Hiding homomorphism

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 ... 

Lemma 8.34. For families of hiding homomorphisms, the hiding property from Definition 8.31c can be replaced by the following shorter statement ... 

Construction 8.46. Let a family of groups of prime order be given. The corresponding family of pair exponentiations as hiding homomorphisms... 

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 ... 

Construction 8.68. Let a function tau N —> N and a pol)momial-time algorithm that computes tau in unary be given. The corresponding family of iterated squaring and doubling as hiding homomorphisms has the following components ... 

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. ... 

Proof. It has to be shown that Definitions 8.31a, b, d, 8.33a, and the simpler version of the hiding property that replaces Definition 8.31c for homomorphisms according to Lemma 8.34, are fulfilled. 

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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