As a preface, forgive me for some of the links being from Wikipedia. I realize that academia frowns upon this.
I came across this article about "nothing up my sleeve numbers". In it, it says:
In cryptography, nothing up my sleeve numbers are any numbers which, by their construction, are above suspicion of hidden properties. [...] Such numbers can be viewed as the opposite extreme of Chaitin–Kolmogorov random numbers in that they appear random but have very low information entropy.
In looking for an example of one of these numbers, I came across this Crypto Beta Q&A, where the answerer says:
[M]athematical constants like binary expansions of irrational numbers like 2√ (or roots of other numbers), e, π can be used, to show that one didn't select the numbers to create a back door.
This is what Wikipedia has to say about information entropy:
Entropy is typically measured in bits, nats, or bans. Shannon entropy is the average unpredictability in a random variable, which is equivalent to its information content. Shannon entropy provides an absolute limit on the best possible lossless encoding or compression of any communication, assuming that the communication may be represented as a sequence of independent and identically distributed random variables.
Based on this, I don't see why these numbers (or any possible number) can have low entropy. It seems like all numbers have equal likelihood of being generated as a "random variable" as long as that number is within the generated range (e.g., if the number is 1564631 [assumed to be one of these kinds of numbers], and I'm looking for number between 1 and 2000000, it has an equal likelihood of being generated, regardless).
Can anyone explain this concept? I looked at the definition of entropy, but I admit that the math is a little over my head. I'm not sure how someone would use a number without these properties to "create a backdoor."