Um, huh? There are 2^1000 1000-character passwords, not 2^4700. Where is the 4700 coming from?
(added after realizing the above was super wrong): Whoops, that’s what I get for looking at comments first thing in the morning. log2(26^1000) = 4700 Still, the following bit stands:
I’d also like to register that, in my opinion, if it turns out that your comment is wrong and not my original statement, it’s really bad manners to have said it so confidently.
(I’m now not sure if you made an error or if I did, though)
Update: I think you’re actually totally right. The entropy gives a lower bound for the average, not the average itself. I’ll update the post shortly.
As for why I was confident, well, this was a clear (in my eye at least) example of Jensen’s inequality: we are comparing the mean of the log to the log of the mean. And, if you see this inequality come up often, you know that the inequality is always strict, except for a constant distribution. That’s how I knew.
As a last note, I must praise the fact that you left your original comments (while editing them of course) instead of removing them. I respect you a lot for that.
Um, huh? There are 2^1000 1000-character passwords, not 2^4700. Where is the 4700 coming from?(added after realizing the above was super wrong): Whoops, that’s what I get for looking at comments first thing in the morning. log2(26^1000) = 4700 Still, the following bit stands:
I’d also like to register that, in my opinion, if it turns out that your comment is wrong and not my original statement, it’s really bad manners to have said it so confidently.
(I’m now not sure if you made an error or if I did, though)
Update: I think you’re actually totally right. The entropy gives a lower bound for the average, not the average itself. I’ll update the post shortly.
I apologize, my wording was indeed rude.
As for why I was confident, well, this was a clear (in my eye at least) example of Jensen’s inequality: we are comparing the mean of the log to the log of the mean. And, if you see this inequality come up often, you know that the inequality is always strict, except for a constant distribution. That’s how I knew.
As a last note, I must praise the fact that you left your original comments (while editing them of course) instead of removing them. I respect you a lot for that.