Abelian Groups

Commutative groups, those groups in which operand order does not change an equation's result, form Abelian groups that commute: "7 × 3 = 3 × 7". When this condition is not satisfied, we say the expression is non-commutative.

Cyclic groups are a special case of commutative Abelian groups—sets that are monogenous—generated by a single element—and invertible with a single operation.

Consider a set that, if we iterated over every other element with a particular operation, we'd be able to derive all of the elements of the set.

For a finite cyclic group, let G be the group, n be the size of the set, and e be the identity element, such that gⁱ = g^j whenever i ≡ j (mod n); like so.

G = {e, g, g², ... , gⁿ⁻¹}

The commutative property also holds over the additive group of ℤ, or the integers, which are isomorphic to any infinite cyclic group. Similarly, the additive group of ℤ/nℤ, integers modulo n, is isomorphic to the finite cyclic group of order n.

Since all cyclic groups commute, they are all abelian groups, and all finitely produced abelian groups are the direct products of cyclic groups.

For example, the powers of 10 form an infinite subset G = {…, 0.001, 0.01, 0.1, 1, 10, 100, 1000, …} over the rational numbers.

With 10 as a generator, set G is a multiplicative cyclic group. For any element a of the group, one can derive log₁₀ a.

Our set contains 10 and 100. The product \(10^1 \cdot 10^2\) is equivalent to \(10^{1+2} = 1000\). Every cyclic group G is Abelian because if \({\displaystyle x}, {\displaystyle y}\) are in \({\displaystyle G}\), then:

\( {\displaystyle xy=a^{m}a^{n}=a^{m+n}=a^{n}a^{m}=yx}\)

This homomorphic property is relevant in cryptography. It's also useful for computing commitments. For example, we can perform operations to verify information, like so. Let m be a message and r be a random value:

\( {\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=C(m_{1}+m_{2},r_{1}+r_{2})}\)

That is to say, with this special property, one could compute and verify the sums of values without knowing the actual values being committed.