Iterated squaring and doubling

Factoring Case Iterated Squaring and Doubling (Or A Useful Homomorphism on an Ugly Group)... 

Signs c denote inclusion arrows denote constructions. The diagram at the bottom commutes, i.e., iterated squaring and doubling is the special case of iterated permutations where the construction of claw-intractable permutation pairs on the factoring assumption is used. 

The construction of a collision-intractable family of bundling homomorphisms from iterated squaring and doubling is basically Construction 8.56, based on the claw-intractable families of permutation pairs from Construction 8.64. The new points are ... 

Construction 8.66. The family of iterated squaring and doubling as bundling homomorphisms has the following components ... 

Theorem 8.67 (Iterated squaring and doubling as bundling homomor-phisms). On the factoring assumption. Construction 8.66 defines a collision-intractable family of bundling homomorphisms. ... 

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 8.71 (Iterated squaring and doubling as hash functions). 

Figure 11.10. Effects of machine accuracy. Differences between solutions measured by N ax (open S3Tnbols) and Nmean (closed symbols) versus iteration. Squares differences between TCI and TC2 (double precision). Circles differences between TC8 and TC9 (single precision). Triangles differences between TCIO and TCll (quadruple precision)...

The first term in Eq. 6.29 is the absolute location component the second term has the form of the double difference described by Eq. 6.28, and the last term represents the distance from two planes (oriented vertically and horizontally) that describe the reference line of events. The procedure is iterative and the weighting factors (w , w , w ) control the influence of each term on the system of equations and are recalculated for each iteration. The arrival-time difFerences between all pairs of events are used to obtain hybrid locations by minimizing the weighted sum of squares of all arrival-time difFerences. Minimization is done using the standard Gauss-Newton method (Press et al. 1990). 

The RCMT, definitive or quick, is considered stable when (1) a minimum of seismograms of three stations azimuthaUy weU distributed (with an angular distance of about 120° each other) is available, (2) the focal mechanism remains stable during five iterations needed to determine the centroid location, (3) the total root mean square of the misfit between seismograms and synthetics averaged for all station used is lower than 0.4, (4) the difference between initial and final coordinates is lower than 0.3°, and (5) the moment tensor should have a small non-double-couple component. This last point is quantified using thresholds arbitrarily used to define a non-double-couple moment tensor (see entry on... 

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