Authors: Zademn Reviewed by:

Another desired propriety of our cryptographic protocols is **data / message integrity**. This propriety assures that during a data transfer the data has not been modified.

Suppose Alice has a new favourite game and wants to send it to Bob. How can Bob be sure that the file he receives is the same as the one Alice intended to send? One would say to run the game and see. But what if the game is a malware? What if there are changes that are undetectable to the human eye?

**Hashes** are efficient algorithms to check if two files are the same based on the data they contain. The slightest change (a single bit) would change the hash completely.

On the internet, when you download, files you often see a number near the download button called the **hash** of that file. If you download that file, recalculate the hash locally and obtain the same hash you can be sure that the data you downloaded is the intended one.

Another use for hashes is storing passwords. We don't want to store plaintext passwords because in case of a breach the attacker will know our password. If we hash them he will have to reverse the hash (or find a collision) to use our password. Luckily the hashes are very hard to reverse and collision resistant by definition and construction.

Note that hashes need a secure channel for communication. Alice must have a secure way to send her hash to Bob. If Eve intercepts Alice's message and hash she can impersonate Alice by changing the file, computing the hash and sending them to Bob. Hashes do not provide **authenticity.**

**Definition - Hash**

A hash is an

efficient deterministicfunction that takes an arbitrary length input and produces a fixed length output (digest, hash). Let $H:\mathcal{M} \longrightarrow \mathcal{T}$ be a function where

â€‹$\mathcal{M}$ = message space

â€‹$\mathcal{T}$ = digest space

**Desired proprieties**

Deterministic

Fast to compute

Small changes change the hash completely

Preimage, second preimage and collision resistance (Explained below)

*How to use a hash:*

Suppose you want to check if Alice and Bob have the same version of some file (

**File integrity**)They compute $H(a), H(b)$â€‹

They check if $H(a) = H(b)$

from hashlib import sha256m1 = b"Some file"m2 = b"Some file"sha256(m1).digest() == sha256(m2).digest() # -> True

Preimage Image Resistance

Second Preimage resistance

Resistant to collisions

The hash function must be a one way function. Given $t \in \mathcal{T}$ find $m \in \mathcal{M}$ s.t $H(m) = t$â€‹

*Intuition *

It should be unfeasible to reverse a hash function ($\mathcal{O}(2^l)$ time where $l$ is the number of output bits)

This propriety prevents an attacker to find the original message from a hash

Given $m$ it should be hard to find $m' \neq m$ with $H(m') = H(m)$â€‹

**Attack game**

An adversary $\mathcal{A}$ is given a message $m$ and outputs a message $m' \neq m$.

â€‹$\mathcal{A}$ wins the game if he finds $H(m) = H(m')$

His advantage is $Pr[\mathcal{A} \text{ finds a second preimage}]$ where $Pr(\cdot)$is a probability

In practice a hash function with $l$ bits output should need $2^l$ queries before one can find a second preimage

This propriety prevents an attacker to substitute a message with another and get the same hash

*Intuition*

A hash collision happens when we have two different messages that have the same hash

**Why do we care about hash collisions?**

Since hashes are used to fastly verify a message integrity if two messages have the same hash then we can replace one with another => We can play with data

Now, we want to hash big files and big messages so $|\mathcal{M}| >> |\mathcal{T}|$ => It would appear that hash collisions are possible

Natural collisions are normal to happen and we consider them improbable if $\mathcal{T}$ is big enough ($\text{SHA256} \Rightarrow T =$ $\{0,1\}^{256}$)

Yet, we don't want hash collisions to be computable

We don't want an attacker to be able to craft collisions or find collisions given a message

**Attack game**

An adversary $\mathcal{A}$ outputs two messages $m_0 \neq m_1$

â€‹$\mathcal{A}$ wins the game if he finds $H(m_0) = H(m_1)$

His advantage is $Pr[\text{Adversary finds a collision}]$â€‹

**Security**

A hash function $H$ is collision resistant if for all efficient and

explicitadversaries the advantage is negligible

Intuition

We know hash collisions exist (therefore an efficient adversary must exist) and that is easy to prove therefore we request an **explicit** algorithm that finds these collisions

This propriety makes it difficult for an attacker to find 2 input values with the same hash

There is a fundamental difference in how hard it is to break collision resistance and second-preimage resistance.

Breaking collision-resistance is like inviting more people into the room until the room

**contains 2 people with the same birthday**.Breaking second-preimage resistance is like inviting more people into the room until the room

**contains another person with your birthday**.

One of these fundamentally takes longer than the other

**Implications**

**Lemma 1**

Assuming a function $H$ is preimage resistant for every element of the range of $H$ is a

weakerassumption than assuming it is either collision resistant or second preimage resistant.

**Note**

Provisional implication

**Lemma 2**

Assuming a function is second preimage resistant is a

weakerassumption than assuming it is collision resistant.

â€‹

â€‹https://en.wikipedia.org/wiki/Cryptographic_hash_function - Wikipedia entry

â€‹https://www.youtube.com/watch?v=b4b8ktEV4Bg - Computerphile

â€‹https://www.cs.ucdavis.edu/~rogaway/papers/relates.pdf - Good read for more details

Figure 1 - Wikipedia