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Introduction / overview
Authors: Zademn Reviewed by:

Introduction

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.

Definitions and Formalism

Definition - Hash
A hash is an efficient deterministic function that takes an arbitrary length input and produces a fixed length output (digest, hash). Let
H:MTH:\mathcal{M} \longrightarrow \mathcal{T}
be a function where
    M\mathcal{M}
    = message space
    T\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)H(a), H(b)
      They check if
      H(a)=H(b)H(a) = H(b)
Figure 1.
1
from hashlib import sha256
2
m1 = b"Some file"
3
m2 = b"Some file"
4
sha256(m1).digest() == sha256(m2).digest() # -> True
Copied!

Proprieties

    Preimage Image Resistance
    Second Preimage resistance
    Resistant to collisions

1. Preimage Resistance

The hash function must be a one way function. Given
tTt \in \mathcal{T}
find
mMm \in \mathcal{M}
s.t
H(m)=tH(m) = t
Intuition
It should be unfeasible to reverse a hash function (
O(2l)\mathcal{O}(2^l)
time where
ll
is the number of output bits)
This propriety prevents an attacker to find the original message from a hash

2. Second Preimage Resistance

Given
mm
it should be hard to find
mmm' \neq m
with
H(m)=H(m)H(m') = H(m)
Attack game
An adversary
A\mathcal{A}
is given a message
mm
and outputs a message
mmm' \neq m
.
A\mathcal{A}
wins the game if he finds
H(m)=H(m)H(m) = H(m')
His advantage is
Pr[A finds a second preimage]Pr[\mathcal{A} \text{ finds a second preimage}]
where
Pr()Pr(\cdot)
is a probability
Figure 2. Security game - second preimage resistance
    In practice a hash function with
    ll
    bits output should need
    2l2^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

3. Hash Collisions

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
    M>>T|\mathcal{M}| >> |\mathcal{T}|
    => It would appear that hash collisions are possible
    Natural collisions are normal to happen and we consider them improbable if
    T\mathcal{T}
    is big enough (
    SHA256T=\text{SHA256} \Rightarrow T =
    {0,1}256\{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

Let's throw some definitions

Attack game
An adversary
A\mathcal{A}
outputs two messages
m0m1m_0 \neq m_1
A\mathcal{A}
wins the game if he finds
H(m0)=H(m1)H(m_0) = H(m_1)
His advantage is
Pr[Adversary finds a collision]Pr[\text{Adversary finds a collision}]
Figure 3. Security game - Collision resistance
Security
A hash function
HH
is collision resistant if for all efficient and explicit adversaries 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

Difference from 2nd preimage

    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
HH
is preimage resistant for every element of the range of
HH
is a weaker assumption than assuming it is either collision resistant or second preimage resistant.
Note
Lemma 2
Assuming a function is second preimage resistant is a weaker assumption than assuming it is collision resistant.

Resources

Bibliography

    Figure 1 - Wikipedia
Last modified 5mo ago