A basic concept in mathematics is that of a set of elements. A set is usually denoted by accolade signs with the elements separated by commas.

For instance, the set of all integers is . The ellipses here mean that a certain pattern continues in a particular direction. So the set of all integers also includes and so on, as well as and so on. This set of all integers is typically denoted by .

Another example of a set is , or the set of all integers modulo 11. In contrast to the entire set , this set only contains a finite number of elements, namely .

A common mistake is to think that the set actually is . But this is not the case, given the way we defined the modulo operation earlier. Any negative integers reduced by modulo 11 wrap onto . For instance, the expression wraps around to , while the expression wraps around to .

Another basic concept in mathematics is that of a binary operation. This is any operation that takes two elements to produce a third. For instance, from basic arithmetic and algebra, you would be familiar with four fundamental binary operations: addition, subtraction, multiplication, and division.

These two basic mathematical concepts, sets and binary operations, are used to define the notion of a group, the most essential structure in abstract algebra.

Specifically, suppose some binary operation . In addition, suppose some set of elements S equipped with that operation. All “equipped” means here is that the operation can be performed between any two elements in the set S.

The combination is, then, a group if it meets four specific conditions, known as the group axioms.

Let's explore groups a little further. Denote the set of all integers by . This set combined with standard addition, or , clearly fits the definition of a group, as it meets the four axioms above.

Importantly, that the set of integers with addition constitutes a group does not mean that it constitutes a group with multiplication. You can verify this by testing against the four group axioms (where means standard multiplication).

The first two axioms obviously hold. In addition, under multiplication the element 1 can serve as the identity element. Any integer multiplied by 1, namely yields . However, does not meet the inverse condition. That is, there is not a unique element in for every in , so that .

For instance, suppose that . What value from the set multiplied with would yield the identity element 1? The value of would work, but this is not in the set . In fact, you run into this problem for any integer , other than the values of 1 and -1 (where would have to be 1 and -1 respectively).

If we allowed real numbers for our set, then our problems largely disappear. For any element in the set, multiplication by yields 1. As fractions are included in the set of real numbers, an inverse can be found for every real number. The exception is zero, as any multiplication with zero will never yield the identity element 1. Hence, the set of non-zero real numbers equipped with multiplication is indeed a group.

Some groups meet a fifth general condition, known as the commutativity condition. This condition is as follows:

Suppose a group with a set S and a binary operator . Suppose that and are elements of S. If it is the case that for any two elements and in S, then meets the commutativity condition.

Any group that meets the commutativity condition is known as a commutative group, or an Abelian group (after Niels Henrik Abel). It is easy to verify that both the set of real numbers over addition and the set of integers over addition are Abelian groups. The set of integers over multiplication is not a group at all, so ipso facto cannot be an Abelian group. The set of non-zero real numbers over multiplication, by contrast, is also an Abelian group.

You should heed two important conventions on notation. First, the signs “+” or “×” will frequently be employed to symbolize group operations, even when the elements are not, in fact, numbers. In these cases, you should not interpret these signs as standard arithmetic addition or multiplication. Instead, they are operations with only an abstract similarity to these arithmetic operations.

Unless you are specifically referring to arithmetic addition or multiplication, it is easier to use symbols such as and for group operations, as these do not have very culturally ingrained connotations.

Second, for the same reason that “+” and “×” are often used for indicating non-arithmetic operations, the identity elements of groups are frequently symbolized by “0” and “1”, even when the elements in these groups are not numbers. Unless you are referring to the identity element of a group with numbers, it is easier to use a more neutral symbol such as “” to indicate the identity element.

Many different and very important sets of values in mathematics equipped with certain binary operations are groups. Cryptographic applications, however, only work with sets of integers or at least elements that are described by integers, that is, within the domain of number theory. Hence, sets with real numbers other than integers are not employed in cryptographic applications.

Let's finish by providing an example of elements that can be “described by integers”, even though they are not integers. A good example is the points of elliptic curves. Though any point on an elliptic curve is clearly not an integer, such a point is indeed described by two integers.

Elliptic curves are, for instance, crucial to Bitcoin. Any standard Bitcoin private and public key pair is selected from the set of points that is defined by the following elliptic curve:

(which is the largest prime number less than ).

Transactions in Bitcoin typically involve locking outputs to one or more public keys in some way. The value from these transactions can, then, be unlocked making digital signatures with the corresponding private keys.