Bundling property

DTBP also has been used to investigate the dimerization and actin bundling properties of vil-lin (George et al., 2007), the interaction of the Mrell complex with RPA (Olson et al., 2007), the study of gamma-secretase complex assembly (Spasic et al., 2007), and the multi-protein assembly of Kv4.2, KChIP3 and DPP10 (Jerng et al., 2005). 

George, S.P., Wang, Y., Mathew, S., Kamalakkannan, S., and Seema, K. (2007) Dimerization and actin-bundling properties of villin and its role in the assembly of epithelial cell brush borders. J. Biol. Chem. 10.1074/jbc.M703617200. 

Glenney, J. R., Jr., Kaulfus, P., Matsudaira, P., and Weber, K. (1981). F-actin binding and bundling properties of fimbrin, a major cytoskeletal protein of microvillus core filaments. / Biol. Chem. 256, 9283-9288. 

The small grey circles are individual preimage sets the bundling property guarantees that they are of a certain minimum size d, the bundling degree. The range need not be the whole codomain. 

The basic functions with a bundling property in the discrete-logarithm case are simply products of several exponentiations. This is called tuple exponentiation or, if the number of components is known to be jti,/x-tuple exponentiation. Pair exponentiation seems to have been first used like this in [B0CT88], larger tuples in [ChHP92]. 

Assume that a pair (fo,/i) of permutations on a common domain D is given. From this pair, functions B and B (for cr 6 N) with bundling properties will be constructed. ( B stands for bundling.) The construction is due to [G0MR88], but it was only used for collision-intractability there, i.e., no bundling property was shown. 

The property that makes all the functions with bundling properties, and also some other functions such as hash functions, interesting for cryptology is that it is infeasible to find collisions, i.e., two values with the same image see Figure 8.4 (and Figure 6.9). For precision, this notion and two related ones are given formal definitions. 

In abstract constructions, the domains Gg are written additively and the codomains Hg multiplicatively. This notation corresponds to the discrete-logarithm case, where hg is tuple exponentiation. Note that homomorphisms automatically have a bundling property if the domain is sufficiently larger than the codomain, as in Lemma 8.17. 

Theorem 8.57 (Iterated permutations as bundling functions). If a strong claw-intractable family of permutation pairs is given. Construction 8.56 defines a collision-intractable family of bundling functions. If the underlying family is weak, all properties except for the bundling property are still guaranteed. ... 

This statement will be used in the factoring case when n is not a generalized Blum integer In that case, the bundling property can be proved in a different way. 

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. 

The bundling property is shown even for the weak GMR family of permutation pairs. 

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