The probability of a Twt Hash collision depends on the size of the hash and the number of possible values it can take. For the Twt Hash, which uses a Blake2b 256-bit hash, Base32 encoding, and takes the last 7 characters, the space of possible hash values is significantly reduced.
### Breakdown:
1. Base32 encoding: Each character in the Base32 encoding represents 5 bits of information (since \\( 2^5 = 32 \\)).
2. 7 characters: With 7 characters, the total number of possible hashes is:
\\[ 32^7 = 3,518,437,208 \\]
This gives about 3.5 billion possible hash values.
### Probability of Collision:
The probability of a hash collision depends on the number of hashes generated and can be estimated using the Birthday Paradox. The paradox tells us that collisions are more likely than expected when hashing a large number of items.
The approximate formula for the probability of at least one collision after generating
n hashes is:\\[ P(\\text{collision}) \\approx 1 - e^{-\\frac{n^2}{2M}}
\\]
Where:
- \\(n\\) is the number of generated Twt Hashes.
- \\(M = 32^7 = 3,518,437,208\\) is the total number of possible hash values.
For practical purposes, here are some example probabilities for different numbers of hashes (
n):- For 1,000 hashes:
\\[ P(\\text{collision}) \\approx 1 - e^{-\\frac{1000^2}{2 \\cdot 3,518,437,208}} \\approx 0.00014 \\, \\text{(0.014%)}
\\]
- For 10,000 hashes:
\\[ P(\\text{collision}) \\approx 1 - e^{-\\frac{10000^2}{2 \\cdot 3,518,437,208}} \\approx 0.14 \\, \\text{(14%)}
\\]
- For 100,000 hashes:
\\[ P(\\text{collision}) \\approx 1 - e^{-\\frac{100000^2}{2 \\cdot 3,518,437,208}} \\approx 0.999 \\, \\text{(99.9%)}
\\]
### Conclusion:
- For small to moderate numbers of hashes (up to around 1,000–10,000), the collision probability is quite low.
- However, as the number of Twts grows (above 100,000), the likelihood of a collision increases significantly due to the relatively small hash space (3.5 billion).=