Created: 󰃭 2023-03-29
Updated: 󰃭 2023-12-15

The birthday paradox is a surprising statistical phenomenon that shows how even in a small group, it’s very likely that two people share the same birthday.

In this blog post, we’ll explore the math behind the paradox and its practical applications in computer science and cryptography.

As a flagship example, it states that in a group of just 23 people, there is a 50% change that two of them will share the same birthday.

This probability, which seems quite high, may come as a surprise to many, however it can be easily explained by the principles of probability.

The paradox is a classical example of how probability can lead to seemingly counterintuitive conclusions and is a great illustration of how probability can be used in everyday life.

### Probability

The probability of the Birthday Paradox is computed by considering the number of possible pairs of people in a group and the probability that any of these people will have different birthdays.

To calculate the probability that any two people in a group of n will have different birthdays, we can use the formula:

p(different birthdays) = 365/365 ⋅ 364/365 ⋅ ... ⋅ (365-n+1)/365
= 365! / [(365-n)!⋅365ⁿ]


To find the probability that at least two people in the group will have the same birthday, we can take 1 minus the probability that they all have different birthdays

p(same birthday) = 1 - p(different birthdays)


Some results:

n p(n)
10 0.117
20 0.411
23 0.507
40 0.891
70 0.999

## Generalized Problem

Given a year of d days we want to determine the smaller number n such that the probability of a birthday coincidence is at least 50%.

In other words n is the minimal integer such that:

1 - (1-1/d)⋅(1-2/d)..(1-(n-1)/d) ≥ 1/2


Which is equivalent to:

(1-1/d)⋅(1-2/d)..(1-(n-1)/d) ≤ 1/2


As an approximation, ∏ (1 - i/d) ≈ 1/2 for:

n ≈ ⌈√(2·d·ln(2))⌉


This approximation is derived using the Taylor expansion of the exponential function and is valid for large d.

Proof

∏ (1 - i/d) ≈ exp[ ln( ∏ (1 - i/d) ) ] = exp[ ∑ ln(1 - i/d) ]


Taylor expansion for ln(1 - x) around x = 0 which is ln(1-x) ≈ -x.

∑ ln(1- i/d) = -∑ i/d = -n·(n-1)/(2·d)

→ ∏ (1 - i/d) ≈ exp[ -n·(n-1)/(2·d) ]


Equate to 1/2 which is the target value

exp[ -n·(n-1)/(2·d) ] ≈ 1/2
→ -n·(n-1)/(2·d) ≈ ln(1/2) = -ln(2)
→ n²-n ≈ 2·d·ln(2)


For large enough n we can omit the subtraction

→ n² ≈ 2·d·ln(2)
→ n ≈ √(2·d·ln(2))


Note that the classical birthday problem is an instance of this problem with d = 365 and gives as a result n = 23.

### Collision Probability

Given n items drawn from a set of d elements, we look for the probability p(n) that at least two numbers are equal.

The generic formula is derived using the same argument given in the previous section:

p(n) = 1 - (1-1/d)⋅(1-2/d)..(1-(n-1)/d)


Conversely, n(p) denotes the number of items drawn from a set of d elements to obtain a probability p that at least two numbers are the same, then:

n(p) ≈ √(2·d·ln(1/(1-p)))


The proof of this formula is similar to the one given in the previous section for p = 1/2.

When applied to hash functions this is the expected number of N-bit hashes that can be generated before getting a collision with probability p.

Surprisingly, for p = 1/2 this is not O(2ᴺ), but rather only O(√(2ᴺ)).

There are a number of attacks to crypto systems which leverages this fact.

## Attack

The birthday paradox can be particularly insidious if the hash is used as a primitive building block for a more complex scheme which heavily relies on the collision resistance property of some hash function.

For example, if an attacker discovers two messages m₁ and m₂ such that H(m₁) = H(m₂), he can submit m₁ to the victim in order to have it signed, thus obtaining a signature for H(m₁), but this is a valid signature for H(m₂) as well.

If the number of possible outputs of H is 2ᴺ, a technique to find a collision is:

• Construct k = 2^(N/2) variants of a legit message m₁.
• Use the same technique to construct k variants of a malicious message m₂.
• Construct two sets of digests: A = {H(m₁ᵢ)}, B = {H(m₂ᵢ)}.
• Search for any item in A ∩ B.

The variants can be constructed with a technique which generates messages with changes which are not detectable from a typical renderer. For example insert the same number of spaces and backspaces after a word.

By inserting 100 spaces around the message the attacker can construct 2^100 different variants.

The probability of a collision obviously depends on k.

Can be proven that

Pr(A ∩ B ≠ ∅) ≥ 1/2 if k ≥ 2^(N/2)


The number of the elements in each set corresponds to the expected number of elements to withdraw from a set of 2ᴺ before observing a collision with probability 1/2.

As |A ∪ B| ≥ 2^(N/2) then the probability of finding a collision is still ≈ 1/2 (it doesn’t double). Thus, there is a good chance (≥ 1/2) to find a collision for two elements which belong to different sets.

### Mitigations

To reduce birthday attack risk we have to choose a value for N that is sufficiently large.

Hashes like MD5 (N=128) or sha1 (N=160) are not considered secure anymore.

Another way to counter this attack is to use a MAC (a keyed hash) which bounds the output also to a secret key k.

In this way an attacker cannot pre-compute offline a table of collisions, as they do not know the secret key used in the particular context.

As a final note, always keep in mind that for pigeon-hole principle if the hash possible outputs are k, then after k+1 extractions we’re going to have a collision with probability 1.

## References

• Simple PoC in Rust here