Merge pull request #159 from poanetwork/afck-into-fr

Accept more types in threshold crypto API.
This commit is contained in:
Andreas Fackler 2018-07-26 08:31:54 +02:00 committed by GitHub
commit 8d449eceb5
No known key found for this signature in database
GPG Key ID: 4AEE18F83AFDEB23
9 changed files with 205 additions and 115 deletions

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@ -107,7 +107,7 @@ impl<T: Clone + Debug + AsRef<[u8]> + PartialEq + Send + Sync + From<Vec<u8>> +
// keys here. A fully-featured application would need to take appropriately initialized keys
// from elsewhere.
let secret_key_set = SecretKeySet::from(Poly::zero());
let sk_share = secret_key_set.secret_key_share(our_id as u64);
let sk_share = secret_key_set.secret_key_share(our_id);
let pub_key_set = secret_key_set.public_keys();
let sk = SecretKey::default();
let pub_keys = all_ids

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@ -207,16 +207,8 @@ where
fn combine_and_verify_sig(&self) -> Result<Signature> {
// Pass the indices of sender nodes to `combine_signatures`.
let ids_shares: BTreeMap<&NodeUid, &SignatureShare> = self.received_shares.iter().collect();
let ids_u64: BTreeMap<&NodeUid, u64> = ids_shares
.keys()
.map(|&id| (id, self.netinfo.node_index(id).unwrap() as u64))
.collect();
// Convert indices to `u64` which is an interface type for `pairing`.
let shares: BTreeMap<&u64, &SignatureShare> = ids_shares
.iter()
.map(|(id, &share)| (&ids_u64[id], share))
.collect();
let to_idx = |(id, share)| (self.netinfo.node_index(id).unwrap(), share);
let shares = self.received_shares.iter().map(to_idx);
let sig = self.netinfo.public_key_set().combine_signatures(shares)?;
if !self
.netinfo

55
src/crypto/into_fr.rs Normal file
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@ -0,0 +1,55 @@
use pairing::bls12_381::Fr;
use pairing::{Field, PrimeField};
/// A conversion into an element of the field `Fr`.
pub trait IntoFr: Copy {
fn into_fr(self) -> Fr;
}
impl IntoFr for Fr {
fn into_fr(self) -> Fr {
self
}
}
impl IntoFr for u64 {
fn into_fr(self) -> Fr {
Fr::from_repr(self.into()).expect("modulus is greater than u64::MAX")
}
}
impl IntoFr for usize {
fn into_fr(self) -> Fr {
(self as u64).into_fr()
}
}
impl IntoFr for i32 {
fn into_fr(self) -> Fr {
if self >= 0 {
(self as u64).into_fr()
} else {
let mut result = ((-self) as u64).into_fr();
result.negate();
result
}
}
}
impl IntoFr for i64 {
fn into_fr(self) -> Fr {
if self >= 0 {
(self as u64).into_fr()
} else {
let mut result = ((-self) as u64).into_fr();
result.negate();
result
}
}
}
impl<'a, T: IntoFr> IntoFr for &'a T {
fn into_fr(self) -> Fr {
(*self).into_fr()
}
}

