How does timestamp hashing work? - encryption

Question

Suppose you have an existing hash g84t5tw73y487tb38wo4bq8o34q384o7nfw3q434hqa which was created from the original string dont downvote my stupid question
Now I timestamp this hash like this (in JS/pseudo-code):
var hash = 'g84t5tw73y487tb38wo4bq8o34q384o7nfw3q434hqa';
var today= new Date(); // 2017-10-19
var timestamped = hash + today;
var new_hash = SHA256(timestamped);
// new_hash is 34t346tf3847tr8qrot3r8q248rtbrq4brtqti4t
If I wanted to verify my original string I can do:
var verified = goodHash('dont downvote my stupid question',hash); // true
If I wanted to verify the timestamped version I can do:
var original_hash = 'g84t5tw73y487tb38wo4bq8o34q384o7nfw3q434hqa';
var today = '2017-10-19';
var verified = goodHash(original_hash+today, timestamped_hash); // true
But if I tried to verify the original string against the timestamp, I CANT do:
var today = '2017-10-19';
var verified = goodHash('dont downvote my stupid question'+today, timestamped_hash); // FALSE
Now suppose this original string is hashed and timestamped over and over again for n iterations.
I would only ever be able to verify the n-1th timestamp, provided I have the n-1th hash.
But what if I have the original string dont downvote my stupid question and want to verify any ith timestamp, where 0 < i < n.
Basically, I want to verify whether a string that only I should have knowledge of, has been timestamped with a given date, regardless of how many times it may have been timestamped and without increasing the length of the string (too much - although any increase in length would approach infinity as n grows).
Is this even possible? Can a hash even contain all this information?

Answer 1

Let's look at the mathematics involved here:
First, you have a input string s and a sequence t of timestamps. I will use t[i] to denote the ith timestamp. Your repeated hashing is a recurrence relation:
f(i) = hash(f(t[i-1]) + t[i])
Where + denotes string concatenation. Now we want to determine if there is a closed formula F(x) which will calculate the ith hash with less time-complexity than evaluating the recurrence relation f(i).
One way to accomplish this is to find a string x(i) with the same hash as f(t[i-1]) + t[i]. For a good hashing algorithm, these collisions are exceedingly rare. My gut instinct is that finding such a string (other than f(t[i-1]) + t[i] itself) is more difficult than simply calculating directly from the recurrence relation.