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variability in the observations Y on oxygen purity. Thus, when 2 is small, the observed values of Y will fall close to the line, and when 2 is large, the observed values of Y may deviate considerably from the line. Because 2 is constant, the variability in Y at any value of x is the same. The regression model describes the relationship between oxygen purity Y and hydrocarbon level x. Thus, for any value of hydrocarbon level, oxygen purity has a normal distribution with mean 75 15x and variance 2. For example, if x 1.25, Y has mean value Y x 75 15(1.25) 93.75 and variance 2. In most real-world problems, the values of the intercept and slope ( 0, 1) and the error variance 2 will not be known, and they must be estimated from sample data. Then this tted regression equation or model is typically used in prediction of future observations of Y, or for estimating the mean response at a particular level of x. To illustrate, a chemical engineer might be interested in estimating the mean purity of oxygen produced when the hydrocarbon level is x 1.25%. This chapter discusses such procedures and applications for the simple linear regression model. 12 will discuss multiple linear regression models that involve more than one regressor.

11-2

Figure 3.6: Stream cipher using one LFSR. A second approach to constructing a keystream generator from LFSRs is illustrated in Figure 3.7. Again, the purpose of the nonlinear combining function f is to effectively hide the linearity of the underlying LFSRs. In both of these examples, the key is the initial fill of the LFSR or LFSRs.

The case of simple linear regression considers a single regressor or predictor x and a dependent or response variable Y. Suppose that the true relationship between Y and x is a straight line and that the observation Y at each level of x is a random variable. As noted previously, the expected value of Y for each value of x is E1Y 0 x2

0 1x

LFSR,

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where the intercept 0 and the slope 1 are unknown regression coef cients. We assume that each observation, Y, can be described by the model Y

0 1x

11.25.2009 09:58

(11-2)

Figure 3.7: Stream cipher using n LFSRs. The keystream generator in Figure 3.6 is actually a special case of the generator in Figure 3.7. To see that this is indeed the case, suppose that all

where is a random error with mean zero and (unknown) variance 2. The random errors corresponding to different observations are also assumed to be uncorrelated random variables.

Observed value Data ( y)

LFSRs in Figure 3.7, are identical to the single LFSR in Figure 3.6, except for the initial fills. Let the initial fill of LFSRo be identical to the initial fill of the LFSR in Figure 3.6, while LFSRl has as its initial fill the initial fill of LFSRo stepped once and, in general, the initial fill of LFSRi is the initial fill of LFSRo stepped i times. Then the nonlinear combining function f in Figure 3.7 has access to precisely the same bits at each step as t,he function f in Figure 3.6. That is, the stream cipher in Figure 3.6 is a special case of that in Figure 3.7. Consequently, in the discussion below, we restrict our attention to the more general case, as represented by Figure 3.7.

Suppose that we have n pairs of observations (x1, y1), (x2, y2), p (xn, yn). Figure 11-3 shows a typical scatter plot of observed data and a candidate for the estimated regression line. The estimates of 0 and 1 should result in a line that is (in some sense) a best t to the data. The German scientist Karl Gauss (1777 1855) proposed estimating the parameters 0 and 1 in Equation 11-2 to minimize the sum of the squares of the vertical deviations in Fig. 11-3. We call this criterion for estimating the regression coef cients the method of least squares. Using Equation 11-2, we may express the n observations in the sample as yi

0 1 xi i,

In this section we discuss a correlation attack on a shift register-based stream cipher. Consider the keystream generator in Figure 3.8, which consists of three small shift registers and a nonlinear combining function f. The feedback functions of the shift registers X , Y ,and 2 are

(11-3)

(11-4)

0 0, 1

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