frost/frost-core/src/keys.rs

//! FROST keys, keygen, key shares
#![allow(clippy::type_complexity)]

use std::{
    collections::{BTreeMap, BTreeSet, HashSet},
    convert::TryFrom,
    default::Default,
    fmt::{self, Debug},
    iter,
};

use derive_getters::Getters;
#[cfg(any(test, feature = "test-impl"))]
use hex::FromHex;

use rand_core::{CryptoRng, RngCore};
use zeroize::{DefaultIsZeroes, Zeroize};

use crate::{
    serialization::{Deserialize, Serialize},
    Ciphersuite, Element, Error, Field, Group, Header, Identifier, Scalar, SigningKey,
    VerifyingKey,
};

#[cfg(feature = "serde")]
use crate::serialization::{ElementSerialization, ScalarSerialization};

use super::compute_lagrange_coefficient;

pub mod dkg;
pub mod repairable;

/// Sum the commitments from all participants in a distributed key generation
/// run into a single group commitment.
#[cfg_attr(feature = "internals", visibility::make(pub))]
pub(crate) fn sum_commitments<C: Ciphersuite>(
    commitments: &[&VerifiableSecretSharingCommitment<C>],
) -> Result<VerifiableSecretSharingCommitment<C>, Error<C>> {
    let mut group_commitment = vec![
        CoefficientCommitment(<C::Group>::identity());
        commitments
            .get(0)
            .ok_or(Error::IncorrectNumberOfCommitments)?
            .0
            .len()
    ];
    for commitment in commitments {
        for (i, c) in group_commitment.iter_mut().enumerate() {
            *c = CoefficientCommitment(
                c.value()
                    + commitment
                        .0
                        .get(i)
                        .ok_or(Error::IncorrectNumberOfCommitments)?
                        .value(),
            );
        }
    }
    Ok(VerifiableSecretSharingCommitment(group_commitment))
}

/// Return a vector of randomly generated polynomial coefficients ([`Scalar`]s).
pub(crate) fn generate_coefficients<C: Ciphersuite, R: RngCore + CryptoRng>(
    size: usize,
    rng: &mut R,
) -> Vec<Scalar<C>> {
    iter::repeat_with(|| <<C::Group as Group>::Field>::random(rng))
        .take(size)
        .collect()
}

/// Return a list of default identifiers (1 to max_signers, inclusive).
#[cfg_attr(feature = "internals", visibility::make(pub))]
pub(crate) fn default_identifiers<C: Ciphersuite>(max_signers: u16) -> Vec<Identifier<C>> {
    (1..=max_signers)
        .map(|i| Identifier::<C>::try_from(i).expect("nonzero"))
        .collect::<Vec<_>>()
}

/// A secret scalar value representing a signer's share of the group secret.
#[derive(Clone, Copy, PartialEq, Eq)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "serde", serde(bound = "C: Ciphersuite"))]
#[cfg_attr(feature = "serde", serde(try_from = "ScalarSerialization<C>"))]
#[cfg_attr(feature = "serde", serde(into = "ScalarSerialization<C>"))]
pub struct SigningShare<C: Ciphersuite>(pub(crate) Scalar<C>);

impl<C> SigningShare<C>
where
    C: Ciphersuite,
{
    /// Create a new [`SigningShare`] from a scalar.
    #[cfg(feature = "internals")]
    pub fn new(scalar: Scalar<C>) -> Self {
        Self(scalar)
    }

    /// Get the inner scalar.
    #[cfg(feature = "internals")]
    pub fn to_scalar(&self) -> Scalar<C> {
        self.0
    }

    /// Deserialize from bytes
    pub fn deserialize(
        bytes: <<C::Group as Group>::Field as Field>::Serialization,
    ) -> Result<Self, Error<C>> {
        <<C::Group as Group>::Field>::deserialize(&bytes)
            .map(|scalar| Self(scalar))
            .map_err(|e| e.into())
    }

    /// Serialize to bytes
    pub fn serialize(&self) -> <<C::Group as Group>::Field as Field>::Serialization {
        <<C::Group as Group>::Field>::serialize(&self.0)
    }

