feat: basic polynomial arithmetic (#48)

* feat: basic polynomial arithmetic

* fix: remove polynomial const generics

math/field.sage

@@ -1,4 +1,4 @@
-# Define a finite field, our `PlutoField` which is
+# Define a finite field, our `GF101` which is 
 # a field of 101 elements (101 being prime).
 F = GF(101)
 print(F)
@@ -39,6 +39,26 @@ omega_n = primitive_element ^ quotient
 print("The", n, "th root of unity is: ", omega_n)
 
 # Check that this is actually a root of unity:
+assert omega_n^n == 1
+print(omega_n, "^", n, " = ", omega_n^n)
+######################################################################
+
+######################################################################
+# Let's find a mth root of unity (for l = 2)
+# First, check that m divides 101 - 1 = 100
+l = 2
+assert (101 - 1) % l == 0
+quotient = (101 - 1) // l
+print("The quotient is: ", quotient)
+
+# Find a primitive root of unity using the formula:
+# omega = primitive_element^quotient
+omega_l = primitive_element^quotient
+print("The ", l, "th root of unity is: ", omega_l)
+
+# Check that this is actually a root of unity:
+assert omega_l^l == 1
+print(omega_l, "^", l, " = ", omega_l^l)
 assert omega_n ^ n == 1
 print(omega_n, "^", n, " = ", omega_n ^ n)
 ######################################################################

math/polynomial.sage

@@ -0,0 +1,36 @@
+# Define a finite field, our `GF101` which is 
+# a field of 101 elements (101 being prime).
+F = GF(101)
+
+# Define the polynomial ring over GF(101)
+R.<x> = PolynomialRing(F)
+
+# Create the polynomials
+a = 4*x^3 + 3*x^2 + 2*x^1 + 1
+b = 9*x^4 + 8*x^3 + 7*x^2 + 6*x^1 + 5
+
+# Perform polynomial division
+q_ab, r_ab = a.quo_rem(b)
+
+# Print the results
+print("For a / b :")
+print("Quotient:", q_ab)
+print("Remainder:", r_ab)
+
+# Perform polynomial division
+q_ba, r_ba = b.quo_rem(a)
+
+# Print the results
+print("\nFor b / a :")
+print("Quotient:", q_ba)
+print("Remainder:", r_ba)
+
+# Polynomials easy to do by hand
+p = x^2 + 2*x + 1
+q = x + 1
+
+# Perform polynomial division
+quot, rem = p.quo_rem(q)
+print("\nFor p / q :")
+print("Quotient:", quot)
+print("Remainder:", rem)

src/field/mod.rs

@@ -4,6 +4,8 @@ use std::{
   ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Rem, Sub, SubAssign},
 };
 
+use super::*;
+
 pub mod gf_101;
 pub mod gf_101_2;
 
@@ -74,10 +76,10 @@ pub trait FiniteField:
   // - There exists a multiplicative subgroup generated by a primitive element 'a'.
   //
   // According to the Sylow theorems (https://en.wikipedia.org/wiki/Sylow_theorems):
-  // A non-trivial multiplicative subgroup of prime order 'q' exists if and only if
-  // 'p - 1' is divisible by 'q'.
-  // The primitive q-th root of unity 'w' is defined as: w = a^((p - 1) / q),
-  // and the roots of unity are generated by 'w', such that {w^i | i in [0, q - 1]}.
+  // A non-trivial multiplicative subgroup of prime order 'n' exists if and only if
+  // 'p - 1' is divisible by 'n'.
+  // The primitive n-th root of unity 'w' is defined as: w = a^((p - 1) / n),
+  // and the roots of unity are generated by 'w', such that {w^i | i in [0, n - 1]}.
   fn primitive_root_of_unity(n: Self::Storage) -> Self {
     let p_minus_one = Self::ORDER - Self::Storage::from(1);
     assert!(p_minus_one % n == 0.into(), "n must divide p - 1");

src/lib.rs

@@ -12,3 +12,10 @@ pub mod curves;
 pub mod field;
 pub mod kzg;
 pub mod polynomial;
+
+use core::{
+  fmt,
+  hash::Hash,
+  iter::{Product, Sum},
+  ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Rem, Sub, SubAssign},
+};