For instance, a sequence with (potentially huge) period $N=2^n-1$ can be constructed from a LFSR with only $n$ stages. I now wonder if we can now say anything about the minimal FSR that outputs a given sequence. Can that be generalized to any $N$?ĮDIT 2: So, my question was trivial, see joriki's answer below. The linear feedback shift register, most often used in hardware designs, is the basis of the stream ciphers we will examine here. In other words, if a binary sequence has period $N=2^n-1$ for some $n$, there exists a Linear FSR with that ouputs this sequence. It is a novelty to propose coding solutions by means of Reverse Polish Notation, thanks to which the simple mechanism of a stack with automation, realizing a context-free grammar of. ![]() Let denote a root of the primitive polynomial f(x) f0 + f1x + + fnxn 1 + f1x + + fn 1xn 1 + xn Consider any initial nonzero. ![]() From a mathematical point of view, such a sequence is constructed recursively as follows: each state of the LFSR with length at most $n$ (such a state is an element of $^n$ (Golomb, Gong, Signal Design for Good Correlation, 2005). The selection of the register feedback structure to achieve the maximum cycle is a difficult task, especially for the register with a non-linear feedback function. Definition: A primitive polynomial f(x) is an irreducible polynomial of degree n in F2nx with the property that each root of f is a generator of F × 2n, the multiplicative group of F2n. Quick background: The output of a Linear Feedback Shift Register (LFSR) with length $n$ is a binary sequence which is periodic of length dividing $2^n-1$.
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