# Fields

A fundamental component of many cryptographic protocols is the algebraic structure known
as a [field]. Fields are sets of objects (usually numbers) with two associated binary
operators $+$ and $\times$ such that various [field axioms][field-axioms] hold. The real
numbers $\mathbb{R}$ are an example of a field with an uncountably infinite number of
elements.

[field]: https://en.wikipedia.org/wiki/Field_(mathematics)
[field-axioms]: https://en.wikipedia.org/wiki/Field_(mathematics)#Classic_definition

Halo makes use of _finite fields_ which have a finite number of elements. Finite fields
are fully classified as follows:
- if $\mathbb{F}$ is a finite field, it contains $|\mathbb{F}| = p^k$ elements  for some
  integer $k \geq 1$ and some prime $p$;
- any two finite fields with the same number of elements are isomorphic. In particular,
  all of the arithmetic in a prime field $\mathbb{F}_p$ is isomorphic to addition and
  multiplication of integers modulo $p$, i.e. in $\mathbb{Z}_p$. This is why we often
  refer to $p$ as the _modulus_.

We'll write a field as $\mathbb{F}_q$ where $q = p^k$. The prime $p$ is called its
_characteristic_. In the cases where $k \gt 1$ the field $\mathbb{F}_q$ is a $k$-degree
extension of the field $\mathbb{F}_p$. (By analogy, the complex numbers
$\mathbb{C} = \mathbb{R}(i)$ are an extension of the real numbers.) However, in Halo we do
not care about extension fields. Whenever we write $\mathbb{F}_p$ we are referring to what
we call a _prime field_ which has a prime $p$ number of elements, i.e. $k = 1$.

Important notes:
* There are two special elements in any field: $\mathcal{O}$, the additive identity, and
  $1$, the multiplicative identity.
* The least significant bit of a field element, when represented as an integer in binary
  format, can be interpreted as its "sign" to help distinguish it from its additive
  inverse (negation). This is because for some nonzero element $a$ which has a least
  significant bit $0$ we have that $-a = p - a$ has a least significant bit $1$, and vice
  versa. We could also use whether or not an element is larger than $(p - 1) / 2$ to give
  it a "sign."

## Inverses and groups

Any non-zero element $a \in \mathbb{F}_p$ has a _multiplicative inverse_ $b = a^{-1}$,
which is the _unique_ element $b$ such that $ab = 1$.

A quick way of obtaining the inverse is $a^{-1} = a^{p - 2}$. The reason for this stems
from [Fermat's little theorem][fermat-little], which states that $a^p = a \pmod p$ for any
integer $a$. If $a$ is nonzero, we can divide by $a$ twice to get $a^{p-2} = a^{-1}.$

[fermat-little]: https://en.wikipedia.org/wiki/Fermat%27s_little_theorem
     
However, it may be more intuitive to understand the set of nonzero elements of
$\mathbb{F}_p$ as a [group], where the group operation is given by multiplication on the
field. We use the notation $\mathbb{F}_p^\times$ for the multiplicative group over the set
$\mathbb{F}_p - \{\mathcal{O}\}$. Groups are simpler and more limited than fields; they
have only _one_ operator $\cdot$ and fewer axioms.

[group]: https://en.wikipedia.org/wiki/Group_(mathematics)

> #### (aside) Additive vs multiplicative notation 
> If $\cdot$ is written as $+$ and the identity as $0$ or $\mathcal{O}$, then we say the
> group is "written additively". If $\cdot$ is written as $\times$ or omitted (i.e.
> $a \cdot b$ written as $ab$) and the identity as $1$, then we say it is "written
> multiplicatively". It's conventional to use additive notation for
> [elliptic curve groups](curves.md), and multiplicative notation when (as in this case)
> the elements come from a finite field. When additive notation is used, we also write
>
> $$[k] A = \underbrace{A + A + \cdots + A}_{k \text{ times}}$$
>
> for nonnegative $k$ and call this "scalar multiplication"; we also often use uppercase
> letters for variables denoting group elements. When multiplicative notation is used, we
> also write
>
> $$a^k = \underbrace{a \times a \times \cdots \times a}_{k \text{ times}}$$
>
> and call this "exponentiation". In either case we call the scalar $k$ such that
> $[k] g = a$ or $g^k = a$ the "discrete logarithm" of $a$ to base $g$. We can extend
> scalars to negative integers by inversion, i.e. $[-k] A + [k] A = \mathcal{O}$ or
> $a^{-k} \times a^k = 1$.

The _order_ of a group element $a$ is defined (in multiplicative notation) as the smallest
positive integer $k$ such that $a^k = 1$.

Groups always have [generators] which are elements that, when the group operation is
applied repeatedly with the same element some number of times, produce every other element
of the group. That is, a generator has maximal order, which we also call the order of the
group.

[generators]: https://en.wikipedia.org/wiki/Generating_set_of_a_group

There can be many different generators. Let's assume that $\alpha$ is a generator of
$\mathbb{F}_p^\times$, so it has order $p-1$ (equal to the number of elements in
$\mathbb{F}_p^\times$). Therefore, for any element in $a \in \mathbb{F}_p^\times$ there is
a unique integer $i \in \{0..p-2\}$ such that $a = \alpha^i$.

