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Figure 3.4: Linear complexity profile. In [141] a different criteria for cryptographically strong sequences is considered. Although the keystream in (3.3) has the highest possible linear complexity, it differs by only one bit from the all-zero sequence, which has the minimum linear complexity. That is, the sequence in (3.3) is too close (in Hamming distance) to a sequence with small linear complexity. In general, if a sequence is close to a sequence with a relatively low linear complexity, then regardless of the linear complexity of the original sequence, it is an undesirable keystream. The k-error linear complexity is defined to be the smallest linear complexity that can be obtained when any k or fewer bits in one period of a sequence are changed from 0 to 1 or vice versa.

0 1x

E1 2

Given a sequence, we can plot the k-error linear complexity versus k , and for a cryptographically strong sequence, the graph should not have any large drops, particularly for relatively small k , since any such drop would indicate that a sequence with much smaller linear complexity lies close to the given keystream. We refer to the graph of the k-error linear complexity as the k-error linear complexity profile. In Figure 3.5 we have illustrated an undesirable k-error linear complexity profile. This profile shows that the sequence is close to a sequence with a much smaller linear complexity, as indicated by the sharp drop below the dotted line for a relatively small value of k .

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Notice that this is the same relationship that we initially wrote down empirically from inspection of the scatter diagram in Fig. 11-1. The variance of Y given x is V 1Y 0 x2 V1

0 1x

Figure 3.5: k-Error linear complexity profile. In fact, the linear complexity profile in Figure 3.4 and the k-error linear complexity profile in Figure 3.5 were both obtained from the periodic sequence with period

V1 2

Thus, the true regression model Y 0 x 0 1x is a line of mean values; that is, the height of the regression line at any value of x is just the expected value of Y for that x. The slope, 1, can be interpreted as the change in the mean of Y for a unit change in x. Furthermore, the variability of Y at a particular value of x is determined by the error variance 2. This implies that there is a distribution of Y-values at each x and that the variance of this distribution is the same at each x. For example, suppose that the true regression model relating oxygen purity to hydrocarbon level is Y 0 x 75 15x, and suppose that the variance is 2 2. Figure 11-2 illustrates this situation. Notice that we have used a normal distribution to describe the random variation in . Since Y is the sum of a constant 0 1x (the mean) and a normally distributed random variable, Y is a normally distributed random variable. The variance 2 determines the

y (Oxygen purity)

s = 0001 1010 1001 1010 1000 1010 1001 1010.

0 + 1 (1.25)

0 + 1 (1.00)

This option displays a Date/Time screen where you can set the current date, time, and daylight saving status. The daylight saving adjustment is indicated by the sun icon to the right of the display. The Quick Tour describes how to set the date and time.

The linear complexity profile of this particular sequence appears to satisfy Rueppel s criteria, since it follows the n/2 line closely and no regular pattern is evident. However, the k-error linear complexity profile indicates that this particular sequence is probably not a strong keystream, since it lies close to a keystream with low linear complexity. For this example, the k-error linear complexity is more informative than the linear complexity profile. In the general case, no efficient algorithm for computing the k-error linear complexit,y is known. However, for the special case where s is periodic with period length 2n, an efficient algorithm is given in [141].

x = 100

x = 1.25

Due to the Berlekamp-Massey Algorithm, we cannot directly use the output of an LFSR as a stream cipher. The fundamental problem lies with the linearity of LFSRs. However, LFSRs have useful mathematical and statistical properties, so it would be desirable to construct stream ciphers based on LFSRs. There are two generic approaches that are often used to create keystream generators based on LFSRs. One such approach is illustrated in Figure 3.6, where a nonlinear combining function f is applied to the contents of a shift register to yield the keystream sequence. The combining function is intended to mask the linearity of the LFSR, while taking advantage of the long period and good statistical properties of LFSR sequences.

x (Hydrocarbon level)

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