Is there any statistical analysis of the security of a "Brain Wallet"?

Question

Assume that a transaction is paid to a brain wallet address. (not P2SH, or M of N)

Is there any statistical approximation (or tiers of approximation) of how secure a Brain Wallet is to

• A key generated from pure random

• The non-random keys created by a broken random on Android pre 2013

• any other key generation technique

My goal is to get an approximation based on complexity relative to the other methods. A constructive thing that this would help with is what complexity is necessary for the Brain Wallet to have 2^64 bits of pure random?

Some additional question that may come of this line of thinking include:

• In a Brain Wallet, what is the relationship between complexity, length and the random output?

• What patterns are there to avoid? (vowel frequency, other letter statistical distribution)

• Is a Brain Wallet more secure if written in a foreign language that has more characters in it (more entropy)?

Much of this may be in a Cryptanalysis book, but I'm looking for someone who may have studied passwords in the past and can offer suggestions on how to brute force a Brain Wallet.

Answer 1

There are roughly 170,000 words in the english dictionary according to this link. If you combine 4 random words (170,000^4) , you get 8.3*10^20 potential combinations which is more than 2^64 (1.8*10^19). However 64 bits of security is generally considered weak these days. Bitcoin uses 256 bit keys (1.1*10^77 combinations) so if you want that level of security you need to use at least 15 words (2.8*10^78). Altering cases will obviously increase the potential combinations by a significant factor. Different languages have different number of words so that will change the calculations as well.

This calculation only applies if you choose the words completely randomly. People keep ignoring this important detail, and is the reason why brain wallets are not recommended. There are many cases of people getting their bitcoin cleaned out because they choose common phrases or include commonly used words in their private key.