Implement polynomials for distributed key generation.

keygen.rs

@@ -0,0 +1,613 @@
+//! Utilities for distributed key generation.
+//!
+//! A `BivarPoly` can be used for Verifiable Secret Sharing (VSS) and for key generation by a
+//! trusted dealer. In a perfectly synchronous setting, e.g. on a blockchain or other agreed
+//! transaction log, it works like this:
+//!
+//! The dealer generates a `BivarPoly` of degree `t` and publishes the `BivariateCommitment`,
+//! with which the polynomial's values can be publicly verified. They then send _row_ `m > 0` to
+//! node number `m`. Node `m`, in turn, sends _value_ `s` to node number `s`. Then if `2 * t + 1`
+//! nodes confirm that they received a valid row, and there are at most `t` faulty nodes, then at
+//! least `t + 1` honest nodes sent on an entry of every other node's column to that node. So we
+//! know that every node can now reconstruct its column and the value at `0` of its column. These
+//! values all lie on a univariate polynomial of degree `t`, so they can be used as secret keys.
+//!
+//! For Distributed Key Generation (DKG), every node proposes a polynomial via VSS. After a fixed
+//! number (at least `N - 2 * t` if there are `N` nodes and up to `t` faulty ones) of them have
+//! successfully been distributed, every node adds up the resulting secrets. Since the sum of
+//! polynomials of degree `t` is itself a polynomial of degree `t`, these sums are still valid
+//! secret keys, but now nobody knows the master key (number `0`).
+// TODO: Expand this explanation and add examples, once the API is complete and stable.
+
+use std::borrow::Borrow;
+use std::{cmp, iter, ops};
+
+use pairing::{CurveAffine, CurveProjective, Engine, Field, PrimeField};
+use rand::Rng;
+
+/// A univariate polynomial in the prime field.
+#[derive(Clone, Debug)]
+pub struct Poly<E: Engine> {
+    /// The coefficients of a polynomial.
+    coeff: Vec<E::Fr>,
+}
+
+impl<E: Engine> PartialEq for Poly<E> {
+    fn eq(&self, other: &Self) -> bool {
+        self.coeff == other.coeff
+    }
+}
+
+impl<B: Borrow<Poly<E>>, E: Engine> ops::AddAssign<B> for Poly<E> {
+    fn add_assign(&mut self, rhs: B) {
+        let len = cmp::max(self.coeff.len(), rhs.borrow().coeff.len());
+        self.coeff.resize(len, E::Fr::zero());
+        for (self_c, rhs_c) in self.coeff.iter_mut().zip(&rhs.borrow().coeff) {
+            self_c.add_assign(rhs_c);
+        }
+        self.remove_zeros();
+    }
+}
+
+impl<'a, B: Borrow<Poly<E>>, E: Engine> ops::Add<B> for &'a Poly<E> {
+    type Output = Poly<E>;
+
+    fn add(self, rhs: B) -> Poly<E> {
+        (*self).clone() + rhs
+    }
+}
+
+impl<B: Borrow<Poly<E>>, E: Engine> ops::Add<B> for Poly<E> {
+    type Output = Poly<E>;
+
+    fn add(mut self, rhs: B) -> Poly<E> {
+        self += rhs;
+        self
+    }
+}
+
+impl<B: Borrow<Poly<E>>, E: Engine> ops::SubAssign<B> for Poly<E> {
+    fn sub_assign(&mut self, rhs: B) {
+        let len = cmp::max(self.coeff.len(), rhs.borrow().coeff.len());
+        self.coeff.resize(len, E::Fr::zero());
+        for (self_c, rhs_c) in self.coeff.iter_mut().zip(&rhs.borrow().coeff) {
+            self_c.sub_assign(rhs_c);
+        }
+        self.remove_zeros();
+    }
+}
+
+impl<'a, B: Borrow<Poly<E>>, E: Engine> ops::Sub<B> for &'a Poly<E> {
+    type Output = Poly<E>;
+
+    fn sub(self, rhs: B) -> Poly<E> {
+        (*self).clone() - rhs
+    }
+}
+
+impl<B: Borrow<Poly<E>>, E: Engine> ops::Sub<B> for Poly<E> {
+    type Output = Poly<E>;
+
+    fn sub(mut self, rhs: B) -> Poly<E> {
+        self -= rhs;
+        self
+    }
+}
+
+// Clippy thinks using any `+` and `-` in a `Mul` implementation is suspicious.
+#[cfg_attr(feature = "cargo-clippy", allow(suspicious_arithmetic_impl))]
+impl<'a, B: Borrow<Poly<E>>, E: Engine> ops::Mul<B> for &'a Poly<E> {
+    type Output = Poly<E>;
+
+    fn mul(self, rhs: B) -> Self::Output {
+        let coeff = (0..(self.coeff.len() + rhs.borrow().coeff.len() - 1))
+            .map(|i| {
+                let mut c = E::Fr::zero();
+                for j in i.saturating_sub(rhs.borrow().degree())..(1 + cmp::min(i, self.degree())) {
+                    let mut s = self.coeff[j];
+                    s.mul_assign(&rhs.borrow().coeff[i - j]);
+                    c.add_assign(&s);
+                }
+                c
+            })
+            .collect();
+        Poly { coeff }
+    }
+}
+
+impl<B: Borrow<Poly<E>>, E: Engine> ops::Mul<B> for Poly<E> {
+    type Output = Poly<E>;
+
+    fn mul(self, rhs: B) -> Self::Output {
