*	Apps:
	-	Crypto: some cryptosystems, not all
	-	Coding: e.g. eth, flop CRCs; hdd, cd, cell, etc.
	-	Generate pseudo-random sequences, e.g. test patterns
	-	We're acccustomed to applied DEs, etc. but here is applied
		*algebra*.
	-	Ref. Golomb

*	Physical setup
	-	n cells in Fq (F2: bits -- gen theory w/ specific examples)
		form a register
	-	v_{n-1} .. v_1
	-	Clock; v(t+1) = f(v(t)) vs. v := f(v)
	-	Shift, feedback, linear
	-	Draw Fibo but note u-RER
	-	Perhaps note recurring sequences (fibo)
	-	States/ outs / ins
	-	Non-autonomous cf. ODEs (e.g. CRCs); autonomous here

*	Graph
	-	Iteration is a function from a finite set to itself hence
		there must be loops by the pigeonhole principle
	-	Number states & obtain graph:  single trajectory is a rho;
		full graph is rhos with hair
	-	2-to-1; tails; loops.
	-	Lock-up states.
	-	Note lin sys:  zero always locks up.  So max poss is q^n-1 loop
	-	Claim usually desired is one big loop, q^n-1, in particular
		for pseudo-random sequence generation.  Ask if
		constructible; short answer is yes.
	-	[Note:  floppy CRC: 16-bit LFSR; *two* lock-up states @ 0's
		and 1's; two big loops of 32767 each.]

*	Good/bad/ugly
	-	Write matrix; compute charpoly
	-	It turns out that the behavior of the *physical* system reduces
		to an *algebraic* problem.
	-	Ugly: c0 = 0.  Note (a) physically leads to subsys, ->
		graph hair; (b) mx non-invertible; (c) zero eigenvalue.
	-	Rest: c0 != 0.  Matrix is invertible.
	-	T^k * T => T^{k+1} is a function from finite set to self,
		hence dup at T^l = T^k; T^{l-k} is identity.
	-	Cyclic subgroup of GL(Fq, n).  Group action on set S of LFSR
		states => matrix powers are permuations on S. => no hair
		since 1-1; no tails since permutation has cycle decomposition.
	-	Difference between bad & good: number and size of loops.
		Analyze the charpoly
	-	Hierarchy: ugly/singular already seen
	-	Next is reducible; degree of splitting field = lcm of degrees
		of factors.
	-	Next is irreducible but not primitive (define) in which
		case loops are order of u in the multiplicative group
	-	Best is primitive
	-	Obtain primitives (feasible for small degree) by computing
		q^n-1'st cyclotomic polynomial, then factoring over Fq.
		Result is all primitives of degree n over Fq.

		gp: lift(lift(factorff(polcyclo(15), 2, u+1)))
		gp: lift(lift(factorff(polcyclo(15), 2, u^2+u+1)))