Coding for Data and Computer Communications by David Salomon

By David Salomon

Details an important concepts used to make the garage and transmission of information quick, safe, and reliable.

available to either experts and nonspecialists: Avoids complicated arithmetic

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Sample text

6. A binary LSFR of length 5 was used to generate the following stream 101011101100 : : : Recover the feedback polynomial by the Berlekamp-Massey method. 7. 5 Consider the linear recurrence xn = a0 xn;d + a1 xn;d;1 + ad;1 xn;1 (*) with aj 2 F 2 and a0 6= 0. (i) Suppose K is a eld containing F 2 such that the auxiliary polynomial C has a root in K . Then n is a solution of ( ) in K . (ii) Suppose K is a eld containing F 2 such that the auxiliary polynomial C has d distinct roots 1 , 2, : : : , d in K .

For budding cryptologists and cryptographers (as well as those who want a good read) Kahn's The Codebreakers 2] has the same role as is taken by Bell's Men of Mathematics. for budding mathematicians. References 1] U. Eco The Search for the Perfect Language (English translation), Blackwell, Oxford 1995. 2] D. Kahn The Codebreakers: The Story of Secret Writing MacMillan, New York, 1967. ) 3] D. Kahn Seizing the Enigma Houghton Mi in, Boston, 1991. 4] M. Kline Mathematical Thought from Ancient to Modern Times OUP, 1972.

Ii) Suppose K is a eld containing F 2 such that the auxiliary polynomial C has d distinct roots 1, 2, : : : , d in K . Then the general solution of ( ) in K is xn = d X j =1 bj n j for some bj 2 K . If x0 x1 : : : xd;1 2 F 2 then xn 2 F 2 for all n. (iii) Work out the rst few lines of Pascal's triangle modulo 2. Show that the functions fj : Z ! F 2 fj (n) = nj are linearly independent in the sense that m X j =0 aj fj (n) = 0 for all n implies aj = 0 for 1 j m. (iv) Suppose K is a eld containing F 2 such that the auxiliary polynomial C factorises completely into linear factors.

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