commonware_cryptography/bls12381/primitives/

ops.rs

//! Digital signatures over the BLS12-381 curve using G1 as the Public Key (48 bytes)
//! and G2 as the Signature (96 bytes).
//!
//! # Domain Separation Tag (DST)
//!
//! All signatures use the `POP` (Proof of Possession) scheme during signing. For Proof-of-Possession (POP) signatures,
//! the domain separation tag is `BLS_POP_BLS12381G2_XMD:SHA-256_SSWU_RO_POP_`. For signatures over other messages, the
//! domain separation tag is `BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_POP_`. You can read more about DSTs [here](https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-bls-signature-05#section-4.2).

use super::{
    group::{self, Element, Point, Scalar, Share, DST},
    poly::{self, Eval, PartialSignature, Weight},
    variant::Variant,
    Error,
};
use crate::bls12381::primitives::poly::{compute_weights, prepare_evaluations};
#[cfg(not(feature = "std"))]
use alloc::{borrow::Cow, collections::BTreeMap, vec, vec::Vec};
use commonware_codec::Encode;
use commonware_utils::union_unique;
use rand_core::CryptoRngCore;
#[cfg(feature = "std")]
use rayon::{prelude::*, ThreadPoolBuilder};
#[cfg(feature = "std")]
use std::{borrow::Cow, collections::BTreeMap};

/// Computes the public key from the private key.
pub fn compute_public<V: Variant>(private: &Scalar) -> V::Public {
    let mut public = V::Public::one();
    public.mul(private);
    public
}

/// Returns a new keypair derived from the provided randomness.
pub fn keypair<R: CryptoRngCore, V: Variant>(rng: &mut R) -> (group::Private, V::Public) {
    let private = group::Private::from_rand(rng);
    let public = compute_public::<V>(&private);
    (private, public)
}

/// Hashes the provided message with the domain separation tag (DST) to
/// the curve.
pub fn hash_message<V: Variant>(dst: DST, message: &[u8]) -> V::Signature {
    let mut hm = V::Signature::zero();
    hm.map(dst, message);
    hm
}

/// Hashes the provided message with the domain separation tag (DST) and namespace to
/// the curve.
pub fn hash_message_namespace<V: Variant>(
    dst: DST,
    namespace: &[u8],
    message: &[u8],
) -> V::Signature {
    let mut hm = V::Signature::zero();
    hm.map(dst, &union_unique(namespace, message));
    hm
}

/// Signs the provided message with the private key.
pub fn sign<V: Variant>(private: &Scalar, dst: DST, message: &[u8]) -> V::Signature {
    let mut hm = hash_message::<V>(dst, message);
    hm.mul(private);
    hm
}

/// Generates a proof of possession for the private key.
pub fn sign_proof_of_possession<V: Variant>(private: &group::Private) -> V::Signature {
    // Get public key
    let public = compute_public::<V>(private);

    // Sign the public key
    sign::<V>(private, V::PROOF_OF_POSSESSION, &public.encode())
}

/// Verifies the signature with the provided public key.
pub fn verify<V: Variant>(
    public: &V::Public,
    dst: DST,
    message: &[u8],
    signature: &V::Signature,
) -> Result<(), Error> {
    // Create hashed message `hm`
    let hm = hash_message::<V>(dst, message);

    // Verify the signature
    V::verify(public, &hm, signature)
}

/// Verifies a proof of possession for the provided public key.
pub fn verify_proof_of_possession<V: Variant>(
    public: &V::Public,
    signature: &V::Signature,
) -> Result<(), Error> {
    verify::<V>(public, V::PROOF_OF_POSSESSION, &public.encode(), signature)
}

/// Signs the provided message with the private key.
///
/// # Determinism
///
/// Signatures produced by this function are deterministic and are safe
/// to use in a consensus-critical context.
pub fn sign_message<V: Variant>(
    private: &group::Private,
    namespace: Option<&[u8]>,
    message: &[u8],
) -> V::Signature {
    let payload = match namespace {
        Some(namespace) => Cow::Owned(union_unique(namespace, message)),
        None => Cow::Borrowed(message),
    };
    sign::<V>(private, V::MESSAGE, &payload)
}

