Authors: Zademn, ireland Reviewed by:

**Prerequisites**

Probability theory (for the main idea)

Hashes (an application)

**Motivation**

Breaking a hash function (insert story)

*insert story / introduction about why it's called a paradox + use*

**Question 1**

What is the probability that 1 person has the same birthday as you?

*Solution*

Let $A$ be the event that someone has the same birthday as you and $\bar{A}$ be the complementary event

The events are mutually exclusive => $Pr(A) = 1 - Pr(\bar{A})$

Let $E_i$ be the events that person $i$ does not have your birthday

Then

$Pr(A) = 1 - Pr(\bar{A}) = 1 - \prod_{i=1}^n Pr(E_i) = 1 - \left( \dfrac {364} {365}\right)^n$

Reminder: $Pr(A, B) = Pr(A) \cdot Pr(B)$if $A, B$are independent events

**Question 2**

What is the probability that 2 out of $n$ people in a room share the same birthday?

**Suppose the birthdays are distributed independently and uniformly**

*Solution*

Let $A$ be the event that 2 people have the same birthday, let $\bar{A}$ be the complementary event (no 2 people have the same birthday)

Event 1 = Person 1 is born => $Pr(E_1) = \dfrac {365} {365}$

Event 2 = Person 2 is born on a different day than Person 1 => $Pr(E_2) = \dfrac {364} {365}$

$\vdots$

Event n = Person n is born on a different day than Person $1,...,n-1 \Rightarrow$ $\Rightarrow Pr(E_n) = \dfrac {365-(n-1)} {365}$

$Pr(\bar{A}) = Pr(E1) \cdot Pr(E_2) \cdot \dots \cdot Pr(E_n) = \dfrac {365} {365} \cdot \dfrac {364} {365} \cdot \dots \cdot \dfrac {365-(n-1)} {365} = \left( \dfrac {1} {365} \right) ^{n} \cdot \dfrac {365!} {(365-n)!} = \prod_{i=1}^{n-1} \left(1 - \dfrac i {365}\right)$****

**Question 1**

Instead of $365$ days we have $d \Rightarrow \boxed{1 - \left( \dfrac {d-1} {d}\right)^n}$

**Question 2**

Instead of $365$ days we have $d \Rightarrow \boxed{1 - \prod_{i=1}^{n-1} \left(1 - \dfrac i {d}\right)}$

**Code examples**

def my_birthday(n, d):return 1 - pow((d-1)/d , n)def same_birthday(n, d):p = 1for i in range(1, n): #1 -> n-1p*=(1-i/d)return 1 - pprint(same_birthday(23, 365), same_birthday(32, 365), same_birthday(100, 365))# (0.5072972343239854, 0.7533475278503207, 0.9999996927510721)

From the Taylor approximation we know $e^x = 1 + x + \dfrac {x^2} {2!} + \dots => e_x\approx 1 + x$ for $x \ll 1$ Apply for each event: $\Rightarrow x = -\dfrac a d \Rightarrow e^{ -a /d} \approx 1- \dfrac a d \Rightarrow Pr(A) = 1 - \prod_{i=1}^{n-1}e^{-i/d} = 1-e^{-n(n-1) /{2d}} \approx 1-\boxed{e^{-{n^2} / {2d}}}$

If we want to solve for $n$ knowing $Pr(A)$ we take the $\ln$ => $\boxed{n \approx \sqrt{2d \ln \left(\dfrac 1 {1-Pr(A)}\right)}}$

def approx_same_birthday(n, d):return 1 - pow(e, -pow(n, 2) / (2*d)).n()print(approx_same_birthday(23, 365))print(approx_same_birthday(32, 365))print(approx_same_birthday(100, 365))# 0.515509538061517# 0.754077719532824# 0.999998876014983

Let $H:\mathcal{M} \longrightarrow \mathcal{T}$ be a hash function with $|\mathcal{M}| >> |T|$

Let's denote $N = |\mathcal{T}|$

**Algorithm**

1. Choose $s \approx \sqrt{N}$ random distinct messages in $\mathcal{M}$

2. Compute $t_i = H(m_i)$ for $1\leq i \leq \sqrt{N}$

3. Look for $(t_i = t_j) \to$ If not found go to step 1

**Example: **

Consider the following hash function:

import hashlibimport randomfrom Crypto.Util.number import long_to_bytes, bytes_to_longdef small_hash(m, hash_bits):'''Argumentsm: bytes -- inputhash_bits: int -- number of bits of the hashReturns:{bytes} - hash of the message of dimension `hash_bits`'''assert hash_bits > 0, "no negative number of bits"mask = (1 << hash_bits) - 1 # Mask of bitst = hashlib.sha256(m).digest() # the hash in bytest = bytes_to_long(t)t = t & mask # get the last 12 bitsreturn long_to_bytes(t)

We make the following function to find the hashes:

def small_hash_colision(M_bits, T_bits):'''ArgumentsM_bits: int -- dimension of the message spaceT_bits: int -- dimension of the hash spaceReturns:{(bytes, bytes, bytes)} -- message1, message2, hashor{(b'', b'', b'')} -- if no collision found'''N = 1<<T_bitsprint('Hash size: ', N)# This is `s`num_samples = 1 * isqrt(N)num_samples += num_samples//5 + 1 # num_samples = 1.2 * isqrt(N) + 1print(f'Making a list of {num_samples} hashes')print(f'Probability of finding a collision is {same_birthday(num_samples, N)}')m_list = []t_list = []# Make a list of hashesfor i in range(num_samples):m = random.getrandbits(M_bits) # Get random message#m = random.randint(0, M_bits-1)m = long_to_bytes(m)t = small_hash(m, T_bits) # Hash itif m not in m_list:m_list.append(m)t_list.append(t)# Check for collisionnfor i in range(len(t_list)):for j in range(i+1, len(t_list)):if t_list[i] == t_list[j]:print('Collision found!')return m_list[i], m_list[j], t_list[i]else:print('Collision not found :(')return b"", b"", b""