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@ -3,6 +3,7 @@
#![cfg_attr(feature = "cargo-clippy", allow(derive_hash_xor_eq))]
pub mod error;
mod into_fr;
pub mod poly;
#[cfg(feature = "serialization-protobuf")]
pub mod protobuf_impl;
@ -14,12 +15,13 @@ use std::ptr::write_volatile;
use byteorder::{BigEndian, ByteOrder};
use init_with::InitWith;
use pairing::bls12_381::{Bls12, Fr, FrRepr, G1, G1Affine, G2, G2Affine};
use pairing::{CurveAffine, CurveProjective, Engine, Field, PrimeField};
use pairing::bls12_381::{Bls12, Fr, G1, G1Affine, G2, G2Affine};
use pairing::{CurveAffine, CurveProjective, Engine, Field};
use rand::{ChaChaRng, OsRng, Rng, SeedableRng};
use ring::digest;
use self::error::{ErrorKind, Result};
use self::into_fr::IntoFr;
use self::poly::{Commitment, Poly};
use fmt::HexBytes;
@ -330,26 +332,26 @@ impl PublicKeySet {
}
/// Returns the `i`-th public key share.
pub fn public_key_share<T: Into<FrRepr>>(&self, i: T) -> PublicKeyShare {
let value = self.commit.evaluate(from_repr_plus_1::<Fr>(i.into()));
pub fn public_key_share<T: IntoFr>(&self, i: T) -> PublicKeyShare {
let value = self.commit.evaluate(into_fr_plus_1(i));
PublicKeyShare(PublicKey(value))
}
/// Combines the shares into a signature that can be verified with the main public key.
pub fn combine_signatures<'a, ITR, IND>(&self, shares: ITR) -> Result<Signature>
pub fn combine_signatures<'a, T, I>(&self, shares: I) -> Result<Signature>
where
ITR: IntoIterator<Item = (&'a IND, &'a SignatureShare)>,
IND: Into<FrRepr> + Clone + 'a,
I: IntoIterator<Item = (T, &'a SignatureShare)>,
T: IntoFr,
{
let samples = shares.into_iter().map(|(i, share)| (i, &(share.0).0));
Ok(Signature(interpolate(self.commit.degree() + 1, samples)?))
}
/// Combines the shares to decrypt the ciphertext.
pub fn decrypt<'a, ITR, IND>(&self, shares: ITR, ct: &Ciphertext) -> Result<Vec<u8>>
pub fn decrypt<'a, T, I>(&self, shares: I, ct: &Ciphertext) -> Result<Vec<u8>>
where
ITR: IntoIterator<Item = (&'a IND, &'a DecryptionShare)>,
IND: Into<FrRepr> + Clone + 'a,
I: IntoIterator<Item = (T, &'a DecryptionShare)>,
T: IntoFr,
{
let samples = shares.into_iter().map(|(i, share)| (i, &share.0));
let g = interpolate(self.commit.degree() + 1, samples)?;
@ -386,8 +388,8 @@ impl SecretKeySet {
}
/// Returns the `i`-th secret key share.
pub fn secret_key_share<T: Into<FrRepr>>(&self, i: T) -> SecretKeyShare {
let value = self.poly.evaluate(from_repr_plus_1::<Fr>(i.into()));
pub fn secret_key_share<T: IntoFr>(&self, i: T) -> SecretKeyShare {
let value = self.poly.evaluate(into_fr_plus_1(i));
SecretKeyShare(SecretKey(value))
}
@ -446,15 +448,15 @@ 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`.
fn interpolate<'a, C, ITR, IND>(t: usize, items: ITR) -> Result<C>
fn interpolate<'a, C, T, I>(t: usize, items: I) -> Result<C>
where
C: CurveProjective,
ITR: IntoIterator<Item = (&'a IND, &'a C)>,
IND: Into<<C::Scalar as PrimeField>::Repr> + Clone + 'a,
C: CurveProjective<Scalar = Fr>,
I: IntoIterator<Item = (T, &'a C)>,
T: IntoFr,
{
let samples: Vec<_> = items
.into_iter()
.map(|(i, sample)| (from_repr_plus_1::<C::Scalar>(i.clone().into()), sample))
.map(|(i, sample)| (into_fr_plus_1(i), sample))
.collect();
if samples.len() < t {
return Err(ErrorKind::NotEnoughShares.into());
@ -480,10 +482,10 @@ where
Ok(result)
}
fn from_repr_plus_1<F: PrimeField>(repr: F::Repr) -> F {
let mut x = F::one();
x.add_assign(&F::from_repr(repr).expect("invalid index"));
x
fn into_fr_plus_1<I: IntoFr>(x: I) -> Fr {
let mut result = Fr::one();
result.add_assign(&x.into_fr());
result
}
#[cfg(test)]