+        &self * rhs
+    }
+}
+
+impl<B: Borrow<Self>, E: Engine> ops::MulAssign<B> for Poly<E> {
+    fn mul_assign(&mut self, rhs: B) {
+        *self = &*self * rhs;
+    }
+}
+
+impl<E: Engine> Poly<E> {
+    /// Creates a random polynomial.
+    pub fn random<R: Rng>(degree: usize, rng: &mut R) -> Self {
+        Poly {
+            coeff: (0..(degree + 1)).map(|_| rng.gen()).collect(),
+        }
+    }
+
+    /// Returns the polynomial with constant value `0`.
+    pub fn zero() -> Self {
+        Poly { coeff: Vec::new() }
+    }
+
+    /// Returns the polynomial with constant value `1`.
+    pub fn one() -> Self {
+        Self::monomial(0)
+    }
+
+    /// Returns the polynomial with constant value `c`.
+    pub fn constant(c: E::Fr) -> Self {
+        Poly { coeff: vec![c] }
+    }
+
+    /// Returns the identity function, i.e. the polynomial "`x`".
+    pub fn identity() -> Self {
+        Self::monomial(1)
+    }
+
+    /// Returns the (monic) monomial "`x.pow(degree)`".
+    pub fn monomial(degree: usize) -> Self {
+        Poly {
+            coeff: iter::repeat(E::Fr::zero())
+                .take(degree)
+                .chain(iter::once(E::Fr::one()))
+                .collect(),
+        }
+    }
+
+    /// Returns the unique polynomial `f` of degree `samples.len() - 1` with the given values
+    /// `(x, f(x))`.
+    pub fn interpolate<'a, T, I>(samples_repr: I) -> Self
+    where
+        I: IntoIterator<Item = (&'a T, &'a E::Fr)>,
+        T: Into<<E::Fr as PrimeField>::Repr> + Clone + 'a,
+    {
+        let convert = |(x_repr, y): (&T, &E::Fr)| {
+            let x = E::Fr::from_repr(x_repr.clone().into()).expect("invalid index");
+            (x, *y)
+        };
+        let samples: Vec<(E::Fr, E::Fr)> = samples_repr.into_iter().map(convert).collect();
+        Self::compute_interpolation(&samples)
+    }
+
+    /// Returns the degree.
+    pub fn degree(&self) -> usize {
+        self.coeff.len() - 1
+    }
+
+    /// Returns the value at the point `i`.
+    pub fn evaluate<T: Into<<E::Fr as PrimeField>::Repr>>(&self, i: T) -> E::Fr {
+        let mut result = match self.coeff.last() {
+            None => return E::Fr::zero(),
+            Some(c) => *c,
+        };
+        let x = E::Fr::from_repr(i.into()).expect("invalid index");
+        for c in self.coeff.iter().rev().skip(1) {
+            result.mul_assign(&x);
+            result.add_assign(c);
+        }
+        result
+    }
+
+    /// Returns the corresponding commitment.
+    pub fn commitment(&self) -> Commitment<E> {
+        let to_g1 = |c: &E::Fr| E::G1Affine::one().mul(*c);
+        Commitment {
+            coeff: self.coeff.iter().map(to_g1).collect(),
+        }
+    }
+
+    /// Removes all trailing zero coefficients.
+    fn remove_zeros(&mut self) {
+        let zeros = self.coeff.iter().rev().take_while(|c| c.is_zero()).count();
+        let len = self.coeff.len() - zeros;
+        self.coeff.truncate(len)
+    }
+
+    /// Returns the unique polynomial `f` of degree `samples.len() - 1` with the given values
+    /// `(x, f(x))`.
+    fn compute_interpolation(samples: &[(E::Fr, E::Fr)]) -> Self {
+        if samples.is_empty() {
+            return Poly::zero();
+        } else if samples.len() == 1 {
+            return Poly::constant(samples[0].1);
+        }
+        // The degree is at least 1 now.
+        let degree = samples.len() - 1;
+        // Interpolate all but the last sample.
+        let prev = Self::compute_interpolation(&samples[..degree]);
+        let (x, mut y) = samples[degree]; // The last sample.
+        y.sub_assign(&prev.evaluate(x));
+        let step = Self::lagrange(x, &samples[..degree]);
+        prev + step * Self::constant(y)
+    }
+
+    /// Returns the Lagrange base polynomial that is `1` in `p` and `0` in every `samples[i].0`.
+    fn lagrange(p: E::Fr, samples: &[(E::Fr, E::Fr)]) -> Self {
+        let mut result = Self::one();
+        for &(sx, _) in samples {
+            let mut denom = p;
+            denom.sub_assign(&sx);
+            denom = denom.inverse().expect("sample points must be distinct");
+            result *= (Self::identity() - Self::constant(sx)) * Self::constant(denom);
+        }
+        result
+    }
+}
+
+/// A commitment to a univariate polynomial.
+#[derive(Debug, Clone)]
+#[cfg_attr(feature = "serialization-serde", derive(Serialize, Deserialize))]
+pub struct Commitment<E: Engine> {
+    /// The coefficients of the polynomial.
+    #[cfg_attr(feature = "serialization-serde", serde(with = "super::serde_impl::projective_vec"))]
