List-shaped tree

The abbreviated names of the constructions mean bottom-up tree authentication (10.9), top-down tree authentication (10.13), top-down tree authentication with a small amount of private storage (10.19), the discrete-logarithm scheme with minimized secret key (10.22) without combination with tree authentication, and the construction with a list-shaped tree for a fixed recipient from Section 10.6. The first column of lower bounds is for standard fail-stop signature schemes (Sections 11.3 and 11.4), the second one for standard information-theoretically secure signature schemes (Section 11.5) here the length of a test key has been entered in the row with the public keys. 

Such a task description invites task analysis, which would lead naturally to human reliability analysis (HRA). Indeed, perhaps the earliest work in this field applied HRA techniques to construct fault trees for aircraft structural inspection (Lock and Strutt 1985). The HRA tradition lists task steps, such as expanded versions of the generic functions above, lists possible errors for each step, then compiles performance shaping factors for each error. Such an approach was tried early in the FAA s human factors initiative (Drury et al. 1990) but was ultimately seen as difficult to use because of the sheer number of possible errors and PSFs. It is occasionally revised, such as in the current FRANCIE project (Haney 1999), using a much expanded framework that incorporates inspection as one of a number of possible maintenance tasks. Other attempts have been made to apply some of the richer human error models (e.g.. Reason 1990 Hollnagel 1997 Rouse 1985) to inspection activities (La-toreUa and Drury 1992 Prabhu and Drury 1992 Latorella and Prabhu 2000) to inspection tasks. These have given a broader understanding of the possible errors but have not helped better define the PoD curve needed to ensure continuing airworthiness of the civil air fleet. 

The square lattice is only one of a myriad possible representations of space. Every lattice has an associated coordination number, z, which describes the number of bonds emanating from each site for example, the square lattice in Figure 4 has a coordination number of 4. In addition there are lattices which have no obvious dimensionality, like the Bethe lattice. The Bethe lattice is a homogeneous tree structure, the number of sites on the surface of the tree increases without bound as the size of the tree grows. The coordination number of the Bethe lattice can be from 2 to oo. There are also lattice representations that are irregular each site does not have the same characteristic shape. Voronoi lattices, both two- and three-dimensional, are constructed by placing points randomly in space and tessellating around these points to construct an internal surface [39, 40]. Some relevant properties of each lattice-dimensionality D, coordination number z, critical probability p for site and bond percolation--are listed in Table 1. 

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