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@ -21,10 +21,12 @@ use std::hash::{Hash, Hasher};
use std::ptr::write_volatile;
use std::{cmp, iter, ops};
use pairing::bls12_381::{Fr, FrRepr, G1, G1Affine};
use pairing::{CurveAffine, CurveProjective, Field, PrimeField};
use pairing::bls12_381::{Fr, G1, G1Affine};
use pairing::{CurveAffine, CurveProjective, Field};
use rand::Rng;
use super::IntoFr;
/// A univariate polynomial in the prime field.
#[derive(Clone, Debug, Serialize, Deserialize, PartialEq, Eq)]
pub struct Poly {
@ -61,6 +63,30 @@ impl<B: Borrow<Poly>> ops::Add<B> for Poly {
}
}
impl<'a> ops::Add<Fr> for Poly {
type Output = Poly;
fn add(mut self, rhs: Fr) -> Self::Output {
if self.coeff.is_empty() {
if !rhs.is_zero() {
self.coeff.push(rhs);
}
} else {
self.coeff[0].add_assign(&rhs);
self.remove_zeros();
}
self
}
}
impl<'a> ops::Add<u64> for Poly {
type Output = Poly;
fn add(self, rhs: u64) -> Self::Output {
self + rhs.into_fr()
}
}
impl<B: Borrow<Poly>> ops::SubAssign<B> for Poly {
fn sub_assign(&mut self, rhs: B) {
let len = cmp::max(self.coeff.len(), rhs.borrow().coeff.len());
@ -89,6 +115,25 @@ impl<B: Borrow<Poly>> ops::Sub<B> for Poly {
}
}
// Clippy thinks using `+` in a `Sub` implementation is suspicious.
#[cfg_attr(feature = "cargo-clippy", allow(suspicious_arithmetic_impl))]
impl<'a> ops::Sub<Fr> for Poly {
type Output = Poly;
fn sub(self, mut rhs: Fr) -> Self::Output {
rhs.negate();
self + rhs
}
}
impl<'a> ops::Sub<u64> for Poly {
type Output = Poly;
fn sub(self, rhs: u64) -> Self::Output {
self - rhs.into_fr()
}
}
// 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>> ops::Mul<B> for &'a Poly {
@ -124,6 +169,27 @@ impl<B: Borrow<Self>> ops::MulAssign<B> for Poly {
}
}
impl<'a> ops::Mul<Fr> for Poly {
type Output = Poly;
fn mul(mut self, rhs: Fr) -> Self::Output {
if rhs.is_zero() {
self.coeff.clear();
} else {
self.coeff.iter_mut().for_each(|c| c.mul_assign(&rhs));
}
self
}
}
impl<'a> ops::Mul<u64> for Poly {
type Output = Poly;
fn mul(self, rhs: u64) -> Self::Output {
self * rhs.into_fr()
}
}
impl Drop for Poly {
fn drop(&mut self) {
let start = self.coeff.as_mut_ptr();
@ -176,15 +242,13 @@ impl Poly {
/// 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
pub fn interpolate<T, U, I>(samples_repr: I) -> Self
where
I: IntoIterator<Item = (&'a T, &'a Fr)>,
T: Into<FrRepr> + Clone + 'a,
I: IntoIterator<Item = (T, U)>,
T: IntoFr,
U: IntoFr,
{
let convert = |(x_repr, y): (&T, &Fr)| {
let x = Fr::from_repr(x_repr.clone().into()).expect("invalid index");
(x, *y)
};
let convert = |(x, y): (T, U)| (x.into_fr(), y.into_fr());
let samples: Vec<(Fr, Fr)> = samples_repr.into_iter().map(convert).collect();
Self::compute_interpolation(&samples)
}
@ -195,12 +259,12 @@ impl Poly {
}
/// Returns the value at the point `i`.
pub fn evaluate<T: Into<FrRepr>>(&self, i: T) -> Fr {
pub fn evaluate<T: IntoFr>(&self, i: T) -> Fr {
let mut result = match self.coeff.last() {
None => return Fr::zero(),
Some(c) => *c,
};
let x = Fr::from_repr(i.into()).expect("invalid index");
let x = i.into_fr();
for c in self.coeff.iter().rev().skip(1) {
result.mul_assign(&x);
result.add_assign(c);
@ -306,12 +370,12 @@ impl Commitment {
}
/// Returns the `i`-th public key share.
pub fn evaluate<T: Into<FrRepr>>(&self, i: T) -> G1 {
pub fn evaluate<T: IntoFr>(&self, i: T) -> G1 {
let mut result = match self.coeff.last() {
None => return G1::zero(),
Some(c) => *c,
};
let x = Fr::from_repr(i.into()).expect("invalid index");
let x = i.into_fr();
for c in self.coeff.iter().rev().skip(1) {
result.mul_assign(x);
result.add_assign(c);
@ -367,7 +431,7 @@ impl BivarPoly {
}
/// Returns the polynomial's value at the point `(x, y)`.
pub fn evaluate<T: Into<FrRepr>>(&self, x: T, y: T) -> Fr {
pub fn evaluate<T: IntoFr>(&self, x: T, y: T) -> 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?
@ -384,7 +448,7 @@ impl BivarPoly {
}
/// Returns the `x`-th row, as a univariate polynomial.
pub fn row<T: Into<FrRepr>>(&self, x: T) -> Poly {
pub fn row<T: IntoFr>(&self, x: T) -> Poly {
let x_pow = self.powers(x);
let coeff: Vec<Fr> = (0..=self.degree)
.map(|i| {
@ -410,8 +474,8 @@ impl BivarPoly {
}
/// Returns the `0`-th to `degree`-th power of `x`.