    /// Computes the signing share from a list of coefficients.
    #[cfg_attr(feature = "internals", visibility::make(pub))]
    pub(crate) fn from_coefficients(coefficients: &[Scalar<C>], peer: Identifier<C>) -> Self {
        Self(evaluate_polynomial(peer, coefficients))
    }
}

impl<C> Debug for SigningShare<C>
where
    C: Ciphersuite,
{
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> std::fmt::Result {
        f.debug_tuple("SigningShare").field(&"<redacted>").finish()
    }
}

impl<C> Default for SigningShare<C>
where
    C: Ciphersuite,
{
    fn default() -> Self {
        Self(<<C::Group as Group>::Field>::zero())
    }
}

// Implements [`Zeroize`] by overwriting a value with the [`Default::default()`] value
impl<C> DefaultIsZeroes for SigningShare<C> where C: Ciphersuite {}

#[cfg(any(test, feature = "test-impl"))]
impl<C> FromHex for SigningShare<C>
where
    C: Ciphersuite,
{
    type Error = &'static str;

    fn from_hex<T: AsRef<[u8]>>(hex: T) -> Result<Self, Self::Error> {
        let v: Vec<u8> = FromHex::from_hex(hex).map_err(|_| "invalid hex")?;
        match v.try_into() {
            Ok(bytes) => Self::deserialize(bytes).map_err(|_| "malformed secret encoding"),
            Err(_) => Err("malformed secret encoding"),
        }
    }
}

#[cfg(feature = "serde")]
impl<C> TryFrom<ScalarSerialization<C>> for SigningShare<C>
where
    C: Ciphersuite,
{
    type Error = Error<C>;

    fn try_from(value: ScalarSerialization<C>) -> Result<Self, Self::Error> {
        Self::deserialize(value.0)
    }
}

#[cfg(feature = "serde")]
impl<C> From<SigningShare<C>> for ScalarSerialization<C>
where
    C: Ciphersuite,
{
    fn from(value: SigningShare<C>) -> Self {
        Self(value.serialize())
    }
}

/// A public group element that represents a single signer's public verification share.
#[derive(Copy, Clone, PartialEq, Eq)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "serde", serde(bound = "C: Ciphersuite"))]
#[cfg_attr(feature = "serde", serde(try_from = "ElementSerialization<C>"))]
#[cfg_attr(feature = "serde", serde(into = "ElementSerialization<C>"))]
pub struct VerifyingShare<C>(pub(super) Element<C>)
where
    C: Ciphersuite;

impl<C> VerifyingShare<C>
where
    C: Ciphersuite,
{
    /// Create a new [`VerifyingShare`] from a element.
    #[cfg(feature = "internals")]
    pub fn new(element: Element<C>) -> Self {
        Self(element)
    }

    /// Get the inner element.
    #[cfg(feature = "internals")]
    pub fn to_element(&self) -> Element<C> {
        self.0
    }

    /// Deserialize from bytes
    pub fn deserialize(bytes: <C::Group as Group>::Serialization) -> Result<Self, Error<C>> {
        <C::Group as Group>::deserialize(&bytes)
            .map(|element| Self(element))
            .map_err(|e| e.into())
    }

    /// Serialize to bytes
    pub fn serialize(&self) -> <C::Group as Group>::Serialization {
        <C::Group as Group>::serialize(&self.0)
    }

    /// Computes a verifying share for a peer given the group commitment.
    #[cfg_attr(feature = "internals", visibility::make(pub))]
    pub(crate) fn from_commitment(
        identifier: Identifier<C>,
        commitment: &VerifiableSecretSharingCommitment<C>,
    ) -> VerifyingShare<C> {
        // DKG Round 2, Step 4
        //
        // > Any participant can compute the public verification share of any
        // > other participant by calculating
        // > Y_i = ∏_{j=1}^n ∏_{k=0}^{t−1} φ_{jk}^{i^k mod q}.
        //
        // Rewriting the equation by moving the product over j to further inside
        // the equation:
        // Y_i = ∏_{k=0}^{t−1} (∏_{j=1}^n φ_{jk})^{i^k mod q}
        // i.e. we can operate on the sum of all φ_j commitments, which is
        // what is passed to the functions.
        VerifyingShare(evaluate_vss(identifier, commitment))
    }
}

impl<C> Debug for VerifyingShare<C>
where
    C: Ciphersuite,
{
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        f.debug_tuple("VerifyingShare")
            .field(&hex::encode(self.serialize()))
            .finish()
    }
}

impl<C> From<SigningShare<C>> for VerifyingShare<C>
where
    C: Ciphersuite,
{
    fn from(secret: SigningShare<C>) -> VerifyingShare<C> {
        VerifyingShare(<C::Group>::generator() * secret.0 as Scalar<C>)
    }
}

