Cryptographic hash functions
Hashing is commonly used when we want to reduce a message of an arbitrary size to a fixed length “digest”, for purposes ranging from integrity checking of files downloaded from the internet to building blocks for cryptocurrencies. There are many functions one can use to map the input message to its hash. The only requirement is that the output be of a known, fixed size. For example, consider the function
which maps an arbitrarily large value $x$ to a value in the set $\left\{ 0,1,2 \right\}$. Such hash functions (i.e., based on simple modulo arithmetic) are used (with other, more complicated hash functions) in distributed systems (see consistent hashing). However, this simple function has little utility when it comes to integrity checking. For example, we would like to download a file and check if the file we downloaded was indeed the one that we intended to download. This can be solved as follows:

Website offering the file lists the file’s hash along with the hashing function used.

We download the file and use the same hashing function on the contents of the file to get its hash.

Next, we compare the hash published on the site with the answer we got, and conclude that the integrity check passed if the hashes are equal.
It is easy to see that the simple modulo function based hashing will not work at all here, since given any message value $x$ (the contents of the file), every third value, i.e., $x+3, x+6, …$ will have the same hash, and an attacker could fool our integrity checking by adding appropriate content to the original file.
So what do we need from a good hash function for integrity checking?
1. Collision freedom
In the above example, we found collisions under the function $H_{mod3}$ – i.e., we could easily craft a message $x’$ such that $H_{mod3}\left(x\right) = H_{mod3}\left(x’\right)$ (just pick $x’ = x + 3k$ for any $k \gt 1$). We want a hash function $H$ such that it is extremely hard to find collisions under it. Note that since a hash function must transform an arbitrary message into a fixed size output that is often much smaller than the input size, there will necessarily be collisions – a large number of them indeed. What is important is that it is extremely hard to craft a message that does not equal the original message, but whose hash still matches that of the original message.
2. Hiding
It should be extremely hard to figure out the original message $x$ from its hash $H\left(x\right)$. Our modulo function $H_{mod3}$ is actually pretty good at this — given the remainder of a number of when divided by 3, you cannot figure out the original number at all, even though you can reduce the number of candidates by only looking at the numbers that produce the given remainder when divided by 3. As you might notice, the degree of hiding offered by the hashing function depends on the range of values that the message variable $x$ can take up: consider trying to hide the outcome of a coin toss. We’ll encode , and .
such that $H_{toss}\left(\text{HEAD}\right) = 1$ and $H_{toss}\left(\text{TAIL}\right) = 0$.
Now if the domain ($\left\{\text{HEAD},\text{TAIL}\right\}$) is known to the attacker, given a hash value 1, they might simply try hashing each value in the set to find out that the original message was HEAD. The problem here is that the message domain is easily enumerable. Now if we pick a random integer of, say 256 bits, $r$ from a distribution with a high min entropy, and use its bits appended to the original message’s bits, we suddenly have an input space which is extremely hard to enumerate ($2^257$ possibilities). So, now, given a hash value of 1, it is extremely hard for an attacker without the knowledge of the random key $r$ to deduce the outcome of the toss. Min entropy of a distribution is simply the negative logarithm of the probability of the most probable value being taken up by the random variable distributed according to that distribution. For example, if you have a loaded die that shows a 6 50% of the times, with the other faces each having a probability of 10%, the min entropy of this distribution would be $\lg{\frac{1}{2}} = 1$, since the most probable value, 6, comes up with a probability $\frac{1}{2}$. Intuitively, when all values in a distribution are negligibly likely, and no particular value is more likely than others, the distribution has a high min entropy, and it is increasingly difficult to enumerate such a distribution’s values to break the hiding property of a hash function.
3 “Puzzle friendliness”
For the sake of using hashes as challenges, as is done in cryptocurrency implementations, we need an additional property from a good hash function. Given a key $k$ chosen from a distribution with high min entropy, and a set of target hash values $Y$, it should be hard to find a value $x$ such that the concatenation of $k$ and $x$ hashes to a value in $Y$. More precisely, by “hard” we mean that no strategy of picking $x$ values should be better than randomly picking $x$ values and trying if $H(kx) \in Y$.