By Zimmer H.G.

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**Example text**

Thus a and b are not shift equivalent, but they are decimation equivalent. For m-sequences, we have the following interesting results for the decimation operator. Property 4 Assume that a is an m-sequence of period 2n − 1. (a) Any 2i -decimation of a is a shift of a for i = 1, 2, · · · , n − 1. , a sequence forms by the even bits of a is a itself. (b) If gcd(s, 2n − 1) = 1, but s is not a power of 2, then s-decimation of a is an m-sequence of period 2n − 1 which is shift distinct from a. , starting with one m-sequence of period 2n − 1, the rest of them can be obtained by performing the decimation operation.

However, {ai } could be generated by an LFSR with smaller degree. For example {ai } = 011 can be generated by a 3-stage LFSR with characteristic polynomial f (x) = x3 + 1 and by a 2-stage LFSR with characteristic polynomial h(x) = x2 + x + 1. Note that f (x) = (x + 1)(x2 + x + 1). Thus h(x) is a divisor of f (x). The characteristic polynomial with the smallest degree which generates {ai } is called the minimal polynomial of {ai }. This is a general result, which is stated as follows. Property 2 The minimal polynomial of an LFSR sequence is a divisor of its characteristic polynomial.

1) then the output sequence is called a linear feedback shift register (LFSR) sequence. Otherwise, it is called a nonlinear feedback shift register (NLFSR) sequence. Sometimes, we also say that the sequence a = a0 , a1 , · · · is generated by an LFSR (or NLFSR). In the LFSR case, the recursive relation becomes the following linear recursive relation n−1 ak+n = i=0 ci ak+i , k = 0, 1, · · · . 2) A State Diagram of an FSR The states of an FSR can also be represented in a graph, called a state (vector) graph (or diagram).

### Computational aspects of theory of elliptic curves by Zimmer H.G.

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