#[cfg(feature = "serde")]
impl<C> TryFrom<ElementSerialization<C>> for VerifyingShare<C>
where
    C: Ciphersuite,
{
    type Error = Error<C>;

    fn try_from(value: ElementSerialization<C>) -> Result<Self, Self::Error> {
        Self::deserialize(value.0)
    }
}

#[cfg(feature = "serde")]
impl<C> From<VerifyingShare<C>> for ElementSerialization<C>
where
    C: Ciphersuite,
{
    fn from(value: VerifyingShare<C>) -> Self {
        Self(value.serialize())
    }
}

/// A [`Group::Element`] newtype that is a commitment to one coefficient of our secret polynomial.
///
/// This is a (public) commitment to one coefficient of a secret polynomial used for performing
/// verifiable secret sharing for a Shamir secret share.
#[derive(Clone, Copy, PartialEq, Eq)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "serde", serde(bound = "C: Ciphersuite"))]
#[cfg_attr(feature = "serde", serde(try_from = "ElementSerialization<C>"))]
#[cfg_attr(feature = "serde", serde(into = "ElementSerialization<C>"))]
pub struct CoefficientCommitment<C: Ciphersuite>(pub(crate) Element<C>);

impl<C> CoefficientCommitment<C>
where
    C: Ciphersuite,
{
    /// returns serialized element
    pub fn serialize(&self) -> <C::Group as Group>::Serialization {
        <C::Group>::serialize(&self.0)
    }

    /// Creates a new commitment from a coefficient input
    pub fn deserialize(
        coefficient: <C::Group as Group>::Serialization,
    ) -> Result<CoefficientCommitment<C>, Error<C>> {
        Ok(Self(<C::Group as Group>::deserialize(&coefficient)?))
    }

    /// Returns inner element value
    pub fn value(&self) -> Element<C> {
        self.0
    }
}

impl<C> Debug for CoefficientCommitment<C>
where
    C: Ciphersuite,
{
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> std::fmt::Result {
        f.debug_tuple("CoefficientCommitment")
            .field(&hex::encode(self.serialize()))
            .finish()
    }
}

#[cfg(feature = "serde")]
impl<C> TryFrom<ElementSerialization<C>> for CoefficientCommitment<C>
where
    C: Ciphersuite,
{
    type Error = Error<C>;

    fn try_from(value: ElementSerialization<C>) -> Result<Self, Self::Error> {
        Self::deserialize(value.0)
    }
}

#[cfg(feature = "serde")]
impl<C> From<CoefficientCommitment<C>> for ElementSerialization<C>
where
    C: Ciphersuite,
{
    fn from(value: CoefficientCommitment<C>) -> Self {
        Self(value.serialize())
    }
}

/// Contains the commitments to the coefficients for our secret polynomial _f_,
/// used to generate participants' key shares.
///
/// [`VerifiableSecretSharingCommitment`] contains a set of commitments to the coefficients (which
/// themselves are scalars) for a secret polynomial f, where f is used to
/// generate each ith participant's key share f(i). Participants use this set of
/// commitments to perform verifiable secret sharing.
///
/// Note that participants MUST be assured that they have the *same*
/// [`VerifiableSecretSharingCommitment`], either by performing pairwise comparison, or by using
/// some agreed-upon public location for publication, where each participant can
/// ensure that they received the correct (and same) value.
#[derive(Clone, Debug, PartialEq, Eq)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "serde", serde(bound = "C: Ciphersuite"))]
pub struct VerifiableSecretSharingCommitment<C: Ciphersuite>(
    pub(crate) Vec<CoefficientCommitment<C>>,
);

impl<C> VerifiableSecretSharingCommitment<C>
where
    C: Ciphersuite,
{
    /// Returns serialized coefficent commitments
    pub fn serialize(&self) -> Vec<<C::Group as Group>::Serialization> {
        self.0
            .iter()
            .map(|cc| <<C as Ciphersuite>::Group as Group>::serialize(&cc.0))
            .collect()
    }

