By T. Korner

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**Extra resources for Coding and Cryptography**

**Example 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.