Reverse iteration as a means to find points in a Julia set is known.
The process of iteration: Z_new=Z_old(1-Z_old) can be reversed by
solving the quadratic and equating real to real, imaginary to
imaginary components. In the reverse case, any value of Z_new has two
preceding values for Z_old. For a sequence of iterations, a binary
series of bits can encode the choices of predecessor values at each
iteration. Said in another way, a sequence of reverse iterations from
a given point in the complex plane will produce a binary tree of
possible preceding points, any path through which leads back to the
original point by a simple set of iterations. The choice of a path
through the binary tree can be expressed as a binary vector, v.
Thus we have: Z_m = f(Z_o,v) while Z_o = g(Z_m, m) Z_m is easily
confirmed as related to Z_o but determining Z_m from Z_o requires
knowing a binary vector v, which is m bits long.
(I posted the above on alt.math.recreational but for a week, no
Is the above well known? I reached these conclusions without benefit
of any references. They may be useful, for example, for concealing a
family of pass codes in software. Other instances of usefulness elude
me lacking a back door for the function.
Has anyone seen similar or related approaches for creating one-way
functions? Suggestions on where or how to search would be
appreciated. In particular, are there known back-doors?