    /// Returns VerifiableSecretSharingCommitment from a vector of serialized CoefficientCommitments
    pub fn deserialize(
        serialized_coefficient_commitments: Vec<<C::Group as Group>::Serialization>,
    ) -> Result<Self, Error<C>> {
        let mut coefficient_commitments = Vec::new();
        for cc in serialized_coefficient_commitments {
            coefficient_commitments.push(CoefficientCommitment::<C>::deserialize(cc)?);
        }

        Ok(Self(coefficient_commitments))
    }

    /// Get the VerifyingKey matching this commitment vector (which is the first
    /// element in the vector), or an error if the vector is empty.
    pub(crate) fn verifying_key(&self) -> Result<VerifyingKey<C>, Error<C>> {
        Ok(VerifyingKey::new(
            self.0.get(0).ok_or(Error::MissingCommitment)?.0,
        ))
    }

    /// Returns the coefficient commitments.
    #[cfg_attr(feature = "internals", visibility::make(pub))]
    pub(crate) fn coefficients(&self) -> &[CoefficientCommitment<C>] {
        &self.0
    }
}

/// A secret share generated by performing a (t-out-of-n) secret sharing scheme,
/// generated by a dealer performing [`generate_with_dealer`].
///
/// `n` is the total number of shares and `t` is the threshold required to reconstruct the secret;
/// in this case we use Shamir's secret sharing.
///
/// As a solution to the secret polynomial _f_ (a 'point'), the `identifier` is the x-coordinate, and the
/// `value` is the y-coordinate.
///
/// To derive a FROST keypair, the receiver of the [`SecretShare`] *must* call
/// .into(), which under the hood also performs validation.
#[derive(Clone, Debug, Zeroize, PartialEq, Eq, Getters)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "serde", serde(bound = "C: Ciphersuite"))]
#[cfg_attr(feature = "serde", serde(deny_unknown_fields))]
pub struct SecretShare<C: Ciphersuite> {
    /// Serialization header
    #[getter(skip)]
    pub(crate) header: Header<C>,
    /// The participant identifier of this [`SecretShare`].
    #[zeroize(skip)]
    pub(crate) identifier: Identifier<C>,
    /// Secret Key.
    pub(crate) signing_share: SigningShare<C>,
    #[zeroize(skip)]
    /// The commitments to be distributed among signers.
    pub(crate) commitment: VerifiableSecretSharingCommitment<C>,
}

impl<C> SecretShare<C>
where
    C: Ciphersuite,
{
    /// Create a new [`SecretShare`] instance.
    pub fn new(
        identifier: Identifier<C>,
        signing_share: SigningShare<C>,
        commitment: VerifiableSecretSharingCommitment<C>,
    ) -> Self {
        SecretShare {
            header: Header::default(),
            identifier,
            signing_share,
            commitment,
        }
    }

    /// Verifies that a secret share is consistent with a verifiable secret sharing commitment,
    /// and returns the derived group info for the participant (their public verification share,
    /// and the group public key) if successful.
    ///
    /// This ensures that this participant's share has been generated using the same
    /// mechanism as all other signing participants. Note that participants *MUST*
    /// ensure that they have the same view as all other participants of the
    /// commitment!
    ///
    /// An implementation of `vss_verify()` from the [spec].
    /// This also implements `derive_group_info()` from the [spec] (which is very similar),
    /// but only for this participant.
    ///
    /// [spec]: https://www.ietf.org/archive/id/draft-irtf-cfrg-frost-14.html#appendix-C.2-4
    pub fn verify(&self) -> Result<(VerifyingShare<C>, VerifyingKey<C>), Error<C>> {
        let f_result = <C::Group>::generator() * self.signing_share.0;
        let result = evaluate_vss(self.identifier, &self.commitment);

        if !(f_result == result) {
            return Err(Error::InvalidSecretShare);
        }

        Ok((VerifyingShare(result), self.commitment.verifying_key()?))
    }
}

#[cfg(feature = "serialization")]
impl<C> SecretShare<C>
where
    C: Ciphersuite,
{
    /// Serialize the struct into a Vec.
    pub fn serialize(&self) -> Result<Vec<u8>, Error<C>> {
        Serialize::serialize(&self)
    }

    /// Deserialize the struct from a slice of bytes.
    pub fn deserialize(bytes: &[u8]) -> Result<Self, Error<C>> {
        Deserialize::deserialize(bytes)
    }
}

