Squashed commit of the following:

commit f9aa66e
Author: Waylon Jepsen <[email protected]>
Date:   Tue May 7 09:00:35 2024 -0600

    curves in sage (#39)

commit 63ce62f
Author: Waylon Jepsen <[email protected]>
Date:   Mon May 6 17:46:22 2024 -0600

    fix: lock

commit d1c84eb
Author: Colin Roberts <[email protected]>
Date:   Mon May 6 16:39:25 2024 -0700

    feat: home-baked `FiniteField` trait (#38)

    * feat: new `FiniteField` trait

    Now everything compiles again. Will work to clean this all up and get all the tests to pass.

    * fix: `GF101` tests pass

    * fix: reimplement monty optimizations

    * clean: udeps

    ---------

    Co-authored-by: Waylon Jepsen <[email protected]>

commit 96c8b66
Merge: fb27e5f 62a9a57
Author: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>
Date:   Mon May 6 23:26:56 2024 +0000

    Merge pull request #42 from pluto/dependabot/cargo/anyhow-1.0.83

    Bump anyhow from 1.0.82 to 1.0.83

commit 62a9a57
Author: dependabot[bot] <49699333+dependabot[bot]@users.noreply.github.com>
Date:   Mon May 6 23:26:43 2024 +0000

    Bump anyhow from 1.0.82 to 1.0.83

    Bumps [anyhow](https://github.com/dtolnay/anyhow) from 1.0.82 to 1.0.83.
    - [Release notes](https://github.com/dtolnay/anyhow/releases)
    - [Commits](dtolnay/anyhow@1.0.82...1.0.83)

    ---
    updated-dependencies:
    - dependency-name: anyhow
      dependency-type: direct:production
      update-type: version-update:semver-patch
    ...

    Signed-off-by: dependabot[bot] <[email protected]>

math/curve.sage

@@ -0,0 +1,61 @@
+# our prime modulus
+F101 = IntegerModRing(101)
+
+# A number 5 in our prime modulus, should be 5
+print(IntegerMod(F101, 5))
+
+# Should be 96
+print(IntegerMod(F101, -5))
+
+# should be 81
+print(IntegerMod(F101, 1/5))
+
+# should be 20
+print(IntegerMod(F101, -1/5))
+
+# should be 100
+print(IntegerMod(F101, -1))
+
+# Lets make our elliptic curve
+E = EllipticCurve(F101, [0, 3])
+
+# lets print out the points, notice they print (x,y,z) the difference between homogenious points and affine points is that to use affine you just divide x,y by z. 
+# We can see here that for all points in the curve group z = 1 except the zero point at infinity. So for this field they are the same
+print(E.points())
+
+# Define polynomial ring
+R.<X> = PolynomialRing(F101)
+
+# Lets make an extension field
+# niavely: we could pick x^2 + 1 but
+# x^2 + 1 = x^2 + 100 = (x+10)(x-10) -> There is a root in the field
+# lets pick x^2 + 2 which is irreducible in our field
+
+# Extended polynomial ring
+K.<X> = GF(101**2, modulus = x^2 + 2)
+
+# Curve group over polynomial ring
+E2 = EllipticCurve(K, [0, 3])
+print(E2.points())
+
+# G1 is the generator for E1
+G1 = E(1,2)
+print(G1)
+
+# N is the order of the group E1    
+N = 17
+
+# G2 is the generator for E2
+G2 = E2([36, 31 *X])
+print(G2)
+
+# Now Lets generate the structured refrence string (SRS), 
+# we will use the "random" number 2 for the example but in practice it should be strong random. 
+# a circuit with n gates requires an SRS with at least
+# n + 5 elements as below
+# We will let it be of length 9, pythagorean triple uses 4 gates
+g1SRS = [(2**i)*G1 for i in range(7)]
+print(g1SRS)
+
+g2SRS = [(2**i)*G2 for i in range(2)]
+print(g2SRS)

src/field/gf_101.rs

@@ -26,13 +26,8 @@ impl fmt::Display for GF101 {
   fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { write!(f, "{}", self.value) }
 }
 
-impl From<u32> for GF101 {
-  fn from(val: u32) -> Self { Self::new(val) }
-}
-
 impl GF101 {
-  // pub const fn new(value: u32) -> Self { Self { value: to_monty(value) } }
-  pub const fn new(value: u32) -> Self { Self { value: value % Self::ORDER } }
+  pub const fn new(value: u32) -> Self { Self { value: to_monty(value) } }
 }
 
 impl FiniteField for GF101 {
@@ -119,8 +114,7 @@ impl SubAssign for GF101 {
 impl Mul for GF101 {
   type Output = Self;
 
-  // fn mul(self, rhs: Self) -> Self { Self { value: from_monty(self.value * rhs.value) } }
-  fn mul(self, rhs: Self) -> Self::Output { Self::new(self.value * rhs.value) }
+  fn mul(self, rhs: Self) -> Self { Self { value: from_monty(self.value * rhs.value) } }
 }
 
 impl MulAssign for GF101 {
@@ -428,18 +422,6 @@ mod tests {
 
   #[test]
   fn primitive_root_of_unity() {
-    let n = 2;
-    let omega = GF101::primitive_root_of_unity(n);
-    println!("omega: {:?}", omega);
-    assert_eq!(omega, F::new(95));
-    let omega_n = omega.pow(n);
-    for i in 1..n {
-      let omega_i = omega.pow(i);
-      println!("omega^{}: {:?}", i, omega_i);
-      assert_ne!(omega_i, F::new(1));
-    }
-    assert_eq!(omega_n, F::new(1));
-
     let n = 5;
     let omega = GF101::primitive_root_of_unity(n);
     println!("omega: {:?}", omega);

src/field/mod.rs

@@ -25,7 +25,6 @@ pub trait FiniteField:
   + DivAssign
   + Neg<Output = Self>
   + Rem<Output = Self>
-  + From<u32>
   + Hash
   + 'static {
   type Storage: From<u32>