Cryptographic Hash Functions explained

Code Security Math Writing

Hash function is a mathematical function that takes arbitrary input and maps to fixed-size value. Here’s a little example of a hash function that maps input to 4 digits number.

h("dog") = 1571
h("Hey, that's a beautiful cat you've got over there!") = 9801
h("c") = 0130

Cryptographic hash function is a hash function with more restrictive properties that has many applications in cryptography, encryption and security in general. Here’s an example of modern hash function that is used in some way to secure almost all the internet traffic.

sha-3("turtle") = 38b33b6f08b5cd5428ddc7aa698d7e8a0a2f15830ef17b9dbb518699f850a141

Before we get into hash function characteristics, lets try to design a hash function ourselves. We will start with something simple, for example a function that takes a number and returns its reminder after dividing it by 10 (this operation is also called modulo).

h(x) = x mod 10

A few example:

h(33) = 3
h(182) = 2
h(10) = 0

Does this function take an arbitrary input and returns fixed-size output? Yes, it does. Is it a good cryptographic hash function? Well, no it isn’t. To make a hash functions useful in cryptography we introduce further restrictions.


First one is Pre-Image Resistance. What this property means is, that it should be hard to reverse the hash function. In other words, knowing the value that came out of the hash function should provide zero to none information about what came in. Here our previously defined function fails. Knowing the value that came out means knowing the last digit of the number that came in.

Let’s try to fix that and change our hash function. Now we will add random number to the input before doing what we did before, taking the reminder of division by 10. This function still returns fixed-size value, furthermore, it is no longer possible to tell anything about the input. Do we have a good hash function? Not yet.

Another property that good hash functions should have is being Deterministic. Deterministic function always maps the same input to the same output. Our function doesn’t do that. Number 100 can be mapped to 1, 4, or any other number depending on the random value we add.

There are two more properties that we need that are hard to show on easy functions. First one being Efficiency. Hash functions must be fast to compute to be usable. It should be easy to compute the output for any value but really hard to find input for given output. This is partially overlapping with Pre-Image Resistance, and we call such function a One-way Function.

The last property of hash functions is Collision Resistant. That means that it must be extremely unlikely (read “virtually impossible”) for two inputs to produce the same output. That’s hard to imagine on the example that we set with a function that has 10 possible outputs but take for example SHA-3, modern secure hash function that is the current standard. This function has

2^256 = 115792089237316195423570985008687907853269984665640564039457584007913129639936

possible output values. Such a huge output space makes it practically impossible to find two inputs that map to the same output.

We have covered the 4 base properties of good and secure hash function. Pre-Image Resistance, being Deterministic, Efficiency and Collision Resistance. Now lets show why these properties are important on some real life examples.


Cryptographic hash functions are a basic tool of modern cryptography and have many applications in fields such as digital signatures, fingerprinting, message authentication and other forms of authentication, and more.

Data integrity

First application that we will look into is data integrity verification. How that works is that we take a message, file, or any other digital medium, we create message digest (compute the hash function value for given message or file) and any time we need to verify the integrity we compute the message digest again and compare it with the original. This way we can determine whether any changes of modifications have been made to the original message or file during transmission.

Message digests are for example published by many websites to allow verification of integrity of downloaded files. The whole process looks like this: an author creates application or file that he wants to publish securely for public. He creates message digest for given file and publishes it alongside the file on his website. User downloads the file, computes message digest using the same cryptographic hash function as the author did and compares the digest with the one published by the author. This way he can verify that the files hasn’t been tempered with during the download. This method isn’t bulletproof because the hash on the website could be tempered but as long as we trust the website we can trust the downloaded file as well.

Digital Signatures

Other application of cryptographic hash functions are Digital signatures. Digital signatures are closely related to Data integrity, they again allow you to verify data integrity. Furthermore, digital signatures tie the signer to the document making it hard for the signer to deny signing the document. This property is called non-repudiation and refers to a situation where author cannot dispute validity or authorship of associated document.

To create a Digital signature the author creates message digest and signs it using his private key. Digital signature is then distributed alongside the document or file. If anyone wants to verify the signature he needs to compute the message digest on his side, decrypt the signature using authors public key, and compare the two message digests. This way he can verify not only the integrity of the data but also the authorship. Private key in this case works sort of like a credit card pin, it is something that only the owner knows so signing the message with his private key ties him indisputably to the signed document.

Digital signatures are used for example by The United States Government Printing Office (GPO) to sign electronic versions of the budget, public and private laws, and congressional bills.


Cryptographic hash functions are also the safest way of storing passwords. All major websites and applications (facebook, google, etc.) use hashing for password storage and user authentication, and if they don’t, they should.

To show how that works lets start with a bad example of how passwords shouldn’t be handled. A bad but straightforward approach to password authentication is prompting the user for a password and then storing the password as the user entered it alongside his username or email on the server. To authenticate the user all that is need is to compare the entered password with the stored one, if they match, the user successfully authenticates. Where this approach lacks though is when things go south and malicious party steals the credentials from the server - suddenly they have information about all users tied with their passwords.

Good approach is hashing the password before storing it on the server. This way at any point in time the only person that knows the password is you. To authenticate you the server computes the hash again and compares it with the stored one. Now even if someone steals the credentials he has no way of retrieving your password from the hash (thanks to the Pre-Image Resistance property).