+    coeff: Vec<E::G1>,
+}
+
+impl<E: Engine> PartialEq for Commitment<E> {
+    fn eq(&self, other: &Self) -> bool {
+        self.coeff == other.coeff
+    }
+}
+
+impl<B: Borrow<Commitment<E>>, E: Engine> ops::AddAssign<B> for Commitment<E> {
+    fn add_assign(&mut self, rhs: B) {
+        let len = cmp::max(self.coeff.len(), rhs.borrow().coeff.len());
+        self.coeff.resize(len, E::G1::zero());
+        for (self_c, rhs_c) in self.coeff.iter_mut().zip(&rhs.borrow().coeff) {
+            self_c.add_assign(rhs_c);
+        }
+        self.remove_zeros();
+    }
+}
+
+impl<'a, B: Borrow<Commitment<E>>, E: Engine> ops::Add<B> for &'a Commitment<E> {
+    type Output = Commitment<E>;
+
+    fn add(self, rhs: B) -> Commitment<E> {
+        (*self).clone() + rhs
+    }
+}
+
+impl<B: Borrow<Commitment<E>>, E: Engine> ops::Add<B> for Commitment<E> {
+    type Output = Commitment<E>;
+
+    fn add(mut self, rhs: B) -> Commitment<E> {
+        self += rhs;
+        self
+    }
+}
+
+impl<E: Engine> Commitment<E> {
+    /// Returns the polynomial's degree.
+    pub fn degree(&self) -> usize {
+        self.coeff.len() - 1
+    }
+
+    /// Returns the `i`-th public key share.
+    pub fn evaluate<T: Into<<E::Fr as PrimeField>::Repr>>(&self, i: T) -> E::G1 {
+        let mut result = match self.coeff.last() {
+            None => return E::G1::zero(),
+            Some(c) => *c,
+        };
+        let x = E::Fr::from_repr(i.into()).expect("invalid index");
+        for c in self.coeff.iter().rev().skip(1) {
+            result.mul_assign(x);
+            result.add_assign(c);
+        }
+        result
+    }
+
+    /// Removes all trailing zero coefficients.
+    fn remove_zeros(&mut self) {
+        let zeros = self.coeff.iter().rev().take_while(|c| c.is_zero()).count();
+        let len = self.coeff.len() - zeros;
+        self.coeff.truncate(len)
+    }
+}
+
+/// A symmetric bivariate polynomial in the prime field.
+///
+/// This can be used for Verifiable Secret Sharing and Distributed Key Generation. See the module
+/// documentation for details.
+#[derive(Debug, Clone)]
+pub struct BivarPoly<E: Engine> {
+    /// The polynomial's degree in each of the two variables.
+    degree: usize,
+    /// The coefficients of the polynomial. Coefficient `(i, j)` for `i <= j` is in position
+    /// `j * (j + 1) / 2 + i`.
+    coeff: Vec<E::Fr>,
+}
+
+impl<E: Engine> BivarPoly<E> {
+    /// Creates a random polynomial.
+    pub fn random<R: Rng>(degree: usize, rng: &mut R) -> Self {
+        BivarPoly {
+            degree,
+            coeff: (0..coeff_pos(degree + 1, 0)).map(|_| rng.gen()).collect(),
+        }
+    }
+
+    /// Returns the polynomial's degree: It is the same in both variables.
+    pub fn degree(&self) -> usize {
+        self.degree
+    }
+
+    /// Returns the polynomial's value at the point `(x, y)`.
+    pub fn evaluate<T: Into<<E::Fr as PrimeField>::Repr>>(&self, x: T, y: T) -> E::Fr {
+        let x_pow = self.powers(x);
+        let y_pow = self.powers(y);
+        // TODO: Can we save a few multiplication steps here due to the symmetry?
+        let mut result = E::Fr::zero();
+        for (i, x_pow_i) in x_pow.into_iter().enumerate() {
+            for (j, y_pow_j) in y_pow.iter().enumerate() {
+                let mut summand = self.coeff[coeff_pos(i, j)];
+                summand.mul_assign(&x_pow_i);
+                summand.mul_assign(y_pow_j);
+                result.add_assign(&summand);
+            }
+        }
+        result
+    }
+
+    /// Returns the `x`-th row, as a univariate polynomial.
+    pub fn row<T: Into<<E::Fr as PrimeField>::Repr>>(&self, x: T) -> Poly<E> {
+        let x_pow = self.powers(x);
+        let coeff: Vec<E::Fr> = (0..=self.degree)
+            .map(|i| {
+                let mut result = E::Fr::zero();
+                for (j, x_pow_j) in x_pow.iter().enumerate() {
+                    let mut summand = self.coeff[coeff_pos(i, j)];
+                    summand.mul_assign(x_pow_j);
+                    result.add_assign(&summand);
+                }
+                result
+            })
+            .collect();
+        Poly { coeff }
+    }
+
+    /// Returns the corresponding commitment. That information can be shared publicly.
+    pub fn commitment(&self) -> BivarCommitment<E> {
+        let to_pub = |c: &E::Fr| E::G1Affine::one().mul(*c);
+        BivarCommitment {
+            degree: self.degree,
+            coeff: self.coeff.iter().map(to_pub).collect(),
+        }
+    }
+
