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By Tom W. Körner

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Using the Berlekamp–Massey method we see that stream codes based on linear feedback registers are unsafe at level (2). 5. Suppose that an unknown cipher stream k0 , k1 , k2 . . is produced by an unknown linear feedback register f of unknown length d ≤ D. The plain text stream p0 , p1 , p2 , . . is enciphered as the cipher text stream z0 , z1 , z2 , . . given by zn = pn + kn . If we are given p0 , p1 , . . p2D−1 and z0 , z1 , . . z2D−1 then we can find kr for all r. Thus if we have a message of length twice the length of the linear feedback register together with its encipherment the code is broken.

Iii) Let K be a field containing F2 in which X 7 − 1 factorises into linear factors. If β is a root of X 3 + X + 1 in K, then β is a primitive root of unity and β 2 is also a root of X 3 + X + 1. (iv) We continue with the notation (iii). The BCH code with {β, β 2} as defining set is Hamming’s original (7,4) code. The next theorem contains the key fact about BCH codes. 15. The minimum distance for a BCH code is at least as great as the design distance. 12 is of full rank for a BCH. To do this we use a result which every undergraduate knew in 1950.

Nowadays, level (3) is considered a minimal requirement for a really secure system. 60 18 Stream ciphers One natural way of enciphering is to use a stream cipher. We work with streams (that is, sequences) of elements of F2 . We use a cipher stream k0 , k1 , k2 . . The plain text stream p0 , p1 , p2 , . . is enciphered as the cipher text stream z0 , z1 , z2 , . . given by zn = pn + kn . This is an example of a private key or symmetric system. The security of the system depends on a secret (in our case the cipher stream) k shared between the cipherer and the encipherer.

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