs to *&*, unlike the unclonability that requires the nonexistence of a generic function that is able to substitute *в*. Moreover, the uniqueness requires to be valid only for a subset of *C*.

Apart from the theoretical meaning, the uniqueness has a practical usefulness. Indeed, let *c e C* be a random picked value and *r = в (c*). The pair *(c, в (c))* induces a partition into *&* such that *&* U *& = &* and *в (c) = r, Ув* e *&* and consequently *в (c) = r, VO e &* .A successive picking of c e *C* defines another pair (*c, в (c)) *which causes a partition in *&*, and so on. By nesting this approach, so successive applications of other *c* values on в, the picked CRPs set *Ф = {(c, в* (c))} will cause increasingly smaller partitions, to have a singleton set in which only one function, в, is characterized by such CRPs. The success of this procedure and the required steps to complete it depends on the *&* cardinality, the characteristics of the PUF *в* and on the picked *c*.