The Birthday Attack
by aoum, Apr 3, 2025, 10:47 PM
The Birthday Attack: A Cryptographic Vulnerability
1. Introduction
The Birthday Attack is a well-known method in cryptography that exploits the Birthday Paradox to find hash collisions much faster than brute force. The attack is particularly relevant in digital security, where cryptographic hash functions are used for password hashing, digital signatures, and message integrity verification.
2. The Birthday Paradox: A Probability Insight
The birthday problem asks:
What is the probability that at least two people in a room of
people share the same birthday?
Surprisingly, with just
people, the probability exceeds 
, and with 
people, it surpasses 
.
To compute this probability, we assume there are people and 
possible birthdays (ignoring leap years). The probability that no two people share a birthday is:
The complementary probability (at least one shared birthday) is then:
Approximating this using exponentials, we get:
3. The Birthday Attack in Cryptography
Cryptographic hash functions, such as SHA-256, map arbitrary-length inputs to fixed-size outputs. A good hash function should be collision-resistant, meaning it should be computationally infeasible to find two different inputs
and 
such that:
However, the birthday attack leverages the birthday paradox to find such collisions in approximately
attempts, where 
is the number of possible hash values.
4. The Mathematics of the Attack
If a hash function produces
-bit outputs, there are 
possible hash values. The probability of a collision after generating random hashes is:
Setting
and solving for , we find that a collision is likely after about:
This means that instead of the full brute-force complexity of
, the birthday attack reduces the complexity to approximately 
.
5. Implications for Cryptography
The birthday attack has serious consequences for hash security. For example:
6. Countermeasures
To mitigate birthday attacks, cryptographic protocols use:
7. Conclusion
The birthday attack exploits the surprising probability result of the birthday paradox to find hash collisions efficiently. This attack highlights the importance of strong cryptographic design, ensuring that hash functions remain resistant to such vulnerabilities.
References
1. Introduction
The Birthday Attack is a well-known method in cryptography that exploits the Birthday Paradox to find hash collisions much faster than brute force. The attack is particularly relevant in digital security, where cryptographic hash functions are used for password hashing, digital signatures, and message integrity verification.
Comparison of the birthday problem (1) and birthday attack (2):
In (1), collisions are found within one set, in this case, 3 out of 276 pairings of the 24 lunar astronauts.
In (2), collisions are found between two sets, in this case, 1 out of 256 pairings of only the first bytes of SHA-256 hashes of 16 variants each of benign and malicious contracts.
In (1), collisions are found within one set, in this case, 3 out of 276 pairings of the 24 lunar astronauts.
In (2), collisions are found between two sets, in this case, 1 out of 256 pairings of only the first bytes of SHA-256 hashes of 16 variants each of benign and malicious contracts.
2. The Birthday Paradox: A Probability Insight
The birthday problem asks:
What is the probability that at least two people in a room of
people share the same birthday?Surprisingly, with just
people, the probability exceeds 
, and with 
people, it surpasses 
.To compute this probability, we assume there are
people and 
possible birthdays (ignoring leap years). The probability that no two people share a birthday is:
The complementary probability (at least one shared birthday) is then:
Approximating this using exponentials, we get:
3. The Birthday Attack in Cryptography
Cryptographic hash functions, such as SHA-256, map arbitrary-length inputs to fixed-size outputs. A good hash function should be collision-resistant, meaning it should be computationally infeasible to find two different inputs
and 
such that:
However, the birthday attack leverages the birthday paradox to find such collisions in approximately
attempts, where 
is the number of possible hash values.4. The Mathematics of the Attack
If a hash function produces
-bit outputs, there are 
possible hash values. The probability of a collision after generating random hashes is:
Setting
and solving for , we find that a collision is likely after about:
This means that instead of the full brute-force complexity of
, the birthday attack reduces the complexity to approximately 
.5. Implications for Cryptography
The birthday attack has serious consequences for hash security. For example:
- A 128-bit hash function can be attacked in
steps, which is computationally feasible with modern hardware. - SHA-1 (160-bit) was broken due to a birthday attack, leading to its deprecation in favor of SHA-256.
- To resist birthday attacks, cryptographic hashes now use at least 256-bit outputs, ensuring
complexity, which is infeasible.
steps, which is computationally feasible with modern hardware.
complexity, which is infeasible.6. Countermeasures
To mitigate birthday attacks, cryptographic protocols use:
- Longer hash outputs: SHA-256, SHA-3, and BLAKE2 offer 256-bit security.
- Salting: Randomizing hash inputs prevents precomputed collision attacks.
- HMAC (Hash-based Message Authentication Codes): Ensures hash integrity even under collision attacks.
7. Conclusion
The birthday attack exploits the surprising probability result of the birthday paradox to find hash collisions efficiently. This attack highlights the importance of strong cryptographic design, ensuring that hash functions remain resistant to such vulnerabilities.
References
- Menezes, A., van Oorschot, P., & Vanstone, S. Handbook of Applied Cryptography.
- AoPS Wiki: Birthday Paradox
- NIST Cryptographic Standards: https://csrc.nist.gov/
April 2025
March 2025