+    /// Returns the `0`-th to `degree`-th power of `x`.
+    fn powers<T: Into<<E::Fr as PrimeField>::Repr>>(&self, x_repr: T) -> Vec<E::Fr> {
+        powers(x_repr, self.degree)
+    }
+}
+
+/// A commitment to a bivariate polynomial.
+#[derive(Debug, Clone)]
+#[cfg_attr(feature = "serialization-serde", derive(Serialize, Deserialize))]
+pub struct BivarCommitment<E: Engine> {
+    /// The polynomial's degree in each of the two variables.
+    degree: usize,
+    /// The commitments to the coefficients.
+    #[cfg_attr(feature = "serialization-serde", serde(with = "super::serde_impl::projective_vec"))]
+    coeff: Vec<E::G1>,
+}
+
+impl<E: Engine> BivarCommitment<E> {
+    /// Returns the polynomial's degree: It is the same in both variables.
+    pub fn degree(&self) -> usize {
+        self.degree
+    }
+
+    /// Returns the commitment's value at the point `(x, y)`.
+    pub fn evaluate<T: Into<<E::Fr as PrimeField>::Repr>>(&self, x: T, y: T) -> E::G1 {
+        let x_pow = self.powers(x);
+        let y_pow = self.powers(y);
+        // TODO: Can we save a few multiplication steps here due to the symmetry?
+        let mut result = E::G1::zero();
+        for (i, x_pow_i) in x_pow.into_iter().enumerate() {
+            for (j, y_pow_j) in y_pow.iter().enumerate() {
+                let mut summand = self.coeff[coeff_pos(i, j)];
+                summand.mul_assign(x_pow_i);
+                summand.mul_assign(*y_pow_j);
+                result.add_assign(&summand);
+            }
+        }
+        result
+    }
+
+    /// Returns the `x`-th row, as a commitment to a univariate polynomial.
+    pub fn row<T: Into<<E::Fr as PrimeField>::Repr>>(&self, x: T) -> Commitment<E> {
+        let x_pow = self.powers(x);
+        let coeff: Vec<E::G1> = (0..=self.degree)
+            .map(|i| {
+                let mut result = E::G1::zero();
+                for (j, x_pow_j) in x_pow.iter().enumerate() {
+                    let mut summand = self.coeff[coeff_pos(i, j)];
+                    summand.mul_assign(*x_pow_j);
+                    result.add_assign(&summand);
+                }
+                result
+            })
+            .collect();
+        Commitment { coeff }
+    }
+
+    /// Returns the `0`-th to `degree`-th power of `x`.
+    fn powers<T: Into<<E::Fr as PrimeField>::Repr>>(&self, x_repr: T) -> Vec<E::Fr> {
+        powers(x_repr, self.degree)
+    }
+}
+
+/// Returns the `0`-th to `degree`-th power of `x`.
+fn powers<P: PrimeField, T: Into<P::Repr>>(x_repr: T, degree: usize) -> Vec<P> {
+    let x = &P::from_repr(x_repr.into()).expect("invalid index");
+    let mut x_pow_i = P::one();
+    iter::once(x_pow_i)
+        .chain((0..degree).map(|_| {
+            x_pow_i.mul_assign(x);
+            x_pow_i
+        }))
+        .collect()
+}
+
+/// Returns the position of coefficient `(i, j)` in the vector describing a symmetric bivariate
+/// polynomial.
+fn coeff_pos(i: usize, j: usize) -> usize {
+    // Since the polynomial is symmetric, we can order such that `j >= i`.
+    if j >= i {
+        j * (j + 1) / 2 + i
+    } else {
+        i * (i + 1) / 2 + j
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use std::collections::BTreeMap;
+
+    use super::{coeff_pos, BivarPoly, Poly};
+
+    use pairing::bls12_381::Bls12;
+    use pairing::{CurveAffine, Engine, Field, PrimeField};
+    use rand;
+
+    type Fr = <Bls12 as Engine>::Fr;
+
+    fn fr(x: i64) -> Fr {
+        let mut result = Fr::from_repr((x.abs() as u64).into()).unwrap();
+        if x < 0 {
+            result.negate();
+        }
+        result
+    }
+
+    #[test]
+    fn test_coeff_pos() {
+        let mut i = 0;
+        let mut j = 0;
+        for n in 0..100 {
+            assert_eq!(n, coeff_pos(i, j));
+            if i >= j {
+                j += 1;
+                i = 0;
+            } else {
+                i += 1;
+            }
+        }
+    }
+
+    #[test]
+    fn poly() {
+        // The polynomial "`5 * x.pow(3) + x.pow(1) - 2`".
+        let poly: Poly<Bls12> =
+            Poly::monomial(3) * Poly::constant(fr(5)) + Poly::monomial(1) - Poly::constant(fr(2));
+        let coeff = vec![fr(-2), fr(1), fr(0), fr(5)];
+        assert_eq!(Poly { coeff }, poly);
+        let samples = vec![
+            (fr(-1), fr(-8)),
+            (fr(2), fr(40)),
+            (fr(3), fr(136)),
+            (fr(5), fr(628)),
+        ];
+        for &(x, y) in &samples {
+            assert_eq!(y, poly.evaluate(x));
+        }
+        let sample_iter = samples.iter().map(|&(ref x, ref y)| (x, y));
