Constructions in the Factoring Case

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].) 

Hence this section starts with the construction of claw-intractable families of permutation pairs from these functions. In addition to the results that are then a consequence of Section 8.5.4, most function families in this section are families of homomorphisms, and one can use the results of Section 8.2.3 to show a bundling property even if only a weak claw-intractable family of permutation pairs is given. 

Squaring and Doubling as Claw-Intractable Permutation Pairs 

The constructions in Section 8.2.3 had different properties depending on whether n was a generalized Blum integer or any element of 4N -1- 1. Thus two different families of good keys. Good and Good weak are provided they lead to a strong and a weak claw-intractable family of permutation pairs, respectively. 

Construction 8.64. The strong and weak GMR family of permutation pairs are defined as follows  

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