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By Tom W. Körner

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If random bits can be safely communicated, so can ordinary messages and the exercise becomes pointless. In practice, we would like to start from a short shared secret ‘seed’ and generate a ciphering string k that ‘behaves like a random sequence’. This leads us straight into deep philosophical waters37 . As might be expected, there is an illuminating discussion in Chapter III of Knuth’s marvellous The Art of Computing Programming [7]. Note, in particular, his warning: . . random numbers should not be generated with a method chosen at random.

If a stream (xn ) comes from a linear feedback generator with auxiliary polynomial C of degree d, then C is determined by the condition G(Z)C(Z) ≡ B(Z) mod Z 2d with B a polynomial of degree at most d − 1. We thus have the following problem. Problem Given a generating function G for a stream and knowing that G(Z) = B(Z) C(Z) with B a polynomial of degree less than that of C and the constant term in C is c0 = 1, recover C. The Berlekamp–Massey method In this method we do not assume that the degree d of C is known.

Xd−1 ∈ F2 , show that xn ∈ F2 for all n. (iii) Work out the first few lines of Pascal’s triangle modulo 2. Show that the functions fj : Z → F2 n fj (n) = j 54 are linearly independent in the sense that m bj fj (n) = 0 j=0 for all n implies bj = 0 for 0 ≤ j ≤ m. (iv) Suppose K is a field containing F2 such that the auxiliary polynomial C factorises completely into linear factors. If the root αu has multiplicity m(u), [1 ≤ u ≤ q], show that the general solution of ⋆ in K is q m(u)−1 xn = bu,v u=1 v=0 n n α v u for some bu,v ∈ K.

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