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In this form, bits 22 and 23 are known on the right-hand side. But these correspond to bits 30 and 31 of 2i on the left-hand side, which allows us to fill in these (previously) unknown bits on the left-hand side. In this way, for each putative 2-list, we can determine bits 30 and 31 of Zi, for i = 1 , 2 , . . . , n, and we can thereby complete each ZZ., except for 2 0 . Note that we cannot determine bits 30 and 31 of 20 using (3.17), since Z Z . -is required on the right-hand side. Also note ~ that the expected number of putative 2-lists is equal to the number of putative Z,, that is, about 222. The number of putative 2-lists can be reduced as follows. For all putative ZZ., can determine the corresponding we values, then sort the resulting lists based on Zz-1, removing any duplicates. The savings provided by this refinement are explored further in Problem 29. In [13] it is suggested to carry out this reduction for 28 steps, which, it is claimed, reduces the expected number of 2-lists from 222 to 218. While this reduces the work by a factor of 24, the price that is paid is that 28 additional known plaintext bytes must be available. For simplicity, in the remainder of our discussion of this attack, we ignore this duplicate-reduction step. Consequently, we expect to have 222 putative 2-lists.

where Y represents the tool life, x1 represents the cutting speed, x2 represents the tool angle, and is a random error term. This is a multiple linear regression model with two regressors. The term linear is used because Equation 12-1 is a linear function of the unknown parameters 0, 1, and 2. The regression model in Equation 12-1 describes a plane in the three-dimensional space of Y, x1, and x2. Figure 12-1(a) shows this plane for the regression model E1Y 2 50 10x1 7x2

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where we have assumed that the expected value of the error term is zero; that is E( ) 0. The parameter 0 is the intercept of the plane. We sometimes call 1 and 2 partial regression coefficients, because 1 measures the expected change in Y per unit change in x1 when x2 is held constant, and 2 measures the expected change in Y per unit change in x2 when x1 is held constant. Figure 12-1(b) shows a contour plot of the regression model that is, lines of constant E(Y ) as a function of x1 and x2. Notice that the contour lines in this plot are straight lines.

Sometimes a partition of the question into successive picks is an easier method to solve the problem. EXAMPLE 2-18 A day s production of 850 manufactured parts contains 50 parts that do not meet customer requirements. Two parts are selected randomly without replacement from the batch. What is the probability that the second part is defective given that the rst part is defective Let A denote the event that the rst part selected is defective, and let B denote the event that the second part selected is defective. The probability needed can be expressed as P1B A2. If the rst part is defective, prior to selecting the second part, the batch contains 849 parts, of which 49 are defective, therefore P1B A2 EXAMPLE 2-19 49 849

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2 . At this point we have about 222 putative 2-lists, each of which is of the form Z1,Z2,. . . ,Z,. Now we rewrite (3.16) as

Continuing the previous example, if three parts are selected at random, what is the probability that the rst two are defective and the third is not defective This event can be described in shorthand notation as simply P(ddn). We have P1ddn2 50 850 49 849 800 848 0.0032

The third term is obtained as follows. After the rst two parts are selected, there are 848 remaining. Of the remaining parts, 800 are not defective. In this example, it is easy to obtain the solution with a conditional probability for each selection. EXERCISES FOR SECTION 2-4

@ CRCinverse[(Z,)o ...7],

2-57. Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: shock resistance high low 70 9 16 5 Let A denote the event that a sample has excellent surface nish, and let B denote the event that a sample has excellent length. Determine: (a) P1A2 (b) P1B2 (c) P1A B2 (d) P1B A2 (e) If the selected part has excellent surface nish, what is the probability that the length is excellent (f) If the selected part has good length, what is the probability that the surface nish is excellent 2-59. The analysis of shafts for a compressor is summarized by conformance to speci cations: roundness conforms yes no 345 5 12 8

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