An Introduction to Cryptology by Henk C. A. van Tilborg (auth.) PDF

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

ISBN-10: 1461289556

ISBN-13: 9781461289555

ISBN-10: 1461316936

ISBN-13: 9781461316930

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Co + CIX + C~2 + ... + C.. _IX .. 9) i=O where c" = 1 by definition and Co = 1 byassumption. 9). 8a) ) . In words il (f) is the set of all output sequences of the LFSR with characteristic polynomial f (x ). 6 Let f be the characteristic polynomial of an n -stage LFSR. Then il (f) is a binary vector space of dimension n. 8a) is a linear recurrence relation, il (f) is obviously a vectorspace. •• ,S.. -l> (Si) i~O so the dimension of il (f) is at most n. On the other hand the n different sequences starting with ~i----+) ~n-i-l~ 00 ...........

30 k sk O Lk c(k)(x) O O 2 1+x ~ x ~ 3 O 2 1+x 2 4 O 2 1+x 2 3 x2 5 LFSR with initial state ~ ~ ~ The problem with non-linear feedback shift registers in general is the difficulty of their analysis. One has to answer questions like: how many different cycles of output sequences are there, what is their length, what is their linear equivalence, etc. The following will make it clear that it is possible to say at least a little bit about general non-linear feedback shift registers. Clearly the output sequence of a non-linear FSR does not have maximal period if there are two different states with the same successor state.

Sk-I and "predict" Sk with this LFSR. 25 Lk «( Si }i"O) is the length of the shortest LFSR that generates SO,SI • . . • Sk-l- When it is clear from the context which {Si );;'0 is involved we shall simply write L/e. The characteristic polynomial of an L/e -stage LFSR that generates Si. 26 Let (Si bo SO,SI • . . • be such that So :5; k - 1, will be denoted by c (/e) (x). (SO,SI' . . • S/e-I) taken as starting state of any S/e-l- = SI = ... = Sk-2 = O and St-I = 1. Then Lt((Si }i"O) = k. Proof: An LFSR of length n < k will only output n consecutive zeros if the initial state is the allzero vector.

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An Introduction to Cryptology by Henk C. A. van Tilborg (auth.)


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