fn powers<T: Into<FrRepr>>(&self, x_repr: T) -> Vec<Fr> {
powers(x_repr, self.degree)
fn powers<T: IntoFr>(&self, x: T) -> Vec<Fr> {
powers(x, self.degree)
}
}
@ -441,7 +505,7 @@ impl BivarCommitment {
}
/// Returns the commitment's value at the point `(x, y)`.
pub fn evaluate<T: Into<FrRepr>>(&self, x: T, y: T) -> G1 {
pub fn evaluate<T: IntoFr>(&self, x: T, y: T) -> 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?
@ -458,7 +522,7 @@ impl BivarCommitment {
}
/// Returns the `x`-th row, as a commitment to a univariate polynomial.
pub fn row<T: Into<FrRepr>>(&self, x: T) -> Commitment {
pub fn row<T: IntoFr>(&self, x: T) -> Commitment {
let x_pow = self.powers(x);
let coeff: Vec<G1> = (0..=self.degree)
.map(|i| {
@ -475,18 +539,18 @@ impl BivarCommitment {
}
/// Returns the `0`-th to `degree`-th power of `x`.
fn powers<T: Into<FrRepr>>(&self, x_repr: T) -> Vec<Fr> {
powers(x_repr, self.degree)
fn powers<T: IntoFr>(&self, x: T) -> Vec<Fr> {
powers(x, 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();
fn powers<T: IntoFr>(into_x: T, degree: usize) -> Vec<Fr> {
let x = into_x.into_fr();
let mut x_pow_i = Fr::one();
iter::once(x_pow_i)
.chain((0..degree).map(|_| {
x_pow_i.mul_assign(x);
x_pow_i.mul_assign(&x);
x_pow_i
}))
.collect()
@ -507,20 +571,12 @@ fn coeff_pos(i: usize, j: usize) -> usize {
mod tests {
use std::collections::BTreeMap;
use super::{coeff_pos, BivarPoly, Poly};
use super::{coeff_pos, BivarPoly, IntoFr, Poly};
use pairing::bls12_381::{Fr, G1Affine};
use pairing::{CurveAffine, Field, PrimeField};
use pairing::{CurveAffine, Field};
use rand;
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;
@ -538,22 +594,15 @@ mod tests {
#[test]
fn poly() {
// The polynomial "`5 * x.pow(3) + x.pow(1) - 2`".
let poly =
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)];
// The polynomial 5 X³ + X - 2.
let poly = Poly::monomial(3) * 5 + Poly::monomial(1) - 2;
let coeff: Vec<_> = [-2, 1, 0, 5].into_iter().map(IntoFr::into_fr).collect();
assert_eq!(Poly { coeff }, poly);
let samples = vec![
(fr(-1), fr(-8)),
(fr(2), fr(40)),
(fr(3), fr(136)),
(fr(5), fr(628)),
];
let samples = vec![(-1, -8), (2, 40), (3, 136), (5, 628)];
for &(x, y) in &samples {
assert_eq!(y, poly.evaluate(x));
assert_eq!(y.into_fr(), poly.evaluate(x));
}
let sample_iter = samples.iter().map(|&(ref x, ref y)| (x, y));
assert_eq!(Poly::interpolate(sample_iter), poly);
assert_eq!(Poly::interpolate(samples), poly);
}
#[test]
@ -571,7 +620,7 @@ mod tests {
.collect();
let pub_bi_commits: Vec<_> = bi_polys.iter().map(BivarPoly::commitment).collect();
let mut sec_keys = vec![fr(0); node_num];
let mut sec_keys = vec![Fr::zero(); 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
@ -579,20 +628,20 @@ mod tests {
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);
let row_poly = bi_poly.row(m);
let row_commit = bi_commit.row(m);
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 = row_poly.evaluate(s);
let val_g1 = G1Affine::one().mul(val);
assert_eq!(bi_commit.evaluate(m as u64, s as u64), val_g1);
assert_eq!(bi_commit.evaluate(m, s), 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);
assert_eq!(bi_poly.evaluate(m, s), val);
}
// A cheating dealer who modified the polynomial would be detected.
let wrong_poly = row_poly.clone() + Poly::monomial(2) * Poly::constant(fr(5));
let wrong_poly = row_poly.clone() + Poly::monomial(2) * Poly::constant(5.into_fr());
assert_ne!(wrong_poly.commitment(), row_commit);
// If `2 * faulty_num + 1` nodes confirm that they received a valid row, then at
@ -604,15 +653,15 @@ mod tests {
// `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)))
.map(|&i| (i, bi_poly.evaluate(m, i)))
.collect();
let my_row = Poly::interpolate(&received);
assert_eq!(bi_poly.evaluate(m as u64, 0), my_row.evaluate(0));
let my_row = Poly::interpolate(received);
assert_eq!(bi_poly.evaluate(m, 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));
sec_keys[m - 1].add_assign(&my_row.evaluate(Fr::zero()));
}
}
@ -626,7 +675,7 @@ mod tests {
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]);
assert_eq!(sec_key_set.evaluate(m), sec_keys[m - 1]);
}
// The sum of the first rows of the public commitments is the commitment to the secret key