/// The identifier list to use when generating key shares.
pub enum IdentifierList<'a, C: Ciphersuite> {
    /// Use the default values (1 to max_signers, inclusive).
    Default,
    /// A user-provided list of identifiers.
    Custom(&'a [Identifier<C>]),
}

/// Allows all participants' keys to be generated using a central, trusted
/// dealer.
///
/// Under the hood, this performs verifiable secret sharing, which itself uses
/// Shamir secret sharing, from which each share becomes a participant's secret
/// key. The output from this function is a set of shares along with one single
/// commitment that participants use to verify the integrity of the share.
///
/// Implements [`trusted_dealer_keygen`] from the spec.
///
/// [`trusted_dealer_keygen`]: https://www.ietf.org/archive/id/draft-irtf-cfrg-frost-14.html#appendix-C
pub fn generate_with_dealer<C: Ciphersuite, R: RngCore + CryptoRng>(
    max_signers: u16,
    min_signers: u16,
    identifiers: IdentifierList<C>,
    rng: &mut R,
) -> Result<(BTreeMap<Identifier<C>, SecretShare<C>>, PublicKeyPackage<C>), Error<C>> {
    let mut bytes = [0; 64];
    rng.fill_bytes(&mut bytes);

    let key = SigningKey::new(rng);

    split(&key, max_signers, min_signers, identifiers, rng)
}

/// Splits an existing key into FROST shares.
///
/// This is identical to [`generate_with_dealer`] but receives an existing key
/// instead of generating a fresh one. This is useful in scenarios where
/// the key needs to be generated externally or must be derived from e.g. a
/// seed phrase.
pub fn split<C: Ciphersuite, R: RngCore + CryptoRng>(
    key: &SigningKey<C>,
    max_signers: u16,
    min_signers: u16,
    identifiers: IdentifierList<C>,
    rng: &mut R,
) -> Result<(BTreeMap<Identifier<C>, SecretShare<C>>, PublicKeyPackage<C>), Error<C>> {
    validate_num_of_signers(min_signers, max_signers)?;

    if let IdentifierList::Custom(identifiers) = &identifiers {
        if identifiers.len() != max_signers as usize {
            return Err(Error::IncorrectNumberOfIdentifiers);
        }
    }

    let verifying_key = VerifyingKey::from(key);

    let coefficients = generate_coefficients::<C, R>(min_signers as usize - 1, rng);

    let secret_shares = match identifiers {
        IdentifierList::Default => {
            let identifiers = default_identifiers(max_signers);
            generate_secret_shares(key, max_signers, min_signers, coefficients, &identifiers)?
        }
        IdentifierList::Custom(identifiers) => {
            generate_secret_shares(key, max_signers, min_signers, coefficients, identifiers)?
        }
    };
    let mut verifying_shares: BTreeMap<Identifier<C>, VerifyingShare<C>> = BTreeMap::new();

    let mut secret_shares_by_id: BTreeMap<Identifier<C>, SecretShare<C>> = BTreeMap::new();

    for secret_share in secret_shares {
        let signer_public = secret_share.signing_share.into();
        verifying_shares.insert(secret_share.identifier, signer_public);

        secret_shares_by_id.insert(secret_share.identifier, secret_share);
    }

    Ok((
        secret_shares_by_id,
        PublicKeyPackage {
            header: Header::default(),
            verifying_shares,
            verifying_key,
        },
    ))
}

/// Evaluate the polynomial with the given coefficients (constant term first)
/// at the point x=identifier using Horner's method.
///
/// Implements [`polynomial_evaluate`] from the spec.
///
/// [`polynomial_evaluate`]: https://www.ietf.org/archive/id/draft-irtf-cfrg-frost-14.html#name-evaluation-of-a-polynomial
fn evaluate_polynomial<C: Ciphersuite>(
    identifier: Identifier<C>,
    coefficients: &[Scalar<C>],
) -> Scalar<C> {
    let mut value = <<C::Group as Group>::Field>::zero();

    let ell_scalar = identifier;
    for coeff in coefficients.iter().skip(1).rev() {
        value = value + *coeff;
        value *= ell_scalar;
    }
    value = value
        + *coefficients
            .get(0)
            .expect("coefficients must have at least one element");
    value
}

