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

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Proof: Let d(x) I gcd ('t(x),/*(x» with degree (d(x» = d ~ 1. _ 't(x)/d(x) Then S (x) - /* (x )/d (x)' SO (Si};;;'O E il (f Id*). 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.

The values of a subset A of the inputs determine an address. 4 Multiplexer In [Jen83] the following multiplexer is proposed. 1) Take two LFSR's with m resp. n stages, both with a primitive characteristic polynomial and with a non-zero starting state. 2) Choose h distinct elements from {O, 1•.... m - 1} and order them, say O~ il < i 2 < . < ih < m. A restriction on the size of h follows in 3). • a;~ (t» of inputs at time t defines the value 35 Shift Register Sequences When h < m, N (t) can have any value in {O, 1, .

3, n = 4, h = 2, il = O,i 2 = 1. multiplexer Let't be defined by 't (O) = 3, 't (1) we get the following sequence: = 2, 't (2) = O, 't (3) = 1. With starting states (1,0,0) resp. (1,0,0,0) t m-register N (t) 't (N (t» n-register u (t) O 1 2 3 4 100 1 2 1000 001 O 2 3 O 0001 010 101 1 2 0100 011 2 O 1 1 2 3 O O 1 O O 1 5 111 6 110 7 100 3 3 1 8 001 O 9 010 2 .. 0010 1001 0011 0110 1101 1010 0101 .... O 1 O O O AN IN1RODUCTION 1'0 CRYPTOLOGY 36 In [Jen83] one can find the following theorems about these multiplexed sequences.