A one way Function provably reversible at N applications with the same seed? I'm looking for a function that is generally one way from some secret $F(s, A) \rightarrow Y$, where $A$ is known, $Y$ is produced (also known), and $s$ is kept secret. But whose repeated application results in the function being reversible (to determine $s$) at some step $F_n$. 

Edit: $A[1,2 \dots n]$ is a set of numbers not known in advance (prior to each successive application of $F$), but of a particular type, say the first 16 bytes of a SHA-256. 

It is essential that it can be proven that the function is not reversible at $n-1$ (or that the work necessary to reverse at $n-1$ is very high) and sufficiently easy to reverse at $n$ ($n+1$ etc.), and that this is provable for any s belonging to a certain group. 

$F(s, A_1) \rightarrow Y_1\\
F(s, A_2) \rightarrow Y_2 \\
\dots \\
F(s, A_3) \rightarrow Y_n$

Would greatly appreciate a pointer in the right direction (what functions to look at, or even some reading on the subject), thanks.
F(s,A) can be

a prefix-free encoding of the number of elements in the finite field encoded by s

|| $\:$ the polynomial R (as described at this link) from that finite field $\;$ ||

the output of the degree-at-most-(N-1) polynomial encoded by s over that finite field at the point A

.
Let SP be the set of polynomials of degree at most N-1 over the finite field. $\:$ N input-output pairs would allow one to efficiently find the encoded element of SP by Lagrange interpolation. $\:$ Observe that for any given field, for all A, evaluate-at-A is a linear map from SP to the field. $\:$ Since Lagrange interpolation will always give a compatible polynomial (and not just find one when there is one), that linear map is surjective. $\:$ Thus, for all integers x in {0,1,2,3,...,N-2,N-1}, for any
given N-x input-output pairs and any given z other inputs, the outputs at those x inputs for such a polynomial chosen uniformly from the subset of SP that produces the given N-x input-output pairs are independent and uniformly distributed. $\:$ Observe that any guess at s can only be correct if it gets those outputs correct.

Therefore, for all integers x in {0,1,2,3,...,N-2,N-1}, the probability of guessing s correctly is

at most [[the sampling error for SP] plus [one over [[the number of elements in the field] to the x]]].

This wiki article describes how to do arithmetic over finite fields given an irreducible polynomial over

their characteristic field. $\:$ This paper gives the "simplest" such polynomials for degree at most 10000

over the field with 2 elements, and these two papers give algorithms which can let you confirm that

those polynomials are in fact irreducible (confirming that they're the simplest would be harder).

More generally, over small prime fields, this paper gives a deterministic algorithm

for constructing irreducible polynomials and this paper gives constructions that

are presumably more efficient for degrees divisible by a not-too-small power of 2.