/// Evaluates the right-hand side of the VSS verification equation, namely
/// ∏^{t−1}_{k=0} φ^{i^k mod q}_{ℓk} (multiplicative notation) using
/// `identifier` as `i` and the `commitment` as the commitment vector φ_ℓ.
///
/// This is also used in Round 2, Step 4 of the DKG.
fn evaluate_vss<C: Ciphersuite>(
    identifier: Identifier<C>,
    commitment: &VerifiableSecretSharingCommitment<C>,
) -> Element<C> {
    let i = identifier;

    let (_, result) = commitment.0.iter().fold(
        (<<C::Group as Group>::Field>::one(), <C::Group>::identity()),
        |(i_to_the_k, sum_so_far), comm_k| (i * i_to_the_k, sum_so_far + comm_k.0 * i_to_the_k),
    );
    result
}

/// A FROST keypair, which can be generated either by a trusted dealer or using
/// a DKG.
///
/// When using a central dealer, [`SecretShare`]s are distributed to
/// participants, who then perform verification, before deriving
/// [`KeyPackage`]s, which they store to later use during signing.
#[derive(Clone, Debug, PartialEq, Eq, Getters, Zeroize)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "serde", serde(bound = "C: Ciphersuite"))]
#[cfg_attr(feature = "serde", serde(deny_unknown_fields))]
pub struct KeyPackage<C: Ciphersuite> {
    /// Serialization header
    #[getter(skip)]
    pub(crate) header: Header<C>,
    /// Denotes the participant identifier each secret share key package is owned by.
    #[zeroize(skip)]
    pub(crate) identifier: Identifier<C>,
    /// This participant's signing share. This is secret.
    pub(crate) signing_share: SigningShare<C>,
    /// This participant's public key.
    #[zeroize(skip)]
    pub(crate) verifying_share: VerifyingShare<C>,
    /// The public verifying key that represents the entire group.
    #[zeroize(skip)]
    pub(crate) verifying_key: VerifyingKey<C>,
    pub(crate) min_signers: u16,
}

impl<C> KeyPackage<C>
where
    C: Ciphersuite,
{
    /// Create a new [`KeyPackage`] instance.
    pub fn new(
        identifier: Identifier<C>,
        signing_share: SigningShare<C>,
        verifying_share: VerifyingShare<C>,
        verifying_key: VerifyingKey<C>,
        min_signers: u16,
    ) -> Self {
        Self {
            header: Header::default(),
            identifier,
            signing_share,
            verifying_share,
            verifying_key,
            min_signers,
        }
    }
}

#[cfg(feature = "serialization")]
impl<C> KeyPackage<C>
where
    C: Ciphersuite,
{
    /// Serialize the struct into a Vec.
    pub fn serialize(&self) -> Result<Vec<u8>, Error<C>> {
        Serialize::serialize(&self)
    }

    /// Deserialize the struct from a slice of bytes.
    pub fn deserialize(bytes: &[u8]) -> Result<Self, Error<C>> {
        Deserialize::deserialize(bytes)
    }
}

impl<C> TryFrom<SecretShare<C>> for KeyPackage<C>
where
    C: Ciphersuite,
{
    type Error = Error<C>;

    /// Tries to verify a share and construct a [`KeyPackage`] from it.
    ///
    /// When participants receive a [`SecretShare`] from the dealer, they
    /// *MUST* verify the integrity of the share before continuing on to
    /// transform it into a signing/verification keypair. Here, we assume that
    /// every participant has the same view of the commitment issued by the
    /// dealer, but implementations *MUST* make sure that all participants have
    /// a consistent view of this commitment in practice.
    fn try_from(secret_share: SecretShare<C>) -> Result<Self, Error<C>> {
        let (verifying_share, verifying_key) = secret_share.verify()?;

        Ok(KeyPackage {
            header: Header::default(),
            identifier: secret_share.identifier,
            signing_share: secret_share.signing_share,
            verifying_share,
            verifying_key,
            min_signers: secret_share.commitment.0.len() as u16,
        })
    }
}

