Being a some constant, further assume that we
are in a factor ring (basically all operations modulo some sumber p).
What do you mean with “a factor ring“? Any ring is a factor ring.
I assume you mean the ring Z/pZ for some nat. number p?
If so, is this p supposed to be prime (as the letter suggests)?
You always have to start with
x=9.
Consider the following recursive formula:
new_x = (x²-1)² / (4*x*(x²+a*x+1))
How often do you have to perform this operation to get a specific x (basically getting the new_x and feeding it back into the formula to get another new_x, and so on)?
Let f:Z/pZ -> Z/pZ be the map given by your assignment
x |-> new_x. Then you basically ask for a map g:Z/pZ -> N
s.t. f^(g(y))(9) = y for all y in Z/pZ, right?
Note: You can start multiple such chains beginning at
x=9, and add the resulting x values
using the addition algorithm from
http://en.wikipedia.org/wiki/Montgomery_curve (
Montgomery arithmetic section).
Note, that the x value, is the value you get at the end of such calculation-chain, and the z value is always 1.
I don't know what you mean with all of this.. Could you put it in more
mathematical terms?
Be also advised that the referenced addition “algorithm“ is not specific
to Montgomery curves, so it would be better to link to the wikipedia article
for elliptic curves where it is also described.
Do you try to consider elliptic curves over F_p? I have the impression,
but you should reformulate your question..