Any collision resistant hashing function H_{k} : {0,1}^{*} → {0,1}^{k} is pre-image resistant regarding the uniform distribution on {0,1}^{2k}.

`A ⇒ B ⇔ ¬B ⇒ ¬A`

: I.e. if a function isn't pre-image resistant it is not collision resistant either.

SHA256 is a function that maps a potentially unlimited set of numbers to a smaller set of numbers.

H_{k} : {0,1}^{*} → {0,1}^{k}

**Collision resistance** is a property that loosely says, it is difficult to find two *inverse images* `X ≠ X'`

that have the same *image* `H(X) = H(X')`

.

More to the point:

A function `H`

is *collision resistant*, if any algorithm can only find a collision with a negligible probability in probabilistic polynomial time.

I.e. if SHA256 would turn out to not be *collision resistant*, one could try to pick hashes that would succeed at the current difficulty, use the above predicted inverse function of SHA256 to calculate *inverse image* candidates, and finally check whether one can satisfy them with the currently available block input.

My gut feeling would be though, that it would be pretty difficult to find *inverse images* that also satisfy the required structure of the block input, especially getting right the hash of the parent block, and matching an address that one controls for the coinbase transaction.

Unfortunately, I have no idea whether the complexity of that would be greater or smaller than bruteforce mining.

I am currently studying for an exam whose topic includes Collision Resistance, pre-image resistance, and so forth. So, please take in account that I am by no means a cryptography expert before building on my answer. :) – Murch – 2014-09-25T15:10:16.973

I thought that Pre-image resistance and Collision resistance are something akin to mutually exclusive? Its more of a relationship between collision resistance and secondary pre-image resistance right? – mcdoomington – 2014-09-25T15:44:33.377

@mcdoomington: Target Collision Resistance (or Second Pre-Image Resistance) is between the two: For big domains hashed to a smaller image sets it is

– Murch – 2014-09-25T20:52:21.927`Collision Resistance ⇒ Second Pre-Image Resistance ⇒ Pre-Image resistance`

. Also see Does collision resistance imply (or not) second-preimage resistance?