Number theory, the oldest branch of mathematics, can be found in the early history of cryptography and number theory continues to have an important role in the subject. Since these first days, the integers have been used to represent the symbols of a message, and the operations of arithmetic have been used to combine these numbers with a cryptographic key to hide the information that the numbers represent. Modern cryptographic systems depend on number theory in a much deeper way by using difficult or unsolved problems of number theory, and other branches of mathematics, to try to hide information. In turn, the adversarial cryptanalyst often attacks those cryptosystems by using deep theorems of mathematics to try to break a cryptosystem and recover the hidden information.

Basic number theory

The set of positive and negative integers (0, ±1, ±2, …}, denoted Z, is closed under the operation of addition, which is an operation that is familiar and has many familiar properties. Integer addition is commutative, meaning that a + b = b + a. Integer addition is associative, meaning that the sum a + b + c can be executed from either side. There is an identity element under integer addition, namely the special integer called zero, and the operation of addition has an inverse operation called subtraction.

The set of integers Z is an early example of a structure called a group.