Group Theory

The values within our circuits are elements of what is known as a finite field. To understand this, we need a bit of abstract algebra background, especially group theory.

Groups

A group is a set equipped with a binary operation that combines any two elements to form a third element, satisfying certain properties. A group is shown as $(G,\cdot )$ and consists of the following components:

• A set of elements $G$.
• A binary operation (denoted as $\cdot$) that takes two elements and produces a third element.

The operation must satisfy the following properties:

• Closure: For any two elements $a,b\in G$, the result of the operation is also in the group: $a\cdot b\in G$. It is said that the group $G$ is closed under its binary operation.

• Identity: There exists an element $e\in G$, called the identity element, such that for any element $a\in G$, the operation $a\cdot e=e\cdot a=a$.

• Inverse: For every element $a\in G$, there exists an element $b\in G$, called the inverse of a, such that $a\cdot b=b\cdot a=e$. The inverse of $a$ is denoted as $a^{-1}$.

• Associativity: For any three elements $a,b,c\in G$, the operation is associative, meaning $(a\cdot b)\cdot c=a\cdot (b\cdot c)$. This property ensures that the order of operations does not matter.

There is an additional property as well, called the commutative property or abelian property. A group is said to be Abelian if the binary operation is commutative, meaning $a\cdot b=b\cdot a$ for all elements $a,b\in G$.

If the group has a finite number of elements, it is called a finite group.

Operation Notation

For the binary operation, we can use the additive notation or multiplicative notation.

• Additive: $a\cdot b=a+b$
• Multiplicative: $a\cdot b=ab$

Examples

• The integers under addition $(\mathbb{Z},+)$.
• The integers modulo $n$ under addition $({\mathbb{Z}}_n,+)$.

Rings

A ring is a set equipped with two binary operations, addition and multiplication, that satisfy certain axioms. A ring $(R,+,\times )$ consists of the following components:

• A set of elements $R$.
• An addition operation (denoted as $+$) that takes two elements and produces a third element.
• A multiplication operation (denoted as $\times$) that takes two elements and produces a third element.

The operations must satisfy the following properties:

• Additive + Multiplicative Closure: For any two elements $a,b\in R$, the result of the addition is also in the ring: $a+b\in R$ and the result of the multiplication is also in the ring: $a\times b\in R$. The ring $R$ is closed under both addition and multiplication.

• Additive + Multiplicative Associativity: For any three elements $a,b,c\in R$, the addition and multiplication operations are associative, meaning $(a+b)+c=a+(b+c)$ and $(a\times b)\times c=a\times (b\times c)$. This property ensures that the order of operations does not matter.

• Additive Identity: There exists an element $0\in R$, called the additive identity, such that for any element $a\in R$, the addition $a+0=0+a=a$. Nothing is said about multiplication yet.

• Additive Inverse: For every element $a\in R$, there exists an element $-a\in R$, called the additive inverse of $a$, such that $a+(-a)=(-a)+a=0$. The inverse of $a$ is denoted as $-a$.

• Addition Commutativity: The addition operation is commutative, meaning $a+b=b+a$ for all elements $a,b\in R$.

• Distributivity: For any three elements $a,b,c\in R$, the ring satisfies the distributive property, meaning $a\times (b+c)=(a\times b)+(a\times c)$ and $(b+c)\times a=(b\times a)+(c\times a)$.

If the ring has a multiplicative identity, i.e., an element $e\in R$ such that $a\times e=e\times a=a$ for all $a\in R$, then the ring is called a ring with unity and that element $e$ is called a unity.

If the multiplication is commutative, then the ring is called a commutative ring.

If the ring has a finite number of elements, it is called a finite ring.

Examples

• The set $\mathbb{Z}$ of all integers, and is a commutative ring with unity.
• The set $\mathbb{Q}$ of all rational numbers.
• The set $\mathbb{R}$ of all real numbers.
• The set $\mathbb{C}$ of all complex numbers.

Fields

A field is a ring $(F,+,\times )$ with the following properties:

• $F$ is a commutative ring.
• There is a non-zero unity $e\in F$.
• Every non-zero element $a\in F$ have a multiplicative inverse $a^{-1}\in F$ such that $a\times a^{-1}=a^{-1}\times a=e$.

If the field has a finite number of elements, it is called a finite field. The ring of integers modulo $p$, denoted as ${\mathbb{Z}}_p$, where $p$ is a prime number, is a finite field. This one is particularly important in cryptography!

In Circom, when you choose a prime with -p or --prime option, you actually choose the order of the finite field that the circuit will be built upon. Here is quick snippet to see the order yourself within the circuit:

template OrderMinus1() {
    log("P - 1 = ", -1);
}

This will print a huge number on your screen, and that number equals $-1$ in the field; adding 1 to that gives you the order.