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@ -407,19 +407,11 @@ where
.get(&self.epoch)
.and_then(|cts| cts.get(&proposer_id))
{
let ids_u64: BTreeMap<&NodeUid, u64> = shares
.keys()
.map(|id| (id, self.netinfo.node_index(id).unwrap() as u64))
.collect();
let indexed_shares: BTreeMap<&u64, _> = shares
.into_iter()
.map(|(id, share)| (&ids_u64[id], share))
.collect();
match self
.netinfo
.public_key_set()
.decrypt(indexed_shares, ciphertext)
{
match {
let to_idx = |(id, share)| (self.netinfo.node_index(id).unwrap(), share);
let share_itr = shares.into_iter().map(to_idx);
self.netinfo.public_key_set().decrypt(share_itr, ciphertext)
} {
Ok(contrib) => {
self.decrypted_contributions.insert(proposer_id, contrib);
}

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@ -242,7 +242,7 @@ impl<NodeUid: Clone + Ord> NetworkInfo<NodeUid> {
.collect();
let public_key_shares = node_indices
.iter()
.map(|(id, idx)| (id.clone(), public_key_set.public_key_share(*idx as u64)))
.map(|(id, idx)| (id.clone(), public_key_set.public_key_share(*idx)))
.collect();
NetworkInfo {
our_uid,
@ -371,7 +371,7 @@ impl<NodeUid: Clone + Ord> NetworkInfo<NodeUid> {
let create_netinfo = |(i, uid): (usize, NodeUid)| {
let netinfo = NetworkInfo::new(
uid.clone(),
sk_set.secret_key_share(i as u64),
sk_set.secret_key_share(i),
pk_set.clone(),
sec_keys[&uid].clone(),
pub_keys.clone(),

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@ -106,18 +106,18 @@
//! let (pks, opt_sks) = node.generate();
//! assert_eq!(pks, pub_key_set); // All nodes now know the public keys and public key shares.
//! let sks = opt_sks.expect("Not an observer node: We receive a secret key share.");
//! secret_key_shares.insert(id as u64, sks);
//! secret_key_shares.insert(id, sks);
//! }
//!
//! // Three out of four nodes can now sign a message. Each share can be verified individually.
//! let msg = "Nodes 0 and 1 does not agree with this.";
//! let mut sig_shares: BTreeMap<u64, SignatureShare> = BTreeMap::new();
//! let mut sig_shares: BTreeMap<usize, SignatureShare> = BTreeMap::new();
//! for (&id, sks) in &secret_key_shares {
//! if id != 0 && id != 1 {
//! let sig_share = sks.sign(msg);
//! let pks = pub_key_set.public_key_share(id as u64);
//! let pks = pub_key_set.public_key_share(id);
//! assert!(pks.verify(&sig_share, msg));
//! sig_shares.insert(id as u64, sig_share);
//! sig_shares.insert(id, sig_share);
//! }
//! }
//!
@ -286,7 +286,7 @@ impl<NodeUid: Ord + Clone + Debug> SyncKeyGen<NodeUid> {
let our_part = BivarPoly::random(threshold, &mut rng);
let commit = our_part.commitment();
let encrypt = |(i, pk): (usize, &PublicKey)| {
let row = our_part.row(i as u64 + 1);
let row = our_part.row(i + 1);
let bytes = bincode::serialize(&row).expect("failed to serialize row");
pk.encrypt(&bytes)
};
@ -334,7 +334,7 @@ impl<NodeUid: Ord + Clone + Debug> SyncKeyGen<NodeUid> {
}
// The row is valid: now encrypt one value for each node.
let encrypt = |(idx, pk): (usize, &PublicKey)| {
let val = row.evaluate(idx as u64 + 1);
let val = row.evaluate(idx + 1);
let wrap = FieldWrap::new(val);
// TODO: Handle errors.
let ser_val = bincode::serialize(&wrap).expect("failed to serialize value");

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@ -68,8 +68,8 @@ fn test_sync_key_gen_with(threshold: usize, node_num: usize) {
let sk = opt_sk.expect("new secret key");
assert_eq!(pks, pub_key_set);
let sig = sk.sign(msg);
assert!(pks.public_key_share(idx as u64).verify(&sig, msg));
(idx as u64, sig)
assert!(pks.public_key_share(idx).verify(&sig, msg));
(idx, sig)
})
.collect();
let sig = pub_key_set