Collision bundling functions

Definition 8.29. A collision-intractable family of bundling functions... 

Definition S.30. A coliision-intractabie family of bundling homomorphisms is a collision-intractable family of bundling functions with the following additional properties and components ... 

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. ... 

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. 

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. 

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 ... 

Movement is an essential function in cellular system. Chemical interactions of proteins are microscopic movements of proteins, namely, diffusion, transport and the collision among the proteins. Vesicles containing proteins move to specific sites within cell for their destined functions. Chromosomes move to separate from each other during cell division. Cytoskeleton dynamically forms bundles and networks to afford routes for the intracellular trafficking and to control the physical architecture of the cell. Cells themselves move by migration. Within multicellular structure, cell movement drives the morphogenesis and preserves the homeostasis. Measurement of these various movements provides quantitative information that is inevitable for understanding the cellular system. 

---