/// Public data that contains all the signers' verifying shares as well as the
/// group verifying key.
///
/// Used for verification purposes before publishing a signature.
#[derive(Clone, Debug, PartialEq, Eq, Getters)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "serde", serde(bound = "C: Ciphersuite"))]
#[cfg_attr(feature = "serde", serde(deny_unknown_fields))]
pub struct PublicKeyPackage<C: Ciphersuite> {
    /// Serialization header
    #[getter(skip)]
    pub(crate) header: Header<C>,
    /// The verifying shares for all participants. Used to validate signature
    /// shares they generate.
    pub(crate) verifying_shares: BTreeMap<Identifier<C>, VerifyingShare<C>>,
    /// The joint public key for the entire group.
    pub(crate) verifying_key: VerifyingKey<C>,
}

impl<C> PublicKeyPackage<C>
where
    C: Ciphersuite,
{
    /// Create a new [`PublicKeyPackage`] instance.
    pub fn new(
        verifying_shares: BTreeMap<Identifier<C>, VerifyingShare<C>>,
        verifying_key: VerifyingKey<C>,
    ) -> Self {
        Self {
            header: Header::default(),
            verifying_shares,
            verifying_key,
        }
    }

    /// Computes the public key package given a list of participant identifiers
    /// and a [`VerifiableSecretSharingCommitment`]. This is useful in scenarios
    /// where the commitments are published somewhere and it's desirable to
    /// recreate the public key package from them.
    pub fn from_commitment(
        identifiers: &BTreeSet<Identifier<C>>,
        commitment: &VerifiableSecretSharingCommitment<C>,
    ) -> Result<PublicKeyPackage<C>, Error<C>> {
        let verifying_keys: BTreeMap<_, _> = identifiers
            .iter()
            .map(|id| (*id, VerifyingShare::from_commitment(*id, commitment)))
            .collect();
        Ok(PublicKeyPackage::new(
            verifying_keys,
            VerifyingKey::from_commitment(commitment)?,
        ))
    }

    /// Computes the public key package given a map of participant identifiers
    /// and their [`VerifiableSecretSharingCommitment`] from a distributed key
    /// generation process. This is useful in scenarios where the commitments
    /// are published somewhere and it's desirable to recreate the public key
    /// package from them.
    pub fn from_dkg_commitments(
        commitments: &BTreeMap<Identifier<C>, &VerifiableSecretSharingCommitment<C>>,
    ) -> Result<PublicKeyPackage<C>, Error<C>> {
        let identifiers: BTreeSet<_> = commitments.keys().copied().collect();
        let commitments: Vec<_> = commitments.values().copied().collect();
        let group_commitment = sum_commitments(&commitments)?;
        Self::from_commitment(&identifiers, &group_commitment)
    }
}

#[cfg(feature = "serialization")]
impl<C> PublicKeyPackage<C>
where
    C: Ciphersuite,
{
    /// Serialize the struct into a Vec.
    pub fn serialize(&self) -> Result<Vec<u8>, Error<C>> {
        Serialize::serialize(&self)
    }

    /// Deserialize the struct from a slice of bytes.
    pub fn deserialize(bytes: &[u8]) -> Result<Self, Error<C>> {
        Deserialize::deserialize(bytes)
    }
}

fn validate_num_of_signers<C: Ciphersuite>(
    min_signers: u16,
    max_signers: u16,
) -> Result<(), Error<C>> {
    if min_signers < 2 {
        return Err(Error::InvalidMinSigners);
    }

    if max_signers < 2 {
        return Err(Error::InvalidMaxSigners);
    }

    if min_signers > max_signers {
        return Err(Error::InvalidMinSigners);
    }

    Ok(())
}

/// Generate a secret polynomial to use in secret sharing, for the given
/// secret value. Also validates the given parameters.
///
/// Returns the full vector of coefficients in little-endian order (including the
/// given secret, which is the first element) and a [`VerifiableSecretSharingCommitment`]
/// which contains commitments to those coefficients.
///
/// Returns an error if the parameters (max_signers, min_signers) are inconsistent.
pub(crate) fn generate_secret_polynomial<C: Ciphersuite>(
    secret: &SigningKey<C>,
    max_signers: u16,
    min_signers: u16,
    mut coefficients: Vec<Scalar<C>>,
) -> Result<(Vec<Scalar<C>>, VerifiableSecretSharingCommitment<C>), Error<C>> {
    validate_num_of_signers(min_signers, max_signers)?;

    if coefficients.len() != min_signers as usize - 1 {
        return Err(Error::InvalidCoefficients);
    }

    // Prepend the secret, which is the 0th coefficient
    coefficients.insert(0, secret.scalar);

