Analysis of birthday paradox bounds & Generalization Birthday Paradox Overview: Addresses likelihood of shared birthdays in a group. Applies to various scenarios, including cryptography. Where and what collisions are Probability of collisions in cryptographic hash functions. Collisions: Two inputs producing the same hash output. Implications for security and system integrity. Probability Approximations:​ Probability approximations are ways to estimate the chance of something happening without doing complex calculations. In the case of the birthday paradox, there are formulas that approximate when the probability of a collision becomes significant. Simple Approximation: P = (N^2) / (2 * M): This is a formula used to estimate the probability of a collision in the context of hash functions. "P" is the probability, "N" is the number of possible hash values, and "M" is the number of different inputs (or messages) being hashed.

1. Analysis of birthday paradox
bounds
&
Generalization
[U-5:6.2]
Mohammed Abdul Lateef
Cryptanalysis
• 22011DA802
• M.tech CFIS

2. 1. Reentrancy
2. Unchecked External​
3. Call Integer Overflow​
4. Uninitialized State Variable​
5. Access Control​
Smart contract vulnerabilities​​

3. Smart
Contract
Vulnerabilities
Blockchain platforms and smart contracts are vulnerable to
security breaches.
Security breaches of smart contracts have led to huge
financial losses in terms of cryptocurrencies and tokens

4. Birthday Paradox Overview:
1.Addresses likelihood of shared birthdays in a group.
2.Applies to various scenarios, including cryptography.
Where and what collisions are
1.Probability of collisions in cryptographic hash functions.
2.Collisions: Two inputs producing the same hash output.
3.Implications for security and system integrity.

5. Birthday Paradox:
The birthday paradox is about how likely it is for two or more people in a
group to share the same birthday.
Applies to Various Scenarios, Including Cryptography:
This concept applies to different situations, and one of them is
cryptography. It helps us understand how likely it is for two different
inputs to produce the same output in a cryptographic hash function.
Collisions: Two Inputs Producing the Same Hash Output:
In cryptography, a collision happens when two different pieces of
information create the same output after being processed by hash
function.

6. Probability Approximations:
• Probability approximations are ways to estimate the chance of
something happening without doing complex calculations.
In the case of the birthday paradox, there are formulas that
approximate when the probability of a collision becomes significant.
Simple Approximation: P = (N^2) / (2 * M):
This is a formula used to estimate the probability of a collision in the
context of hash functions.
• "P" is the probability, "N" is the number of possible hash values, and
• "M" is the number of different inputs (or messages) being hashed.

7. Generalizations
In cryptographic
scenarios, generalizing
the analysis beyond basic
collisions is valuable.
Addressing collisions
between different sets.
Uniform Statistical
distributions
•Collisions Between
Different Sets
Multicollisions
Non - Uniform
Statistical
distributions

8. Collisions Between Different Sets​
• Focusing on collisions between distinct subsets drawn from a larger set.
1. Subset Characteristics:
• Two separate subsets are considered.
• First subset: N1 elements.
• Second subset: N2 elements.
• Larger set: N total elements
2. No Collisions Within Subsets:
• No collisions within individual subsets.
• Focus on collisions between 1st and 2nd subsets.
3. Estimating Expected Collisions:
• Formula introduced to estimate expected collisions between
subsets.
• Estimation: (N1 * N2) / N.
• Formula captures possible collision pairs.
• The formula reflects the number of possible pairs that can lead to
collisions.
Uniform Statistical Distribution: In a
uniform distribution, all outcomes are
equally likely, like drawing candies from a
bag with each candy having the same
chance of being picked.

9. Multi-collisions:
Multiple Elements
Sharing Values
• Involves multiple elements having the same value.
Subcase Distinctions:
• Analysis of multi-collisions has two main subcases,
similar to collision analysis.
• First Subcase: Within a single subset, finding
different elements with the same value.
• Second Subcase: Across distinct subsets,
identifying an element shared by all.
Alternatively, with L subsets, finding an
element common to all. This is called an L-
multicollision.
Non-Uniform Statistical
Distribution: In a non-
uniform distribution,
outcomes have varying
likelihoods, where some
are more probable than
others, as if certain candies
are more likely to be drawn
from the bag.

10. The expected number of L -
multicollisions in a subset of size N
chosen among M elements is
Multi Collision- [Contd.]

11. Thank you