+        assert_eq!(Poly::interpolate(sample_iter), poly);
+    }
+
+    #[test]
+    fn distributed_key_generation() {
+        let mut rng = rand::thread_rng();
+        let dealer_num = 3;
+        let node_num = 5;
+        let faulty_num = 2;
+
+        // For distributed key generation, a number of dealers, only one of who needs to be honest,
+        // generates random bivariate polynomials and publicly commits to them. In partice, the
+        // dealers can e.g. be any `faulty_num + 1` nodes.
+        let bi_polys: Vec<BivarPoly<Bls12>> = (0..dealer_num)
+            .map(|_| BivarPoly::random(faulty_num, &mut rng))
+            .collect();
+        let pub_bi_commits: Vec<_> = bi_polys.iter().map(BivarPoly::commitment).collect();
+
+        let mut sec_keys = vec![fr(0); node_num];
+
+        // Each dealer sends row `m` to node `m`, where the index starts at `1`. Don't send row `0`
+        // to anyone! The nodes verify their rows, and send _value_ `s` on to node `s`. They again
+        // verify the values they received, and collect them.
+        for (bi_poly, bi_commit) in bi_polys.iter().zip(&pub_bi_commits) {
+            for m in 1..=node_num {
+                // Node `m` receives its row and verifies it.
+                let row_poly = bi_poly.row(m as u64);
+                let row_commit = bi_commit.row(m as u64);
+                assert_eq!(row_poly.commitment(), row_commit);
+                // Node `s` receives the `s`-th value and verifies it.
+                for s in 1..=node_num {
+                    let val = row_poly.evaluate(s as u64);
+                    let val_g1 = <Bls12 as Engine>::G1Affine::one().mul(val);
+                    assert_eq!(bi_commit.evaluate(m as u64, s as u64), val_g1);
+                    // The node can't verify this directly, but it should have the correct value:
+                    assert_eq!(bi_poly.evaluate(m as u64, s as u64), val);
+                }
+
+                // A cheating dealer who modified the polynomial would be detected.
+                let wrong_poly = row_poly.clone() + Poly::monomial(2) * Poly::constant(fr(5));
+                assert_ne!(wrong_poly.commitment(), row_commit);
+
+                // If `2 * faulty_num + 1` nodes confirm that they received a valid row, then at
+                // least `faulty_num + 1` honest ones did, and sent the correct values on to node
+                // `s`. So every node received at least `faulty_num + 1` correct entries of their
+                // column/row (remember that the bivariate polynomial is symmetric). They can
+                // reconstruct the full row and in particular value `0` (which no other node knows,
+                // only the dealer). E.g. let's say nodes `1`, `2` and `4` are honest. Then node
+                // `m` received three correct entries from that row:
+                let received: BTreeMap<_, _> = [1, 2, 4]
+                    .iter()
+                    .map(|&i| (i, bi_poly.evaluate(m as u64, i as u64)))
+                    .collect();
+                let my_row = Poly::interpolate(&received);
+                assert_eq!(bi_poly.evaluate(m as u64, 0), my_row.evaluate(0));
+                assert_eq!(row_poly, my_row);
+
+                // The node sums up all values number `0` it received from the different dealer. No
+                // dealer and no other node knows the sum in the end.
+                sec_keys[m - 1].add_assign(&my_row.evaluate(0));
+            }
+        }
+
+        // Each node now adds up all the first values of the rows it received from the different
+        // dealers (excluding the dealers where fewer than `2 * faulty_num + 1` nodes confirmed).
+        // The whole first column never gets added up in practice, because nobody has all the
+        // information. We do it anyway here; entry `0` is the secret key that is not known to
+        // anyone, neither a dealer, nor a node:
+        let mut sec_key_set = Poly::zero();
+        for bi_poly in &bi_polys {
+            sec_key_set += bi_poly.row(0);
+        }
+        for m in 1..=node_num {
+            assert_eq!(sec_key_set.evaluate(m as u64), sec_keys[m - 1]);
+        }
+
+        // The sum of the first rows of the public commitments is the commitment to the secret key
+        // set.
+        let mut sum_commit = Poly::zero().commitment();
+        for bi_commit in &pub_bi_commits {
+            sum_commit += bi_commit.row(0);
+        }
+        assert_eq!(sum_commit, sec_key_set.commitment());
+    }
+}

