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One problem with the sequence in (3.3) is that the linear complexity is, in a sense, concentrated in just a single bit. That is, the linear complexity is zero, until the last bit is processed, then the complexity jumps from the minimum to the maximum possible value. Recognizing this, Rueppel [125] proposes the linear complexity profile as a practical measure of the quality of a keystream. This profile is simply the graph of the linear complexity L of S O , s1,. . , s k for each k = 0 , 1 , 2 , .. .. The required L values are obtained . when the linear complexity of s is computed using the Berlekamp-Massey Algorithm, so it is efficient to determine such a profile. Rueppel has shown that most sequences have a linear complexity profile that follows the n/2 line closely but irregularly, and he proposes this as a criteria for judging the quality of a keystream. Figure 3.4 illustrates a linear complexity profile that satisfies Rueppel s criteria and would therefore be considered cryptographically strong by his definition.
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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.
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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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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
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s = 0001 1010 1001 1010 1000 1010 1001 1010.
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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].
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Shift Register-Based Stream Ciphers
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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.
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