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By Henk C. A. van Tilborg (auth.)

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Since alI elements in (*) are integer. equation (*) also holds modulo 2. (i) . d) Which LFSR of length n gives the same output sequence as the above shift register. What does the initial state have to be in this LFSR to generate the same output sequence. 5. Let a E GF(23) be a zero of f (x) = x 3 + x + 1. 38) = (x - f (x) = (l-ax)(I-a2x)(I-a4x). Prove that a)(x - ( 2)(x - ( 4) and f'" (x) = (x - a~(x - ( 5 )(x - ( 6) = 4 SHANNON THEORY In Chapter II we have seen that the cryptanalysis of a cryptosystem often depen

14) . *(x) .. , = (~ SA:xA:) (~C"-l Xl) = .. , =~ ( ~ C.. -l Sj-l " = 't(x) + ~ (~Ci j~" )x j SU-II)+i = )x j = 't(x) . 15) . a (f) can be written uniquely as < n. On the other hand, a(f) bas cardinality 2" and there are exactly [] 2" binary polynomials 't ~) of degree < n. AN INTRODUCfION TO CRYPTOLOGY 26 It is now easy to prove the foIlowing lemma. 9 Let {Si bo E il (j) and (td i~O E il (g). Let lcm fi, g] denote the least common multiple of I and g. Then {Si + ti }i~O E Proof' Write h = lcm il (lcm fi, g], fi, g D.

22 S(P)(x) i) ii) il (h). The proof in the reverse direction goes exactly the same. (s;};;;,o Let be a = so+ SIX + ... + Sp_IX p - l • [S;}i;;'O 'r/II [] binary, periodic sequence, say with period p. Let Then there exists a unique polynomial m (x) with Eil(m), [(S;}i;;'O E il(h) => m I h ]. 18) , and m (x) is called the minimal characteristic polynomial of Proof: Let (Sdi;;'O E il(f), but (S;}i;;,O ~ il(h) for any proper divisor h of f. 18). 8 implies that for some = S (x) = ~ /* (x) . 21 gcd (f* (x), 't(x» gcd (f* (x), S(P)(x)' /* (x) ) = 1 .

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