mod.rs

@@ -1,5 +1,9 @@
 mod error;
+pub mod keygen;
+#[cfg(feature = "serialization-serde")]
+mod serde_impl;
 
+use self::keygen::{Commitment, Poly};
 use byteorder::{BigEndian, ByteOrder};
 use init_with::InitWith;
 use pairing::{CurveAffine, CurveProjective, Engine, Field, PrimeField};
@@ -149,34 +153,30 @@ impl<E: Engine> PartialEq for DecryptionShare<E> {
 pub struct PublicKeySet<E: Engine> {
     /// The coefficients of a polynomial whose value at `0` is the "master key", and value at
     /// `i + 1` is key share number `i`.
-    coeff: Vec<PublicKey<E>>,
+    commit: Commitment<E>,
+}
+
+impl<E: Engine> From<Commitment<E>> for PublicKeySet<E> {
+    fn from(commit: Commitment<E>) -> PublicKeySet<E> {
+        PublicKeySet { commit }
+    }
 }
 
 impl<E: Engine> PublicKeySet<E> {
     /// Returns the threshold `t`: any set of `t + 1` signature shares can be combined into a full
     /// signature.
     pub fn threshold(&self) -> usize {
-        self.coeff.len() - 1
+        self.commit.degree()
     }
 
     /// Returns the public key.
-    pub fn public_key(&self) -> &PublicKey<E> {
+    pub fn public_key(&self) -> PublicKey<E> {
-        &self.coeff[0]
+        PublicKey(self.commit.evaluate(0))
     }
 
     /// Returns the `i`-th public key share.
-    pub fn public_key_share<T>(&self, i: T) -> PublicKey<E>
+    pub fn public_key_share<T: Into<<E::Fr as PrimeField>::Repr>>(&self, i: T) -> PublicKey<E> {
-    where
+        PublicKey(self.commit.evaluate(from_repr_plus_1::<E::Fr>(i.into())))
-        T: Into<<E::Fr as PrimeField>::Repr>,
-    {
-        let mut x = E::Fr::one();
-        x.add_assign(&E::Fr::from_repr(i.into()).expect("invalid index"));
-        let mut pk = self.coeff.last().expect("at least one coefficient").0;
-        for c in self.coeff.iter().rev().skip(1) {
-            pk.mul_assign(x);
-            pk.add_assign(&c.0);
-        }
-        PublicKey(pk)
     }
 
     /// Combines the shares into a signature that can be verified with the main public key.
@@ -186,7 +186,7 @@ impl<E: Engine> PublicKeySet<E> {
         IND: Into<<E::Fr as PrimeField>::Repr> + Clone + 'a,
     {
         let samples = shares.into_iter().map(|(i, share)| (i, &share.0));
-        Ok(Signature(interpolate(self.coeff.len(), samples)?))
+        Ok(Signature(interpolate(self.commit.degree() + 1, samples)?))
     }
 
     /// Combines the shares to decrypt the ciphertext.
@@ -196,7 +196,7 @@ impl<E: Engine> PublicKeySet<E> {
         IND: Into<<E::Fr as PrimeField>::Repr> + Clone + 'a,
     {
         let samples = shares.into_iter().map(|(i, share)| (i, &share.0));
-        let g = interpolate(self.coeff.len(), samples)?;
+        let g = interpolate(self.commit.degree() + 1, samples)?;
         Ok(xor_vec(&hash_bytes::<E>(g, ct.1.len()), &ct.1))
     }
 }
@@ -205,45 +205,41 @@ impl<E: Engine> PublicKeySet<E> {
 pub struct SecretKeySet<E: Engine> {
     /// The coefficients of a polynomial whose value at `0` is the "master key", and value at
     /// `i + 1` is key share number `i`.
-    coeff: Vec<E::Fr>,
+    poly: Poly<E>,
 }
 
 impl<E: Engine> SecretKeySet<E> {
     /// Creates a set of secret key shares, where any `threshold + 1` of them can collaboratively
     /// sign and decrypt.
-    pub fn new<R: Rng>(threshold: usize, rng: &mut R) -> Self {
+    pub fn random<R: Rng>(threshold: usize, rng: &mut R) -> Self {
         SecretKeySet {
-            coeff: (0..(threshold + 1)).map(|_| rng.gen()).collect(),
+            poly: Poly::random(threshold, rng),
         }
     }
 
     /// Returns the threshold `t`: any set of `t + 1` signature shares can be combined into a full
     /// signature.
     pub fn threshold(&self) -> usize {
-        self.coeff.len() - 1
+        self.poly.degree()
     }
 
     /// Returns the `i`-th secret key share.
-    pub fn secret_key_share<T>(&self, i: T) -> SecretKey<E>
+    pub fn secret_key_share<T: Into<<E::Fr as PrimeField>::Repr>>(&self, i: T) -> SecretKey<E> {
-    where
+        SecretKey(self.poly.evaluate(from_repr_plus_1::<E::Fr>(i.into())))
-        T: Into<<E::Fr as PrimeField>::Repr>,
-    {
-        let x = from_repr_plus_1(i.into());
-        let mut pk = *self.coeff.last().expect("at least one coefficient");
-        for c in self.coeff.iter().rev().skip(1) {
-            pk.mul_assign(&x);
-            pk.add_assign(c);
-        }
-        SecretKey(pk)
     }
 
     /// Returns the corresponding public key set. That information can be shared publicly.
     pub fn public_keys(&self) -> PublicKeySet<E> {
-        let to_pub = |c: &E::Fr| PublicKey(E::G1Affine::one().mul(*c));
         PublicKeySet {
-            coeff: self.coeff.iter().map(to_pub).collect(),
+            commit: self.poly.commitment(),
         }
     }
+
+    /// Returns the secret master key.
+    #[cfg(test)]
+    fn secret_key(&self) -> SecretKey<E> {
+        SecretKey(self.poly.evaluate(0))
+    }
 }
 
 /// Returns a hash of the given message in `G2`.
@@ -287,7 +283,7 @@ fn xor_vec(x: &[u8], y: &[u8]) -> Vec<u8> {
 
