From 5abcd85f840d687c9a77a5759a318bc2c19f0680 Mon Sep 17 00:00:00 2001
From: wborgeaud <williamborgeaud@gmail.com>
Date: Mon, 10 May 2021 18:45:48 +0200
Subject: [PATCH] Started extension field implementation

---
 src/field/extension_field.rs | 91 ++++++++++++++++++++++++++++++++++++
 src/field/mod.rs             |  1 +
 2 files changed, 92 insertions(+)
 create mode 100644 src/field/extension_field.rs

diff --git a/src/field/extension_field.rs b/src/field/extension_field.rs
new file mode 100644
index 00000000..c88d9eb6
--- /dev/null
+++ b/src/field/extension_field.rs
@@ -0,0 +1,91 @@
+use crate::field::crandall_field::CrandallField;
+use crate::field::field::Field;
+use std::fmt::{Debug, Display, Formatter};
+
+pub trait QuarticFieldExtension: Field {
+    type BaseField: Field;
+
+    // Element W of BaseField, such that `X^4 - W` is irreducible over BaseField.
+    const W: Self::BaseField;
+
+    fn to_canonical_representation(&self) -> [Self::BaseField; 4];
+}
+
+pub struct QuarticCrandallField([CrandallField; 4]);
+
+impl QuarticFieldExtension for QuarticCrandallField {
+    type BaseField = CrandallField;
+    // Verifiable in Sage with
+    // ``R.<x> = GF(p)[]; assert (x^4 -3).is_irreducible()`.
+    const W: Self::BaseField = CrandallField::from_canonical_u64(3);
+
+    fn to_canonical_representation(&self) -> [Self::BaseField; 4] {
+        self.0
+    }
+}
+
+impl Field for QuarticCrandallField {
+    const ZERO: Self = Self([CrandallField::ZERO; 4]);
+    const ONE: Self = Self([
+        CrandallField::ONE,
+        CrandallField::ZERO,
+        CrandallField::ZERO,
+        CrandallField::ZERO,
+    ]);
+    const TWO: Self = Self([
+        CrandallField::TWO,
+        CrandallField::ZERO,
+        CrandallField::ZERO,
+        CrandallField::ZERO,
+    ]);
+    const NEG_ONE: Self = Self([
+        CrandallField::NEG_ONE,
+        CrandallField::ZERO,
+        CrandallField::ZERO,
+        CrandallField::ZERO,
+    ]);
+
+    // Does not fit in 64-bits.
+    const ORDER: u64 = 0;
+    const TWO_ADICITY: usize = 30;
+    const MULTIPLICATIVE_GROUP_GENERATOR: Self = Self([
+        CrandallField::from_canonical_u64(3),
+        CrandallField::ONE,
+        CrandallField::ZERO,
+        CrandallField::ZERO,
+    ]);
+    const POWER_OF_TWO_GENERATOR: Self = Self([
+        CrandallField::ZERO,
+        CrandallField::ZERO,
+        CrandallField::ZERO,
+        CrandallField::from_canonical_u64(14096607364803438105),
+    ]);
+
+    fn try_inverse(&self) -> Option<Self> {
+        todo!()
+    }
+
+    fn to_canonical_u64(&self) -> u64 {
+        todo!()
+    }
+
+    fn from_canonical_u64(n: u64) -> Self {
+        todo!()
+    }
+}
+
+impl Display for QuarticCrandallField {
+    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
+        write!(
+            f,
+            "{} + {}*a + {}*a^2 + {}*a^3",
+            self.0[0], self.0[1], self.0[2], self.0[3]
+        )
+    }
+}
+
+impl Debug for QuarticCrandallField {
+    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
+        Display::fmt(self, f)
+    }
+}
diff --git a/src/field/mod.rs b/src/field/mod.rs
index 6c2828a6..179fb10d 100644
--- a/src/field/mod.rs
+++ b/src/field/mod.rs
@@ -1,5 +1,6 @@
 pub(crate) mod cosets;
 pub mod crandall_field;
+pub mod extension_field;
 pub mod fft;
 pub mod field;
 pub(crate) mod lagrange;