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When computing a modular exponentiation ud (mod N ) using the CRT approach, we first compute ad, (mod p ) using the repeated squaring algorithm in Table (7.7), where d, = d (mod ( p - 1)). Suppose that when computing u d p (mod p ) , we have ko multiply steps and kl squaring steps. Note that ko and kl depend only on d, and, therefore, only on d and p , and not on a. Since the probability (7.21) holds for each multiply, and the probability (7.22) holds for each square, the expected number of extra reductions is a (mod p ) P (7.23) +k1-. IC0 2R 3R As a function of a , the expression in (7.23) is piecewise linear-more precisely, it is a linear function with discontinuities a t integer multiples of p . Qualitatively, the graph of (7.23) is similar to that in Figure 7.2 (see Problem 2). Note that the total number of extra reductions in the calculation of Cd (mod N) also include extra reductions due to the factor q. Nevertheless, there would still be a discontinuity in the total number of extra reductions at every integer multiple of p (and also q ) .

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To estimate the importance of fluctuations below the critical temperature, we can compare the fluctuations of m(x) over a distance ~ with its mean value m. For the fluctuations to be small we must have

If you selected an Intuit invoice template, you ll get a warning box after clicking Edit in Step 2. The warning box explains that your editing options are limited. Click OK to bypass it.

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Figure 7.2: Expected number of extra reductions. The idea behind Schindler s timing attack follows directly from the graph in Figure 7.2. Suppose we select ciphertexts CO and C1, with Co < C1. Let T(Co)and T(C1)be the timing measurements for the decryption of Co and C1, respectively. Assuming that timing differences are dominated by the number of extra reductions, we would generally expect T(C1)- T(C0)to be relatively small, since the number of extra reductions grows slowly (linearly) from Co to C1. However, suppose that Co and C1 bracket a multiple of p . For example, suppose that we have Co = 2 p - k and C = 2p l , where k and are 1 reasonably small. Then due to the discontinuity in the number of extra reductions at 2p, the expected difference T(Co)- T(C1)would be relativcly large. Therefore, we can select an initial value z and an offset A and let

(-ro/4u o)

= ( - ro) -1 /2 ,

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The second method is to choose a template to edit from the Templates list. You can make limited modi cations to an existing template. However, if you want to modify more than fonts, and certain limited elds, you must create a Custom Template. (Refer to 8 for more information about lists.) To edit a template from the Templates list, follow these steps: 1. Choose Lists Templates to open the list of templates.

+ PA,


u o[aoltl]<d-4)/2 1

This is known as the "Ginzburg criterion." It can be fulfilled in the limit t ~ 0 only if d> 4. Another way to derive the Ginzburg criterion is to note that there are two length scales in the theory, defined by ro and u o. They may be deduced through dimensional analysis, as follows. The dimension of tf is (Length)-d, since it is the volume integral of the dimensionless energy E[m]:

= 0,1,2,

(17 .94)

If the template you want to customize does not appear, you can t use this method. Instead, you must either use the steps in the preceding method or create a new template of that type (as described in the next section, Creating a new template ).

The dimensionality of m(x) can be deduced from the fact that l'\7mI , which is of dimension [m] 21L 2 , must have the same dimension as tf. The dimensions of ro and U o are then easily deduced. The results are [m(x)] = L(2-d)/2

e = O,1,2,. ..

= L -2


using the chosen ciphertexts Cp. Eventually, we will have C < k p < C!+l for p l, and when this occiirs we should detect a significant increase in T(Cp) -T(Cp+l). Once we have bracketed k p in this manner, we can simply compute gcd(n, N ) for every

2. Highlight the template you want to edit. 3. Click the Templates menu button to open its drop-down list and choose Edit (see Figure 15-4). The template s Customize window appears and you can start to edit text and change some limited options, such as whether the company name and address appear on the invoice.

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