The mathematical branches that underlie modern cryptography are:
Complexity Theory
Information Theory
Numbers Theory
Probability
Abstract Algebra
Most ciphers and other cryptographic operations can be viewed mathematically as some combination of the following basic crypto primitives:
transposition
substitution
split/concatenation
bitoverbit operation
Complexity theory deals with measuring the necessary computational effort for questions involving a large number of entities, and where the number of entities determines the complexity. The results are used to assess how much effort will be required to compromise a cipher.
Information theory measures the content of information by its impact, and after all in cryptography we wish to deny our adversary the impact benefit of our messages.
Numbers theory deals with the properties of the natural numbers: 1,2,3,... It is useful because we believe that we can define numbertheoretical problems that have no mathematical short cuts to allow our adversary to steal our secrets.
probability is used to assess the likelihood of protecting our secret throughout the their lifetime.
Abstract algebra is used to acquire insight into complex problems via similar problems that were already solved.