 /// Given a list of `t` samples `(i - 1, f(i) * g)` for a polynomial `f` of degree `t - 1`, and a
 /// group generator `g`, returns `f(0) * g`.
-pub fn interpolate<'a, C, ITR, IND>(t: usize, items: ITR) -> Result<C>
+fn interpolate<'a, C, ITR, IND>(t: usize, items: ITR) -> Result<C>
 where
     C: CurveProjective,
     ITR: IntoIterator<Item = (&'a IND, &'a C)>,
@@ -352,19 +348,18 @@ mod tests {
     #[test]
     fn test_threshold_sig() {
         let mut rng = rand::thread_rng();
-        let sk_set = SecretKeySet::<Bls12>::new(3, &mut rng);
+        let sk_set = SecretKeySet::<Bls12>::random(3, &mut rng);
         let pk_set = sk_set.public_keys();
 
         // Make sure the keys are different, and the first coefficient is the main key.
-        assert_eq!(*pk_set.public_key(), pk_set.coeff[0]);
+        assert_ne!(pk_set.public_key(), pk_set.public_key_share(0));
-        assert_ne!(*pk_set.public_key(), pk_set.public_key_share(0));
+        assert_ne!(pk_set.public_key(), pk_set.public_key_share(1));
-        assert_ne!(*pk_set.public_key(), pk_set.public_key_share(1));
+        assert_ne!(pk_set.public_key(), pk_set.public_key_share(2));
-        assert_ne!(*pk_set.public_key(), pk_set.public_key_share(2));
 
         // Make sure we don't hand out the main secret key to anyone.
-        assert_ne!(SecretKey(sk_set.coeff[0]), sk_set.secret_key_share(0));
+        assert_ne!(sk_set.secret_key(), sk_set.secret_key_share(0));
-        assert_ne!(SecretKey(sk_set.coeff[0]), sk_set.secret_key_share(1));
+        assert_ne!(sk_set.secret_key(), sk_set.secret_key_share(1));
-        assert_ne!(SecretKey(sk_set.coeff[0]), sk_set.secret_key_share(2));
+        assert_ne!(sk_set.secret_key(), sk_set.secret_key_share(2));
 
         let msg = "Totally real news";
 
@@ -420,7 +415,7 @@ mod tests {
     #[test]
     fn test_threshold_enc() {
         let mut rng = rand::thread_rng();
-        let sk_set = SecretKeySet::<Bls12>::new(3, &mut rng);
+        let sk_set = SecretKeySet::<Bls12>::random(3, &mut rng);
         let pk_set = sk_set.public_keys();
         let msg = b"Totally real news";
         let ciphertext = pk_set.public_key().encrypt(&msg[..]);
@@ -511,76 +506,3 @@ mod tests {
         assert_eq!(sig, deser_sig);
     }
 }
-
-#[cfg(feature = "serialization-serde")]
-mod serde {
-    use pairing::{CurveAffine, CurveProjective, EncodedPoint, Engine};
-
-    use super::{DecryptionShare, PublicKey, Signature};
-    use serde::de::Error as DeserializeError;
-    use serde::{Deserialize, Deserializer, Serialize, Serializer};
-
-    const ERR_LEN: &str = "wrong length of deserialized group element";
-    const ERR_CODE: &str = "deserialized bytes don't encode a group element";
-
-    impl<E: Engine> Serialize for PublicKey<E> {
-        fn serialize<S: Serializer>(&self, s: S) -> Result<S::Ok, S::Error> {
-            serialize_projective(&self.0, s)
-        }
-    }
-
-    impl<'de, E: Engine> Deserialize<'de> for PublicKey<E> {
-        fn deserialize<D: Deserializer<'de>>(d: D) -> Result<Self, D::Error> {
-            Ok(PublicKey(deserialize_projective(d)?))
-        }
-    }
-
-    impl<E: Engine> Serialize for Signature<E> {
-        fn serialize<S: Serializer>(&self, s: S) -> Result<S::Ok, S::Error> {
-            serialize_projective(&self.0, s)
-        }
-    }
-
-    impl<'de, E: Engine> Deserialize<'de> for Signature<E> {
-        fn deserialize<D: Deserializer<'de>>(d: D) -> Result<Self, D::Error> {
-            Ok(Signature(deserialize_projective(d)?))
-        }
-    }
-
-    impl<E: Engine> Serialize for DecryptionShare<E> {
-        fn serialize<S: Serializer>(&self, s: S) -> Result<S::Ok, S::Error> {
-            serialize_projective(&self.0, s)
-        }
-    }
-
-    impl<'de, E: Engine> Deserialize<'de> for DecryptionShare<E> {
-        fn deserialize<D: Deserializer<'de>>(d: D) -> Result<Self, D::Error> {
-            Ok(DecryptionShare(deserialize_projective(d)?))
-        }
-    }
-
-    /// Serializes the compressed representation of a group element.
-    fn serialize_projective<S, C>(c: &C, s: S) -> Result<S::Ok, S::Error>
-    where
-        S: Serializer,
-        C: CurveProjective,
-    {
-        c.into_affine().into_compressed().as_ref().serialize(s)
-    }
-
-    /// Deserializes the compressed representation of a group element.
-    fn deserialize_projective<'de, D, C>(d: D) -> Result<C, D::Error>
-    where
-        D: Deserializer<'de>,
-        C: CurveProjective,
-    {
-        let bytes = <Vec<u8>>::deserialize(d)?;
-        if bytes.len() != <C::Affine as CurveAffine>::Compressed::size() {
-            return Err(D::Error::custom(ERR_LEN));
-        }
-        let mut compressed = <C::Affine as CurveAffine>::Compressed::empty();
-        compressed.as_mut().copy_from_slice(&bytes);
-        let to_err = |_| D::Error::custom(ERR_CODE);
-        Ok(compressed.into_affine().map_err(to_err)?.into_projective())
-    }
-}

