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0.4 0.1 0.4 0.7 0.6 0.5 0.4 0.1 0.0

x2 2

Even with a capture date embedded in the metadata of the le, it can be confusing and images can be lost if you have several les with the same name being imported.

= Xi+3 @

0.16 0.01 0.16 0.49 0.36 0.25 0.16 0.01 1.60

numerical accuracy. A more ef cient computational formula for the sample variance is obtained as follows: a 1xi

x2 2 1

Since we know (Xi+3)24...31 and (Xi+2)24...31, frorn (3.27) we can determine (Xi+2)16.,.2:3 which gives us (Xi+2)16. . . 3 1. Combining this result with (3.26), we find (Xi+1)8...31, which, together with (3.25), allows us to determine all bits of X .We can now use this recovered X i and (3.24) to i firid putative X-lists, of the form X4, Xg, . . . , X,,, as discussed, above.8 We obtain one putative X-list for each putative Y-list, so that the expected number of X-lists is 238.

n n 1 (6-4)

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Note that Equation 6-4 requires squaring each individual xi, then squaring the sum of the xi, subtracting 1 g xi 2 2 n from g x2, and nally dividing by n 1. Sometimes this is called the i shortcut method for calculating s2 (or s). EXAMPLE 6-3 We will calculate the sample variance and standard deviation using the shortcut method, Equation 6-4. The formula gives

4. Finally, to determine the correct X-list from the collection of putative X-lists, we compare the values of (Xz)24 31 obtained from (3.24) with the values obtained from (3.23). We expect each such comparison to reduce the number of remaining lists by a factor of 256. Since we have about 238 X-lists, we need to make five such comparisons before we expect to obtain the correct X-list.

1353.6 10.2286 7

11042 2 8

Y-list and 2-list. Given any triple (X,,Y,,Z,) we can compute the keystream kJ for all 3 2 z and therefore decrypt the ciphertext to recover the (compressed) plaintext. The overall work factor for this att,ack is on the order of 238 (the number of lists generated). The work factor can be reduced by using more known plaintext, as discussed in [13] and Problem 29. We still need to precisely determine the minimum number of known plaintext bytes required, that is, the smallest value of n for which the attack will succeed. From (3.25), (3.26), and (3.27), we see that four consecut,ive (XL)24...31 needed to completely determine the X-lists. Also, five are additional Xi arc needed to find the correct X-list from the set of 238 lists. This means nine X-list elements must be available. However, each X-list is only determined for i = 4,5, . . . , ri, which implies n = 12 is the smallest value for which we obtain the required nine X-list elements. Since our indexing begins at 0, we need a minimum of 13 consecutive known plaintext bytes for the attack described here. Finally, we describe a slightly different implementation of this PKZIP attack, which is easier to program. The attack discussed above requires that we store all putative 2-lists and so on. This is essential if want to do the duplicate-reduction step, where the number of of 2-lists is reduced from 222 to some smaller number using additional known plaintext (see Problem 29). But if wc do not employ this reduction step, we can obtain an algorithm that is easier t,o implement. The idea is essentially to turn the breadth-first approach described above into a depth-first attack.

1.60 7

In fitting a multiple regression model, it is much more convenient to express the mathematical operations using matrix notation. Suppose that there are k regressor variables and n observations, (xi1, xi2, p , xik , yi), i 1, 2, p , n and that the model relating the regressors to the response is yi

This model is a system of n equations that can be expressed in matrix notation as y where y1 y2 o yn 1 1 o 1 x11 x21 o xn1 x12 x22 o xn2 p p p x1k x2k o xnk

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