Collision hiding 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 construction of collision-intractable families of hiding homomorphisms from iterated squaring and doubling is basically Construction 8.58, based on the strong claw-intractable family of permutation pairs from Construction 8.64. Similar to the case with bundling homomorphisms, the functions B are replaced by B ... 

Theorem 8.69 (Iterated squaring and doubling as hiding homomorphisms). On the factoring assumption, Construction 8.68 defines a collision-intractable family of hiding homomorphisms. ... 

The following table summarizes the most important parameters of the constructions of collision-intractable families of bundling homomorphisms, hiding homomor-phisms, and fixed-size hash functions based on the discrete-logarithm assumption. Note that the main use of fixed-size hash functions is in the constmction of real hash functions. 

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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