We use it as follows:

M_bits = 30T_bits = 20m1, m2, t = small_hash_colision(M_bits, T_bits)print(m1, m2, t)print(small_hash(m1, T_bits) == small_hash(m2, T_bits))# Hash size: 1048576# Making a list of 1229 hashes# Probability of finding a collision is 0.513213460854798# Collision found!# b'!\xafB\xc5' b'4F\xb6w' b'\x01y\xf7'# True

While the above algorithm works to find a hash collision in time $O(\sqrt N )$, it also requires storing $O(\sqrt N )$hash values. As such, it represents a classic time-space tradeoff over the naive approach, which involves randomly selecting pairs of inputs until they hash to the same value. While the naive approach does not require any additional storage, it does have runtime $O(N )$.

However, there is a better approach combining the best of both worlds: constant storage requirements and $O(\sqrt N)$runtime. This approach is based on Pollard's Rho algorithm, which is better-known for its application to solving discrete logarithms. The core insight behind the algorithm is that by the Birthday Paradox, we expect to encounter a hash collision after trying $O(\sqrt N)$random inputs. However, it is possible to detect whether a collision has occurred without needing to store all of the inputs and hashes if the inputs are chosen in a clever way.

This is done by choosing the next input based on the hash of the previous input, according to the following sequence:

$x_0 = g(seed) \\
x_1 = g(H(x_0)) \\
x_2 = g(H(x_1)) \\
\dots \\
x_{n+1} = g(H(x_n)) \\$

Where $H:\mathcal{M} \longrightarrow \mathcal{T}$ is our hash function and $g: \mathcal{T} \longrightarrow \mathcal{M}$ is a "sufficiently random" function which takes a hash value and produces a new. We define the composition of the functions to be$f = H \circ g : \mathcal{T} \longrightarrow \mathcal{T}$.

# TODO:# have the two hash functions used in this chapter be the samefrom Crypto.Hash import SHA256from Crypto.Util.number import bytes_to_long, long_to_bytesfrom tqdm.autonotebook import tqdm# bits in hash outputn = 30# Hdef my_hash(x):x = bytes(x, 'ascii')h = SHA256.new()h.update(x)y = h.digest()y = bytes_to_long(y)y = y % (1<<n)y = int(y)return y# gdef get_message(r):x = "Crypto{" + str(r) + "}"return x# fdef f(r):return my_hash(get_message(r))

By the Birthday Paradox, we expect that the sequence will have a collision (where $x_i = x_j$ for two distinct values $i,j$) after $O(\sqrt N)$values. But once this occurs, then the sequence will begin to cycle, because $x_{i+1} = g(H(x_i)) = g(H(x_j)) = x_{j+1}$.

Therefore, we can detect that a collision has occurred by using standard cycle-detection algorithms, such as Floyd's tortoise and hare!

And finally, we can locate the first place in the sequence where the collision occurred, which will let us determine what the colliding inputs to the hash function are. This is done by determining how many iterations apart the colliding inputs are, and then stepping together one iteration at a time until the collision occurs.

For Floyd's tortoise and hare, this is done by noting that when we found our collision after $n$ iterations, we were comparing $H(x_{n}) = H(x_{2 n})$. And because the sequence is cyclic, if the first colliding input is $x_i$, then it collides with$H(x_{i}) = H(x_{i+n})$. So we define the new sequence $y_i = x_{n+i}$, and step through the $x_i$ and $y_i$ sequences together until we find our collision!

This is implemented in the following code snippet.

"""initializationThis routine will find a hash collision in sqrt time with constant space."""seed = 0x0 = get_message(seed)x = x0y = f(x)"""detect collisionusing Floyd / tortoise and hare cycle finding algorithmexpected number of iterations is ~ sqrt(pi/2) * 2^(n/2),we run for up to 4 * 2^(n/2) iterations"""for i in tqdm(range(4 << (n//2))):if f(x) == f(y):breakx = f(x)y = f(y)y = f(y)"""locate collision"""x = x0y = f(y)for j in tqdm(range(i)):if f(x) == f(y):breakx = f(x)y = f(y)m1 = get_message(x)m2 = get_message(y)assert my_hash(m1) == f(x)assert my_hash(m2) == f(y)print("[+] seeds for messages: {}, {}".format(x, y))print("[+] messages: {}, {}".format(m1, m2))print("[+] collided hashes of messages: {}, {}".format(my_hash(m1), my_hash(m2)))# 31% 40391/131072 [00:03<00:08, 10666.20it/s]# 30% 12032/40391 [00:00<00:02, 13112.15it/s]## [+] seeds for messages: 404842900, 254017312# [+] messages: Crypto{404842900}, Crypto{254017312}# [+] collided hashes of messages: 1022927209, 1022927209

Finally, there is a third algorithm for finding hash collisions in $O(\sqrt N)$time, namely the van Oorschot-Wiener algorithm based on Distinguished Points. While it does have additional storage requirements over Pollard's Rho, the main advantage of this algorithm is that it parallelizes extremely well, achieving $O(\frac{\sqrt N}{m})$runtime when run on $m$ processors.

https://en.wikipedia.org/wiki/Birthday_problem - wiki entry

https://en.wikipedia.org/wiki/Birthday_attack - wiki entry

https://www.youtube.com/watch?v=ofTb57aZHZs - vsauce2 video