i get that SHA256 isn’t considered “encryption”, it’s something to do with hashing. so, if i encrypt “hello” into a random string of characters, why is it said to be impossible to decrypt it back to “hello”? if you had a maths equation like P = (k * 10 / G) / (4x * 5gl), it’s possible to work backwards to find the value of k or g eventually. why not with SHA256? is it something to do with random numbers?
In: Mathematics
I want to try to add to the other answers, and hopefully give a bit clearer (if more handwave-y) explanation for your specific question.
So, you are correct that (at its core) you can boil down SHA256 to “just a math equation”. However, I want you to consider a (relatively) simple math equation: the roots of a quintic polynomial ax^(5) + bx^(4) +cx^(3) +dx^(2)+ex+f = 0. Assuming you are given a-f, you only have to find x: and yet, it turns out that there does not exist a general formula you can just plug a-f into and pop out x [(source)](http://mathworld.wolfram.com/QuinticEquation.html). To be completely honest, I do not know the proof of this, I just know that there is a proof nobody has found fault with as of yet (and given that this was known back in the 1800s, its a pretty safe bet that the proof is correct).
That being said, what does this have to do with SHA256? Well, if its impossible to come up with a generic equation to solve something with only 4 exponentiations, 5 multiplications, and 6 additions, imagine how unlikely it is that we have a solution to a mathematical formula so complex that it is written in pseudocode rather than as an equation.
Now, to be clear, this is more an analogy than anything mathematically rigorous. SHA256 doesn’t use exponentiation (in fact, it tends to use boolean operations like AND and bitshifting rather than numerical operations). Also, nobody has *proven* that a formula for reverse-engineering SHA256 doesn’t exist: mostly we are just relying on the fact that its complicated and nobody has (publicly) found a solution so far.
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