serde_impl.rs

@@ -0,0 +1,119 @@
+use std::borrow::Borrow;
+use std::marker::PhantomData;
+
+use pairing::{CurveAffine, CurveProjective, EncodedPoint, Engine};
+
+use super::{DecryptionShare, PublicKey, Signature};
+use serde::de::Error as DeserializeError;
+use serde::{Deserialize, Deserializer, Serialize, Serializer};
+
+const ERR_LEN: &str = "wrong length of deserialized group element";
+const ERR_CODE: &str = "deserialized bytes don't encode a group element";
+
+/// A wrapper type to facilitate serialization and deserialization of group elements.
+struct CurveWrap<C, B>(B, PhantomData<C>);
+
+impl<C, B> CurveWrap<C, B> {
+    fn new(c: B) -> Self {
+        CurveWrap(c, PhantomData)
+    }
+}
+
+impl<C: CurveProjective, B: Borrow<C>> Serialize for CurveWrap<C, B> {
+    fn serialize<S: Serializer>(&self, s: S) -> Result<S::Ok, S::Error> {
+        serialize_projective(self.0.borrow(), s)
+    }
+}
+
+impl<'de, C: CurveProjective> Deserialize<'de> for CurveWrap<C, C> {
+    fn deserialize<D: Deserializer<'de>>(d: D) -> Result<Self, D::Error> {
+        Ok(CurveWrap::new(deserialize_projective(d)?))
+    }
+}
+
+impl<E: Engine> Serialize for PublicKey<E> {
+    fn serialize<S: Serializer>(&self, s: S) -> Result<S::Ok, S::Error> {
+        serialize_projective(&self.0, s)
+    }
+}
+
+impl<'de, E: Engine> Deserialize<'de> for PublicKey<E> {
+    fn deserialize<D: Deserializer<'de>>(d: D) -> Result<Self, D::Error> {
+        Ok(PublicKey(deserialize_projective(d)?))
+    }
+}
+
+impl<E: Engine> Serialize for Signature<E> {
+    fn serialize<S: Serializer>(&self, s: S) -> Result<S::Ok, S::Error> {
+        serialize_projective(&self.0, s)
+    }
+}
+
+impl<'de, E: Engine> Deserialize<'de> for Signature<E> {
+    fn deserialize<D: Deserializer<'de>>(d: D) -> Result<Self, D::Error> {
+        Ok(Signature(deserialize_projective(d)?))
+    }
+}
+
+impl<E: Engine> Serialize for DecryptionShare<E> {
+    fn serialize<S: Serializer>(&self, s: S) -> Result<S::Ok, S::Error> {
+        serialize_projective(&self.0, s)
+    }
+}
+
+impl<'de, E: Engine> Deserialize<'de> for DecryptionShare<E> {
+    fn deserialize<D: Deserializer<'de>>(d: D) -> Result<Self, D::Error> {
+        Ok(DecryptionShare(deserialize_projective(d)?))
+    }
+}
+
+/// Serializes the compressed representation of a group element.
+fn serialize_projective<S, C>(c: &C, s: S) -> Result<S::Ok, S::Error>
+where
+    S: Serializer,
+    C: CurveProjective,
+{
+    c.into_affine().into_compressed().as_ref().serialize(s)
+}
+
+/// Deserializes the compressed representation of a group element.
+fn deserialize_projective<'de, D, C>(d: D) -> Result<C, D::Error>
+where
+    D: Deserializer<'de>,
+    C: CurveProjective,
+{
+    let bytes = <Vec<u8>>::deserialize(d)?;
+    if bytes.len() != <C::Affine as CurveAffine>::Compressed::size() {
+        return Err(D::Error::custom(ERR_LEN));
+    }
+    let mut compressed = <C::Affine as CurveAffine>::Compressed::empty();
+    compressed.as_mut().copy_from_slice(&bytes);
+    let to_err = |_| D::Error::custom(ERR_CODE);
+    Ok(compressed.into_affine().map_err(to_err)?.into_projective())
+}
+
+/// Serialization and deserialization of vectors of projective curve elements.
+pub mod projective_vec {
+    use super::CurveWrap;
+
+    use pairing::CurveProjective;
+    use serde::{Deserialize, Deserializer, Serialize, Serializer};
+
+    pub fn serialize<S, C>(vec: &[C], s: S) -> Result<S::Ok, S::Error>
+    where
+        S: Serializer,
+        C: CurveProjective,
+    {
+        let wrap_vec: Vec<CurveWrap<C, &C>> = vec.iter().map(CurveWrap::new).collect();
+        wrap_vec.serialize(s)
+    }
+
+    pub fn deserialize<'de, D, C>(d: D) -> Result<Vec<C>, D::Error>
+    where
+        D: Deserializer<'de>,
+        C: CurveProjective,
+    {
+        let wrap_vec = <Vec<CurveWrap<C, C>>>::deserialize(d)?;
+        Ok(wrap_vec.into_iter().map(|CurveWrap(c, _)| c).collect())
+    }
+}