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benwr.net
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It’s useful to place this within a larger class of “information collapse” errors.

In general, averages can lead you astray. The fix might be including the standard deviation, or (as here) a min or max statistic, but any time you replace the distribution itself with lossy statistics, you risk errors.

Many cognitive biases and puzzles in probability essentially boil down to this.

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Rather than attempting to estimate the entropy of a scheme, wouldn’t it be better to just define a scheme that represents randomness directly in the encoding? If you target N bits of randomness, then you will not fall into these sorts of traps if you directly encode those bits in a symmetric number scheme, such as base16, base32, or diceware (base `~12.92`).

The math is sort of interesting, but I’m not sure why anyone would in practice want to do anything other than directly encoding randomness.

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I became interested in this because I was trying to develop highly memorable passwords. For example, you might want to choose a random grammatical English sentence as your passphrase. A natural way to do this might be to choose a random parse tree of a given size, but if you do that you’ll have some duplicated sentences (with ambiguous parses).

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There are also the EFF word lists, which are designed to be memorable, real, and chosen by dice roll. I particularly like the list where each word can be uniquely identified by its first three letters: https://www.eff.org/deeplinks/2016/07/new-wordlists-random-passphrases

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I’ve created rpass for this. It generates random mnemonics, ie. `rpass 128` yields:

`juthor kezrem xurvup kindit puxpem vaszun bok`

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Is it really English if the majority of the words are made up?

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You’re right, isn’t English, but “juthor” and “krezrem”, and, “puxpem” are pretty memorable non-words IMO.