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so the moment generating functions of X1 and X2 are MX1 1t2 e 11e 12 and MX2 1t2 e 21e 12, respectively. Using Equation S5-10, we find that the moment generating function of Y X1 X2 is MY 1t2 MX1 1t2 MX2 1t2 e

which is recognized as the moment generating function of a Poisson random variable with parameter 1 2. Therefore, we have shown that the sum of two independent Poisson random variables with parameters 1 and 2 is a Poisson random variable with parameters equal to the sum of the two parameters 1 2.

Before illustrating the use of LFSRs in stream ciphers, we first take a slight detour to briefly consider some of the properties that keystrearn sequences must satisfy. Here, we relate these properties to shift registers. Keystream sequences that are unpredictable, according to some specified set of conditions, are said to be cryptographically strong. However, it is important to realize that this definition is relative to the specified criteria. While we can specify necessary conditions that a keystream sequence must satisfy, there are no known sufficient conditions that ensure that a sequence

EXERCISES FOR SECTION 5-9

S5-13. bution A random variable X has the discrete uniform distrif 1x2 e x!

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is cryptographically strong In a sense, this situation parallels cryptography in general, where the best that can be said about any practical cipher is that, as far as we know, nobody has found an efficient attack That is, we can never prove that a cipher is absolutely secure, but we can show that it satisfies certain criteria that give us some confidence that is likely to be secure in practice Note that if at some point, the fill of an LFSR is all zero, then the register fill will be all zero at every subsequent step Therefore, an upper bound on the period length of any LFSR sequence is 2n - 1, where n is the number of stages in the LFSR In fact, it is possible for an LFSR to attain this upper bound, and the resulting maximal length sequences are known as m-sequences.

1 m,

Use the 4-Way Controller to select the Home or World icon. Select your time zone from the map that is displayed, then press the Function/Set button. Pressing the Menu button twice returns you to Shooting mode.

(b) Use MX (t) to nd the mean and variance of the Poisson random variable. S5-15. The geometric random variable X has probability distribution f 1x2 11 p2 x 1p, x 1, 2, p

For example, a 3-stage LFSR with connection polynomial C ( x ) = 1 x2 will generate a sequence of period length seven for any nonzero initial fill In general, if C ( x )is a primitive polynomial [43], then the resulting LFSR will yield m-sequences for all nonzero initial fills While m-sequences have many nice statistical properties [59],they would be poor choices for keystream generators Suppose we have a 32-bit key and we decide to use a stream cipher that consists of a 32-stage LFSR, with the connection polynomial chosen so that the resulting keystream is an msequence Then if Trudy is able to obtain just 64 consecutive keystream bits, she can use the Berlekamp-Massey Algorithm to determine the entire keystream, which is of length 232 - 1.

(b) Use MX(t) to nd the mean and variance of X. S5-14. A random variable X has the Poisson distribution

5-13

(b) Use MX (t) to nd the mean and variance of X. S5-16. The chi-squared random variable with k degrees of freedom has moment generating function MX (t) (1 2t) k 2. Suppose that X1 and X2 are independent chi-squared random variables with k1 and k2 degrees of freedom, respectively. What is the distribution of Y X1 X2 S5-17. A continuous random variable X has the following probability distribution: f 1x2 4xe

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