Simple Password Entropy
Passwords are easy to generate, easy to record.. but still nobody likes 45-character passwords!
I like to use diceware to generate legible, secure passwords. Generally, each word (out of 7,776) will add ~12~13 bits of entropy. This requires 5 six-sided dice to be rolled for each word.
6 words is plenty, good enough for just about anything. 4 words is enough for maybe your Netflix password1.
Entropies
I’ll use hand-rolled examples to show how weird the words can be. Base64 sprinkled in liberally. Don’t use these passwords, obviously2.
| Pattern | bits of entropy |
|---|---|
| MotocrossVariety-jj+G | |
| = 2·log2(65) + 3·8 | 49.85 bits |
| MotocrossVarietyGaveScroll | |
| = 4·log2(65) | 51.70 bits |
| MotocrossVarietyGave-jj+G | |
| = 3·log2(65) + 3·8 | 62.77 bits |
| MotocrossVarietyGaveScrollFilter | |
| = 5·log2(65) | 64.62 bits |
| MotocrossVariety-reckET89 | |
| = 2·log2(65) + 6·8 | 73.85 bits |
| MotocrossVarietyGaveScroll-jj+G | |
| = 4·log2(65) + 3·8 | 75.70 bits |
| MotocrossVarietyGaveScrollFilterUncombed | |
| = 6·log2(65) | 77.55 bits |
| MotocrossVarietyGave-reckET89 | |
| = 3·log2(65) + 6·8 | 86.77 bits |
| MotocrossVarietyGave-MO+c1RK+jxdF | |
| = 3·log2(65) + 9·8 (base64)[^3] | 110.77 bits |
So a little base64 goes a long way. Too bad it’s hard to write down accurately, let alone remember.
Equivalent entropies
Some more common styles of passwords, for comparison (in increasing entropy):
| Pattern | equation | bits of entropy |
|---|---|---|
| 1234-56-7890 | log2(10,000,000,000) | 33.22 bits |
| wCEHMbIs | 6·8 | 48 bits |
| abcdefghijklm | 13·log2(26) | 61.11 bits |
| H65j/aS5vfmm | 9·8 | 72 bits |
| 0mE07rdje4xzvxUE | 12·8 | 96 bits |
| aT7bubJTM4w2RoyeNPsQ | 15·8 | 120 bits |
64 bits indeed seems “long”.
A lot of bad password policies (6 chars only, 8 chars only..) keep you around 30-40 bits.
Favorites
I like MotocrossVarietyGaveScroll-jj+G (75.70 bits) for long passwords, and MotocrossVariety-jj+G (49.85 bits) for shorter ones. I like to be able to write them down more than I like them super short.
Generating these requires 2-4 dicerolls (& a diceware textfile lookup) and a quick openssl rand -base64 3.
64 bits is IMO “pretty good”. If you need 256 bits, you’re not really talking about a password any more, but a key. You can generate just lots more bits to come out of openssl, if that’s what you’re after
$ openssl rand -base64 32
bIxkkyQAh3GCxbiWurRpV5DsMdvcTTiTyMBWk2lpGcE=
Update
I don’t use a lot of long passwords, but looking back: MotocrossVarietyGaveScroll-jj+G (75.70 bits) is just super long. 31 characters!? I wouldn’t even try to remember those.
Why not go straight base64 at that point? 21 bytes gets you 28 characters. 18 bytes gets you 24 characters. That’s 144-168 bits of entropy. Way beyond normal stuff. Here’s a better “long” password:
If you can use a 20-character password, use 15 random bytes (120 bits) :
$ openssl rand -base64 15
aT7bubJTM4w2RoyeNPsQ # results should vary..
This is way overkill, but a 20-character password is quite long and might as well be used.
You might need to add a special character (pick your favorite..) and maybe a number. I guess, technically, you might land all 1s or something crazy, so maybe append a Az1! or something to fulfill all those nice password requirements. You may have to trim some bits to make it fit, if 24 characters is too many.