Hiding function

The definition is quite similar to that of bundling functions, except that one must decide what can be hidden by the hiding function. The following definition is very general in this respect There is an arbitrary family of domains Sg, i.e., what can be hidden may depend on the key. 

Furthermore, note that not all information-theoretically hiding string commitments have to be constructed from hiding functions in particular, there may be more interaction. 

Figure 8.8. Notation with families of hiding functions...

Definition 8.31. A collision-intractable family of hiding functions has... 

K KeAll called the sets of secrets (their elements are the objects that are hidden by the hiding function),... 

Similar to bundling homomorphisms, hiding homomorphisms are basically hiding functions that are homomorphisms. 

Definition 8.33. A collision-intractable family of hiding homomorphisms is a collision-intractable family of hiding functions with the following additional properties and components ... 

The basic idea is from [G0MR88] it was first applied to hash functions in [Damg88] and to bundling and hiding functions in [Pfit89, PfWa90]. 

Basically, the constructions are special cases of those described in the previous section, based on the permutation pairs from Definition 8.10. (The origin of the constructions is the same as in the previous section, too The basic idea is from [G0MR88] it was first applied to hash functions in [Damg88] and to bundling and hiding functions in [Pfit89, PfWa90].)... 

Proof. The construction is a collision-intractable family of hiding functions according to Theorem 8.59 and Lemma 8.65. (The replacement of B by B can be handled as in the proof of Theorem 8.67.) The functions Hr are homomorphisms between groups Gr and Hj according to Theorem 8.16, and is obviously an Abelian group, too. It remains to be shown that Kr is a homomorphism. This is not completely trivial, although TCg is simply a projection, because Gg as a group is not the direct product of Z2T and RQR , but one can immediately see it from the definition of the operation ... 

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