Unhackable Wallet Security: The Implausibility of Randomly Guessing a 12-Word Seed Phrase

The security of modern cryptocurrency wallets is a result of years of cryptographic research and development. One crucial aspect of this security is the seed phrase, a set of words generated by the wallet that allows for the recovery of the private key. This article explores the mathematical improbability of randomly guessing a 12-word seed phrase and highlights why the chances are almost zero.

Understanding Seed Phrases

Over a decade ago, crypto developers agreed that a 24-word seed phrase would serve as the gold standard for unhackable private keys. Initially, the primary concern was ensuring that these seed phrases were sufficiently unique and secure against brute-force attacks. It was determined that the number of possible seed phrases with 24 words would be astronomically large, making it practically impossible to guess the correct one.

Mathematical Impossible Chances of Random Guessing

The likelihood of randomly guessing a 12-word seed phrase is so minuscule that it can be effectively considered zero. Let's break down the mathematics behind this:

1. Dictionary Words and Probability: According to the English dictionary, approximately 470,000 words are available. Assuming 200,000 of these words are suitable for seed phrases, the probability of selecting the correct first word is 1 in 200,000. The probability of selecting the correct second word is 1 in 199,999.

2. Multiplying the Chances: To calculate the probability of getting the first three words right, we multiply the probabilities as follows:

[ frac{1}{200000} times frac{1}{199999} times frac{1}{199998} approx 5 times 10^{-30} ]

Even by the third word, the probability is already 1 in 7.999999 Quadrillion. Each additional word adds another 5 zeros to the denominator, making the probability astronomically small.

3. Comparison with Everyday Odds: For comparison, the odds of winning a grand lottery might be around 1 in 10 to 100 million. The odds we just calculated are already 6 orders of magnitude beyond this, making the chance of randomly guessing a 12-word seed phrase effectively zero.

Practical Considerations and Limits

For a more practical perspective, we can calculate the number of possible 12-word seed phrases. There are approximately (2^{128}) possible valid 12-word seed phrases. Assuming an upper limit of (2^{12}) (1.099 trillion) used seed phrases, the odds of guessing the correct one in a single attempt are:

[ frac{2^{128}}{2^{12}} 1 text{ in } 2^{116} approx 1 text{ in } 7.999999999999999 times 10^{34} ]

Even if we could guess 1.05 million seed phrases per second, it would still take an impossibly long time to cover all possibilities:

[ frac{2^{116}}{1,050,000 text{ guesses/sec}} approx 7.999999999999999 times 10^{34} text{ seconds} ]

To put this in perspective, this time frame is far beyond the human timescale, considering the age of the universe is approximately 13.8 billion years (or (4.35 times 10^{17}) seconds).

Conclusion

The chances of randomly guessing a 12-word seed phrase are so negligible that they can be effectively considered zero. The mathematical improbability ensures that seed phrases provide a high level of security for cryptocurrency wallets. Understanding and using a seed phrase correctly is crucial for maintaining the security of your digital assets.