mv ecdsa -> ecc, add ecdh

src/ecc/README.md

@@ -0,0 +1,26 @@
+# Elliptic Curve Cryptography
+
+Elliptic curve cryptography takes advantage of the intractability of the elliptic curve discrete logarithm problem.
+
+Let $E$ be an elliptic curve defined over a Galois (finite) field $\mathbb{F}_q$. Point addition forms a cyclic group
+on $E(\mathbb{F}_q)$ with a generator point $G \in E(\mathbb{F}_q)$ and a point at infinite $\mathcal{O}$ such that:
+
+$$
+\forall P, Q \in E(\mathbb{F}_q) : P + Q = R \in E(\mathbb{F}_q)
+\forall P \in E(\mathbb{F}_q) : P + \mathcal{O} = P
+\forall P \in E(\mathbb{F}_q) : P + (-P) = \mathcal{O}
+$$
+
+Scalar multiplication is defined as iterative point addition such that:
+
+$$
+\forall P \in E(\mathbb{F}_q), k \in \mathbb{Z} : k \times P = P_k \in E(\mathbb{F}_q)
+\forall P_k \in E(\mathbb{F}_q), \exists k \in \mathbb{Z} : k \times G = P_k
+$$
+
+## Application
+
+The discrete logarithm problem and algebraic structure of elliptic curve point addition and scalar multiplication
+allows for public key cryptography schemes such as digital signatures
+([ECDSA](https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm)) and key exchanges
+([ECDH](https://en.wikipedia.org/wiki/Elliptic-curve_Diffie%E2%80%93Hellman)).

src/ecc/ecdh.rs

@@ -0,0 +1,56 @@
+//! ECDH key exchange
+
+use self::field::prime::PlutoScalarField;
+use super::*;
+
+// PARAMETERS
+// *******************************************
+// CURVE	the elliptic curve field and equation used
+// G	    a point on the curve that generates a subgroup of large prime order n
+// n	    integer order of G, means that n × G = O, n must also be prime.
+// d_A	    the local private key (randomly selected) (scaler in F_n)
+// Q_A	    the local public key d_A × G = Q_A (point on the curve)
+// d_B	    the foreign private key (randomly selected) (scaler in F_n)
+// Q_B	    the foreign public key d_B × G = Q_B (point on the curve)
+// S	    the shared secret point S = d_A × Q_B = d_B × Q_A
+
+/// SHARED SECRET COMPUTATION
+/// *******************************************
+/// 1. Compute the shared secret point S = d_A × Q_B.
+///
+/// ## Notes:
+/// Elliptic Curve Diffie Hellman (ECDH) exchanges a shared secret over an insecure channel via the
+/// commutativity and associativity of elliptic curve point multiplication.
+///
+/// d_A × (d_B × G) = d_B × (d_A × G)
+///       d_B × Q_A = d_A × Q_B
+pub fn compute_shared_secret(
+  d_a: PlutoScalarField,
+  q_b: AffinePoint<PlutoBaseCurve>,
+) -> AffinePoint<PlutoBaseCurve> {
+  q_b * d_a
+}
+
+#[cfg(test)]
+mod tests {
+  use super::*;
+
+  #[test]
+  fn test_key_exchange() {
+    // secret keys
+    let mut rns = rand::rngs::OsRng;
+    let d_a = PlutoScalarField::new(rand::Rng::gen_range(&mut rns, 1..=PlutoScalarField::ORDER));
+    let d_b = PlutoScalarField::new(rand::Rng::gen_range(&mut rns, 1..=PlutoScalarField::ORDER));
+
+    // public keys
+    let q_a = AffinePoint::<PlutoBaseCurve>::generator() * d_a;
+    let q_b = AffinePoint::<PlutoBaseCurve>::generator() * d_b;
+
+    // shared secret
+    let s_a = compute_shared_secret(d_a, q_b);
+    let s_b = compute_shared_secret(d_b, q_a);
+
+    println!("shared secrets = [\n\t{:?},\n\t{:?}\n]", s_a, s_b);
+    assert_eq!(s_a, s_b);
+  }
+}

src/ecc/mod.rs

@@ -0,0 +1,5 @@
+//! Elliptic curve cryptography primitives
+use super::*;
+
+pub mod ecdh;
+pub mod ecdsa;

src/lib.rs

@@ -25,7 +25,7 @@
 pub mod codes;
 pub mod compiler;
 pub mod curve;
-pub mod ecdsa;
+pub mod ecc;
 pub mod encryption;
 pub mod field;
 pub mod hashes;