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2 Let (Xi,y Z , Zi) denote the 32-bit words (X, Y, ) used to generate the ith keystream byte and let ki be the ith keystream byte, for i = 0 , 1 , 2 , . . .. We are now in a position to discuss the Biham-Kocher attack.

In logistic regression we assume that E(Y) is related to x by the logit function. It is easy to show that E1Y 2 E1Y 2 exp

0 1x

We first summarize the PKZIP attack, then we provide details for each of the points in the summary. We call ko, k l , k2, . . . , k , a k-list. Define plist, X list, Y-list, and Z-list similarly. The attack assumes known plaintext, which implies that the k-list and plist are both known.

(S11-9)

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The quantity exp( 0 1x) on the right-hand side of Equation S11-9 is called the odds ratio. It has a straightforward interpretation: If the odds ratio is 2 for a particular value of x, it means that a success is twice as likely as a failure at that value of the regressor x. Notice that the natural logarithm of the odds ratio is a linear function of the regressor variable. Therefore the slope 1 is the change in the log odds that results from a one-unit increase in x. This means that the odds ratio changes by e 1 when x increases by one unit. The parameters in this logistic regression model are usually estimated by the method of maximum likelihood. For details of the procedure, see Montgomery, Peck, and Vining (2001). Minitab will t logistic regression models and provide useful information on the quality of the t.

1.2 1 0.8 E( Y ) 0.6 0.4 0.2 0

If we can recover any valid triple ( X i , X , & ) , then the internal state of the keystream generator is known and we can determine the keystream k j for all j 2 i . This attack will enable us t o find such a triple (Xi,y ,2i) with a nontrivial, but, feasible, amount of work. In outline form, the attack c0nsist.s of t,he following steps: 1. Use the k-list to find a set of putative 2-lists.

1.2 1 0.8 E( Y) 0.6 0.4 0.2 0

Before printing your images, you must select which images and how many of each to print by choosing Select Images & Quantity In this option, you scroll through your images and add them to the print queue by pressing the Function/Set button when they are displayed Once the image has been selected, use the up and down areas of the 4-Way Controller to choose how many prints from each image you want to print Select Range Selects a range of consecutive images to print Use the Control dial to select the rst image in the range, then press the Function/Set button Scroll through the images using the Control dial and select the last image in the range by pressing the Function/Set button when that image displays Select All Images Choosing this selects all images on the memory card for printing.

6 x (a)

4. The true X-list must be among the putative X-lists. By using the p-list, we can determine the correct X-list, and once we find the correct X-list, we know thc corresponding Y-list and 2-list, so we can determine the keystream.

6 x (b)

Figure S11-7

A total of n 1 known plaintext bytes are r e q ~ i r e d . ~ number the known We plaintext bytes from 0 t o n. These plaintext bytes must be consecutive, but need not be the first n 1 bytes. Then each list (k-list, X-list, etc.) contains n 1 elements, which are numbered from 0 t o n,even though these might not represent the first n 1 el~rnents generated. Eventually, we will show that 13 consecutive known plaintexts is sufficient (that is, rb = l a ) , but for now we leave n unspecified. Next,, we expand on each of the points in thc attack outlined above. For this attack, we assurne that we have available the p l i s t , that is, we have p,, for z = 0 , 1 , 2 . . . . ,n, which implies that we also know the corresponding k-list. We number the steps in the attack below t o correspond with the numbers in the outline of the attack, above.

1 11

6.0 1.0x

1 11

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