Details

Numeral systems are optimal for encoding sequences of symbols (digits) that have uniform distribution.

An asymmetric numeral system is a generalization constructed to be optimal for any given probability distribution of symbols (digits).

With some information stored in a natural number , to add some information stored in a digit that all has the same probability (), take to place this information in the right-most bits.

In the asymmetric case should hold asymptotically.

This cannot usually be done exactly, but for asymptotic behavior the symbols only need to be distributed uniformly. A random number generator can be used to choose a specific distribution. If this generator is initialized with a given key, this additionally encrypts the output.

To encode in this way, would grow to infinity. To prevent that, is restricted to some range and some bits are removed in blocks of "width" bits to a bit stream (compressed file).

Basic information about this coding can be found in:

http://arxiv.org/pdf/0710.3861

More information about using it in cryptography:

http://arxiv.org/abs/0902.0271