    // Create the vector of commitments
    let commitment: Vec<_> = coefficients
        .iter()
        .map(|c| CoefficientCommitment(<C::Group as Group>::generator() * *c))
        .collect();
    let commitment: VerifiableSecretSharingCommitment<C> =
        VerifiableSecretSharingCommitment(commitment);

    Ok((coefficients, commitment))
}

/// Creates secret shares for a given secret using the given coefficients.
///
/// This function accepts a secret from which shares are generated,
/// and a list of threshold-1 coefficients. While in FROST this secret
/// and coefficients should always be generated randomly, we allow them
/// to be specified for this internal function for testability.
///
/// Internally, [`generate_secret_shares`] performs verifiable secret sharing, which
/// generates shares via Shamir Secret Sharing, and then generates public
/// commitments to those shares.
///
/// More specifically, [`generate_secret_shares`]:
/// - Interpret [secret, `coefficients[0]`, ...] as a secret polynomial f
/// - For each participant i, their secret share is f(i)
/// - The commitment to the secret polynomial f is [g^secret, `g^coefficients[0]`, ...]
///
/// Implements [`secret_share_shard`] from the spec.
///
/// [`secret_share_shard`]: https://www.ietf.org/archive/id/draft-irtf-cfrg-frost-14.html#appendix-C.1
pub(crate) fn generate_secret_shares<C: Ciphersuite>(
    secret: &SigningKey<C>,
    max_signers: u16,
    min_signers: u16,
    coefficients: Vec<Scalar<C>>,
    identifiers: &[Identifier<C>],
) -> Result<Vec<SecretShare<C>>, Error<C>> {
    let mut secret_shares: Vec<SecretShare<C>> = Vec::with_capacity(max_signers as usize);

    let (coefficients, commitment) =
        generate_secret_polynomial(secret, max_signers, min_signers, coefficients)?;

    let identifiers_set: HashSet<_> = identifiers.iter().collect();
    if identifiers_set.len() != identifiers.len() {
        return Err(Error::DuplicatedIdentifier);
    }

    for id in identifiers {
        let signing_share = SigningShare::from_coefficients(&coefficients, *id);

        secret_shares.push(SecretShare {
            header: Header::default(),
            identifier: *id,
            signing_share,
            commitment: commitment.clone(),
        });
    }

    Ok(secret_shares)
}

/// Recompute the secret from at least `min_signers` secret shares (inside
/// [`KeyPackage`]s) using Lagrange interpolation.
///
/// This can be used if for some reason the original key must be restored; e.g.
/// if threshold signing is not required anymore.
///
/// This is NOT required to sign with FROST; the point of FROST is being
/// able to generate signatures only using the shares, without having to
/// reconstruct the original key.
///
/// The caller is responsible for providing at least `min_signers` packages;
/// if less than that is provided, a different key will be returned.
pub fn reconstruct<C: Ciphersuite>(
    key_packages: &[KeyPackage<C>],
) -> Result<SigningKey<C>, Error<C>> {
    if key_packages.is_empty() {
        return Err(Error::IncorrectNumberOfShares);
    }
    // There is no obvious way to get `min_signers` in order to validate the
    // size of `secret_shares`. Since that is just a best-effort validation,
    // we don't need to worry too much about adversarial situations where people
    // lie about min_signers, so just get the minimum value out of all of them.
    let min_signers = key_packages
        .iter()
        .map(|k| k.min_signers)
        .min()
        .expect("should not be empty since that was just tested");
    if key_packages.len() < min_signers as usize {
        return Err(Error::IncorrectNumberOfShares);
    }

    let mut secret = <<C::Group as Group>::Field>::zero();

    let identifiers: BTreeSet<_> = key_packages
        .iter()
        .map(|s| s.identifier())
        .cloned()
        .collect();

    if identifiers.len() != key_packages.len() {
        return Err(Error::DuplicatedIdentifier);
    }

    // Compute the Lagrange coefficients
    for key_package in key_packages.iter() {
        let lagrange_coefficient =
            compute_lagrange_coefficient(&identifiers, None, key_package.identifier)?;

        // Compute y = f(0) via polynomial interpolation of these t-of-n solutions ('points) of f
        secret = secret + (lagrange_coefficient * key_package.signing_share().0);
    }

    Ok(SigningKey { scalar: secret })
}