The level of randomness is crucial for hiding information. The higher the level of randomness, the better. The design of cryptosystem is aiming at constructing crypto codes with “perfect avalanche” and good randomness.
Using optimal normal basis (or polynomial basis) math that looks like scrambled bits over an elliptic curve produce a fundamentally nonlinear output. A single bit change in the input will cause a nonlinear and very drastic change in the output.
Well, while symmetric crypto designers are working very hard on this, elliptic curves do it automatically 
Do you mean chaos by randomness? That is, the possibility of a small change in initial conditions creating a large difference in the outcomes, even though every step of the process is deterministic.
Or is there actually a way to have true randomness in cryptography? I know next to nothing about it, so again please pardon the stupid questions.
We say a sequence of numbers or bits is random if there is no causal relationship or corellations between any pair of numbers or bits in the sequence.
Chaos dynamic has the ability to produce random-like sequence, even though every step of the process is deterministic (as tommi said), and because of that, there is always correlation between any pair of numbers or bits which is produced by chaos dynamics.
Most crypto systems we used today are deterministic system so they cannot produce truly random sequence. They can only produce pseudorandom sequence.
“A single bit change in the input will cause a nonlinear and very drastic change in the output” is only a necessary condition for a system to produce sequence with good random-like-property.
So, a sequence is random if there is no way to compress it into shorter form? So, 232323232323…23 is not very random, because it is possible to just say “repeat 23″ n times. Is this similar to the definition you are using?
the randomness that i’m referring here is more about the process of generating the sequences, not about the pattern contained in the sequence.
or did i get the wrong idea?
@ chika,
I agree. The term random apply to a sequence only when we use the crypto system to generate the sequence.
If we refer to the processs: A good cryptosystem is just like a random permutation which maps an input which belong to a set to an output which belong to the same set.
Checking to see if I got it right this time: The point of randomness is to make it hard to see how close a guess was; if the function would be, say, continuous, it would be fairly easy to see if a guess is close to the actual input, but with a near-random system it is pretty hard.
Tommi: yes, you get it right. a near-random system makes it harder to make a guess based on “how close” the input to the output produced