Notice that $a \times b$ where $a, b \in \mathbb{F}_p^\times$ can really be interpreted as
$\alpha^i \times \alpha^j$ where $a = \alpha^i$ and $b = \alpha^j$. Indeed, it holds that
$\alpha^i \times \alpha^j = \alpha^{i + j}$ for all $0 \leq i, j \lt p - 1$. As a result
the multiplication of nonzero field elements can be interpreted as addition modulo $p - 1$
with respect to some fixed generator $\alpha$. The addition just happens "in the exponent."

This is another way to look at where $a^{p - 2}$ comes from for computing inverses in the
field:

$$p - 2 \equiv -1 \pmod{p - 1},$$

so $a^{p - 2} = a^{-1}$.

### Montgomery's Trick

Montgomery's trick, named after Peter Montgomery (RIP) is a way to compute many field
inversions at the same time. Imagine we need to compute the inverses of three nonzero
elements $a, b, c \in \mathbb{F}_p$. Instead, we'll compute the products $x = ab$ and
$y = xc = abc$, and compute the inversion

$$z = y^{p - 2} = \frac{1}{abc}.$$

We can now multiply $z$ by $x$ to obtain $\frac{1}{c}$ and multiply $z$ by $c$ to obtain
$\frac{1}{ab}$, which we can then multiply by $a, b$ to obtain their respective inverses.

This technique generalizes to arbitrary numbers of field elements with just a single
inversion necessary.

## Multiplicative subgroups

A _subgroup_ of a group $G$ with operation $\cdot$, is a subset of elements of $G$ that
also form a group under $\cdot$.

In the previous section we said that $\alpha$ is a generator of the $p - 1$ order
multiplicative group $\mathbb{F}_p^\times$. This is a _composite_ order group, and so by
the Chinese remainder theorem[^chinese-remainder] it has strict subgroups. As an example
let's imagine that $p = 11$, and so $p - 1$ factors into $5 \cdot 2$. Thus, there is a
generator $\beta$ of the $5$-order subgroup and a generator $\gamma$ of the $2$-order
subgroup. All elements in $\mathbb{F}_p^\times$, therefore, can be written uniquely as
$\beta^i \cdot \gamma^j$ for some $i$ (modulo $5$) and some $j$ (modulo $2$).

If we have $a = \beta^i \cdot \gamma^j$ notice what happens when we compute

$$
a^5 = (\beta^i \cdot \gamma^j)^5
    = \beta^{i \cdot 5} \cdot \gamma^{j \cdot 5}
    = \beta^0 \cdot \gamma^{j \cdot 5}
    = \gamma^{j \cdot 5};
$$

we have effectively "killed" the $5$-order subgroup component, producing a value in the
$2$-order subgroup.

[Lagrange's theorem (group theory)][lagrange-group] states that the order of any subgroup
$H$ of a finite group $G$ divides the order of $G$. Therefore, the order of any subgroup
of $\mathbb{F}_p^\times$ must divide $p-1.$

[lagrange-group]: https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)

## Square roots

In a field $\mathbb{F}_p$ exactly half of all nonzero elements are squares; the remainder
are non-squares or "quadratic non-residues". In order to see why, consider an $\alpha$
that generates the $2$-order multiplicative subgroup of $\mathbb{F}_p^\times$ (this always
exists because $p - 1$ is divisible by $2$ since $p$ is prime) and $\beta$ that generates
the $t$-order multiplicative subgroup of $\mathbb{F}_p^\times$ where $p - 1 = 2t$. Then
every element $a \in \mathbb{F}_p^\times$ can be written uniquely as
$\alpha^i \cdot \beta^j$ with $i \in \mathbb{Z}_2$ and $j \in \mathbb{Z}_t$. Half of all
elements will have $i = 0$ and the other half will have $i = 1$.

Let's consider the simple case where $p \equiv 3 \pmod{4}$ and so $t$ is odd (if $t$ is
even, then $p - 1$ would be divisible by $4$, which contradicts $p$ being $3 \pmod{4}$).
If $a \in \mathbb{F}_p^\times$ is a square, then there must exist
$b = \alpha^i \cdot \beta^j$ such that $b^2 = a$. But this means that

$$a = (\alpha^i \cdot \beta^j)^2 = \alpha^{2i} \cdot \beta^{2j} = \beta^{2j}.$$

In other words, all squares in this particular field do not generate the $2$-order
multiplicative subgroup, and so since half of the elements generate the $2$-order subgroup
then at most half of the elements are square. In fact exactly half of the elements are
square (since squaring each nonsquare element gives a unique square). This means we can
assume all squares can be written as $\beta^m$ for some $m$, and therefore finding the
square root is a matter of exponentiating by $2^{-1} \pmod{t}$.

