Integer Factoring

The first assumption is that factoring large integers is infeasible. Only Williams integers will be needed, i.e., those with exactly two prime factors p and q where p = 3 mod 8 and q = 7 mod 8. More precisely, the following assumption from [G0MR88] is used. 

As far as number-theoretic properties are concerned, one could permit generalized Williams integers, similar to generalized Blum integers, of the form p qK However, in a factoring assumption, large exponents s and t would mean small prime factors if n is always of approximately the same length, and numbers with small prime factors are easier to factor. 

The best general-purpose algorithms used in practice until recently have expected miming times of L [ c] with small constants c (between 1 and depending on whether one only considers algorithms whose running time has been proved). As time complexity is usually measured in terms of the length of the input, not of the input n itself, this is superpolynomial, but not strictly exponential because of the square root in the exponent. The number field sieve only has a third root in the exponent, i.e., c] with c around 2. 

In practice, general numbers, i.e., numbers without known special number-theoretic properties, have been factored up to about 130 decimal digits, i.e., 430 bits [AGLL95]. The number field sieve has reached almost 120 digits [DoLe95] and is expected to surpass the other algorithms soon. 

If one wants a deterministically polynomial-time algorithm that only outputs prime numbers, and where the corresponding factoring assumption follows from that made above, one has to rely on Cramer s conjecture (see above) and search for each prime from some random number upwards in steps of two, and test each number with the pure Miller primality test [Mill76], which relies on the extended Riemann hypothesis. 

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Figure 1.2. A number molecular mass distribution N (M) of an ideal chain polymer. N (M) is defined for integer multiples of Mm, the monomer mass. The integer factor, P, is called the degree of polymerization... 

To summarize, the classical method for analyzing sample rate conversion is to assume that the sample rate conversion factor is or can be approximated by a rational number. Then the sample rate conversion can be viewed as a three stage process, up-sample by an integer factor L, filter by h[n, and down-sample by M. Up-sampling by L inserts L — 1 zero valued samples in between the existing samples of x and decimating by M retains only each M-th sample. This approach is an analytical tool rather than an actual implementation strategy, and it allows the comparison of different sample rate conversion methods in a similar framework. 

When a point (or an atom) is placed on a finite symmetry element that converts the point into itself, the multiplicity of the site is reduced by an integer factor when compared to the multiplicity of the general site. Since different finite S5mimetry elements may be present in the same space group symmetry, the total number of different "non-general" sites (they are called special sites or special equivalent positions) may exceed one. Contrary to a general equivalent position, one, two or all three coordinates will be constrained in every atom occupying a special equivalent position. 

Security discussions about RSA initially focused on two questions already raised in [RiSA78] How hard is integer factoring And are there other ways of inverting RSA than by factoring, or can one prove that these two problems are equally hard None of these questions has found a final answer yet. For the first one, see Section 8.4.1, and for an overview of contributions to the second one, in particular the superencryption question, see, e.g., [DaPr89]. 

For each product of sets sizes sP calculate all its integer factors f, such that s. [Pg.199]

Nuclei can also possess spin. Whereas every electron has a spin of half, nuclei can have nuclear spin angular momentum quantum number, I, equal to zero or other integer factors of1/2. 

When two values differ by an integer factor (as in our case) this should stimulate our mind, because it may mean something fundamental that might depend on, e.g., the number of dimensions of our space or something similar. However, one of the most precise experiments ever made by humans gave 2.0023193043737 0.0000000000082 instead of 2. Therefore, our excitement must diminish. A more accurate theory (quantum electrodynamics, some of the effects of this will be described later) gave a result that agreed with the experiment within an experimental error of 0.0000000008. The extreme accuracy achieved witnessed the exceptional status of quantum electrodynamics, because no other theory of mankind has achieved such a level of accuracy. 

Figure 1.4.4(b) shows frequency spectra for corresponding composition profiles (Fig. 1.4.4(c)) at various percent completion values. Frequency spectra show a dominant peak at a wave number index value of around 8.0. Subharmonic peaks are also evident at wave number index values that integer factors of the wave number of the main harmonic thus, there are subharmonics at wave number indices of 16, 24, etc. In Fig. 1.4.4(c), composition profiles are similar to the early-stage behavior for... 

Exp>eriments 1 and 3 indicate that increasing [NO] from 10.0 X 10 M to 14.0 X 10 M at constant [H2] increases the rate from 5.0 X 10 M s to 9.8 X 10 M s. Because the concentration is not increased by a simple integer factor, determining the pwwer x is more complex and must be determined using logarithms. Using the rate law (with y = 1), the ratio of the rate in experiment 3 to that of experiment 1 is given by... 

Consider the process of integer factoring, which impacts modem computer security. It is well appreciated that an integer N is either prime or composite. If prime, N... 

The oxidation half-reaction involves one electron, and the reduction half-reaction involves five electrons. Now we balance the number of electrons transferred in each half-reaction by multiplying each half-reaction by an integer factor. Then we add the two half-reactions and... 

If no elementary reaction predominates for the rate of the overall conversion, the stoichiometry of each elementary reaction is multiplied by an integer factor v, so that the sum of these varied elementary stoichiometries corresponds to the overall stoichiometry, i.e. the intermediate products in the sum are eliminated. For the thermodynamic equilibrium constant it follows that... 

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