How likely is it that Bitcoin wallet private keys are duplicated?
The possibility of duplication of Bitcoin wallet private keys is extremely low and almost negligible. Here are some relevant technical details and calculations to help understand how low the odds are:
1. Length of Bitcoin private key:
âą The Bitcoin private key is a 256-bit random number.
âą This means that the number of possible private keys is (2^{256}), which is approximately equal to (1.1579 \times 10^{77}).
2. Number of Bitcoin addresses:
âą A Bitcoin address is generated by performing a series of cryptographic hash operations on a private key.
âą Theoretically, the number of Bitcoin addresses is also (2^{160}), which is approximately equal to (1.46 \times 10^{48}).
3. Probability calculation:
âą To calculate the probability of two randomly generated private keys colliding (i.e. generating the same private key), we can use the principle of birthday attack. Assuming we randomly generate ( n ) private keys, the probability of collision ( P ) can be approximately calculated by the following formula:
P \approx 1 - e^{-\frac{n^2}{2N}}
where ( N ) is the number of possible private keys (i.e. ( 2^{256} )).
âą Since ( N ) is very large, the probability of generating two identical private keys is almost zero. Specifically, even if trillions of private keys are generated, the probability of collision is still negligible.
4. Specific values:
âą The collision probability of generating ( 10^{18} ) private keys can be calculated as:
P \approx 1 - e^{-\frac{(10^{18})^2}{2 \times 2^{256}}}
Since 2^{256} is approximately equal to 1.1579 \times 10^{77} , this probability is almost zero.
5. Metaphor:
âą The probability of generating the same Bitcoin private key is approximately equivalent to finding two identical grains of sand on Earth.
âą Or, equivalent to the probability that if you pick an atom number at random, you pick two identical atoms.
In short, due to the extremely large number of Bitcoin private keys, the probability of generating two identical private keys is almost zero. The Bitcoin network is designed to be extremely secure, making private key collisions almost impossible to occur in reality.
