hash - Is it safe to ignore the possibility of SHA collisions in practice?

Question

Let's say we have a billion unique images, one megabyte each. We calculate the SHA-256 hash for the contents of each file. The possibility of collision depends on:

• the number of files
• the size of the single file

How far can we go ignoring this possibility, assuming it is zero?

Answer 1

The usual answer goes thus: what is the probability that a rogue asteroid crashes on Earth within the next second, obliterating civilization-as-we-know-it, and killing off a few billion people? It can be argued that any unlucky event with a probability lower than that is not actually very important.

If we have a "perfect" hash function with output size n, and we have p messages to hash (individual message length is not important), then probability of collision is about p²/2ⁿ⁺¹ (this is an approximation which is valid for "small" p, i.e. substantially smaller than 2^n/2). For instance, with SHA-256 (n=256) and one billion messages (p=10⁹) then the probability is about 4.3*10^-60.

A mass-murderer space rock happens about once every 30 million years on average. This leads to a probability of such an event occurring in the next second to about 10^-15. That's 45 orders of magnitude more probable than the SHA-256 collision. Briefly stated, if you find SHA-256 collisions scary then your priorities are wrong.

In a security setup, where an attacker gets to choose the messages which will be hashed, then the attacker may use substantially more than a billion messages; however, you will find that the attacker's success probability will still be vanishingly small. That's the whole point of using a hash function with a 256-bit output: so that risks of collision can be neglected.

Of course, all of the above assumes that SHA-256 is a "perfect" hash function, which is far from being proven. Still, SHA-256 seems quite robust.