In the event that $p \equiv 1 \pmod{4}$ then things get more complicated because
$2^{-1} \pmod{t}$ does not exist. Let's write $p - 1$ as $2^k \cdot t$ with $t$ odd. The
case $k = 0$ is impossible, and the case $k = 1$ is what we already described, so consider
$k \geq 2$. $\alpha$ generates a $2^k$-order multiplicative subgroup and $\beta$ generates
the odd $t$-order multiplicative subgroup. Then every element $a \in \mathbb{F}_p^\times$
can be written as $\alpha^i \cdot \beta^j$ for $i \in \mathbb{Z}_{2^k}$ and
$j \in \mathbb{Z}_t$. If the element is a square, then there exists some $b = \sqrt{a}$
which can be written $b = \alpha^{i'} \cdot \beta^{j'}$ for $i' \in \mathbb{Z}_{2^k}$ and
$j' \in \mathbb{Z}_t$. This means that $a = b^2 = \alpha^{2i'} \cdot \beta^{2j'}$,
therefore we have $i \equiv 2i' \pmod{2^k}$, and $j \equiv 2j' \pmod{t}$. $i$ would have
to be even in this case because otherwise it would be impossible to have
$i \equiv 2i' \pmod{2^k}$ for any $i'$. In the case that $a$ is not a square, then $i$ is
odd, and so half of all elements are squares.

In order to compute the square root, we can first raise the element
$a = \alpha^i \cdot  \beta^j$ to the power $t$ to "kill" the $t$-order component, giving

$$a^t = \alpha^{it \pmod 2^k} \cdot \beta^{jt \pmod t} = \alpha^{it \pmod 2^k}$$

and then raise this result to the power $t^{-1} \pmod{2^k}$ to undo the effect of the
original exponentiation on the $2^k$-order component:

$$(\alpha^{it \bmod 2^k})^{t^{-1} \pmod{2^k}} = \alpha^i$$

(since $t$ is relatively prime to $2^k$). This leaves bare the $\alpha^i$ value which we
can trivially handle. We can similarly kill the $2^k$-order component to obtain
$\beta^{j \cdot 2^{-1} \pmod{t}}$, and put the values together to obtain the square root.

It turns out that in the cases $k = 2, 3$ there are simpler algorithms that merge several
of these exponentiations together for efficiency. For other values of $k$, the only known
way is to manually extract $i$ by squaring until you obtain the identity for every single
bit of $i$. This is the essence of the [Tonelli-Shanks square root algorithm][ts-sqrt] and
describes the general strategy. (There is another square root algorithm that uses
quadratic extension fields, but it doesn't pay off in efficiency until the prime becomes
quite large.)

> TODO: describe more recent algorithms, e.g. Bernstein's table-based method and
> [eprint 2020/1407](https://eprint.iacr.org/2020/1407).

[ts-sqrt]: https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm

## Roots of unity

In the previous sections we wrote $p - 1 = 2^k \cdot t$ with $t$ odd, and stated that an
element $\alpha \in \mathbb{F}_p^\times$ generated the $2^k$-order subgroup. For
convenience, let's denote $n := 2^k.$ The elements $\{1, \alpha, \alpha^2, \alpha^{n-1}\}$
are known as the $n$th [roots of unity](https://en.wikipedia.org/wiki/Root_of_unity).

The **primitive root of unity**, $\omega,$ is an $n$th root of unity such that
$\omega^i \neq 1$ except when $i \equiv 0 \pmod{n}$.

Important notes:
- if $\alpha$ is an $n$th root of unity, $\alpha$ satisfies $\alpha^n - 1 = 0.$ If
  $\alpha \neq 1,$ then
  $$1 + \alpha + \alpha^2 + \cdots + \alpha^{n-1} = 0.$$
- equivalently, the roots of unity are solutions to the equation
  $$X^n - 1 = (X - 1)(X - \alpha)(X - \alpha^2) \cdots (X - \alpha^{n-1}).$$
- **$\boxed{\omega^{\frac{n}{2}+i} =  -\omega^i}$ ("Negation lemma")**. Proof:
  $$
  \begin{aligned}
  \omega^n = 1 &\implies \omega^n - 1 = 0 \\
  &\implies (\omega^{n/2} + 1)(\omega^{n/2} - 1) = 0.
  \end{aligned}
  $$
  Since the order of $\omega$ is $n$, $\omega^{n/2} \neq 1.$ Therefore, $\omega^{n/2} = -1.$

- **$\boxed{(\omega^{\frac{n}{2}+i})^2 =  (\omega^i)^2}$ ("Halving lemma")**. Proof:
  $$
  (\omega^{\frac{n}{2}+i})^2 = \omega^{n + 2i} = \omega^{n} \cdot \omega^{2i} = \omega^{2i} = (\omega^i)^2.
  $$
  In other words, if we square each element in the $n$th roots of unity, we would get back
  only half the elements, $\{(\omega_n^i)^2\} = \{\omega_{n/2}\}$ (i.e. the $n/2$th roots
  of unity). There is a two-to-one mapping between the elements and their squares.

## References
[^chinese-remainder]: Friedman, R. (n.d.) "Cyclic Groups and Elementary Number Theory II" (p. 5). http://www.math.columbia.edu/~rf/numbertheory2.pdf