/// Verifies the signature with the provided public key.
///
/// # Warning
///
/// This function assumes a group check was already performed on
/// `public` and `signature`.
pub fn verify_message<V: Variant>(
    public: &V::Public,
    namespace: Option<&[u8]>,
    message: &[u8],
    signature: &V::Signature,
) -> Result<(), Error> {
    let payload = match namespace {
        Some(namespace) => Cow::Owned(union_unique(namespace, message)),
        None => Cow::Borrowed(message),
    };
    verify::<V>(public, V::MESSAGE, &payload, signature)
}

/// Generates a proof of possession for the private key share.
pub fn partial_sign_proof_of_possession<V: Variant>(
    public: &poly::Public<V>,
    private: &Share,
) -> PartialSignature<V> {
    // Get public key
    let threshold_public = poly::public::<V>(public);

    // Sign the public key
    let sig = sign::<V>(
        &private.private,
        V::PROOF_OF_POSSESSION,
        &threshold_public.encode(),
    );
    Eval {
        value: sig,
        index: private.index,
    }
}

/// Verifies the proof of possession for the provided public polynomial.
///
/// # Warning
///
/// This function assumes a group check was already performed on `signature`.
pub fn partial_verify_proof_of_possession<V: Variant>(
    public: &poly::Public<V>,
    partial: &PartialSignature<V>,
) -> Result<(), Error> {
    let threshold_public = poly::public::<V>(public);
    let public = public.evaluate(partial.index);
    verify::<V>(
        &public.value,
        V::PROOF_OF_POSSESSION,
        &threshold_public.encode(),
        &partial.value,
    )
}

/// Signs the provided message with the key share.
pub fn partial_sign_message<V: Variant>(
    private: &Share,
    namespace: Option<&[u8]>,
    message: &[u8],
) -> PartialSignature<V> {
    let sig = sign_message::<V>(&private.private, namespace, message);
    Eval {
        value: sig,
        index: private.index,
    }
}

/// Verifies the partial signature against the public polynomial.
///
/// # Warning
///
/// This function assumes a group check was already performed on `signature`.
pub fn partial_verify_message<V: Variant>(
    public: &poly::Public<V>,
    namespace: Option<&[u8]>,
    message: &[u8],
    partial: &PartialSignature<V>,
) -> Result<(), Error> {
    let public = public.evaluate(partial.index);
    verify_message::<V>(&public.value, namespace, message, &partial.value)
}

/// Aggregates multiple partial signatures into a single signature.
///
/// # Warning
///
/// This function assumes a group check was already performed on each `signature` and
/// that each `signature` is unique. If any of these assumptions are violated, an attacker can
/// exploit this function to verify an incorrect aggregate signature.
pub fn partial_aggregate_signatures<'a, V, I>(partials: I) -> Option<(u32, V::Signature)>
where
    V: Variant,
    I: IntoIterator<Item = &'a PartialSignature<V>>,
    V::Signature: 'a,
{
    let mut iter = partials.into_iter().peekable();
    let index = iter.peek()?.index;
    let mut s = V::Signature::zero();
    for partial in iter {
        if partial.index != index {
            return None;
        }
        s.add(&partial.value);
    }
    Some((index, s))
}

/// Verifies the signatures from multiple partial signatures over multiple unique messages from a single
/// signer.
///
/// # Warning
///
/// This function assumes a group check was already performed on each `signature`.
pub fn partial_verify_multiple_messages<'a, V, I, J>(
    public: &poly::Public<V>,
    index: u32,
    messages: I,
    signatures: J,
) -> Result<(), Error>
where
    V: Variant,
    I: IntoIterator<Item = &'a (Option<&'a [u8]>, &'a [u8])>,
    J: IntoIterator<Item = &'a PartialSignature<V>>,
    V::Signature: 'a,
{
    // Aggregate the partial signatures
    let (parsed_index, signature) =
        partial_aggregate_signatures::<V, _>(signatures).ok_or(Error::InvalidSignature)?;
    if index != parsed_index {
        return Err(Error::InvalidSignature);
    }
    let public = public.evaluate(index).value;

    // Sum the hashed messages
    let mut hm_sum = V::Signature::zero();
    for (namespace, msg) in messages {
        let hm = match namespace {
            Some(namespace) => hash_message_namespace::<V>(V::MESSAGE, namespace, msg),
            None => hash_message::<V>(V::MESSAGE, msg),
        };
        hm_sum.add(&hm);
    }

    // Verify the signature
    V::verify(&public, &hm_sum, &signature)
}

/// Verify a list of [PartialSignature]s by performing aggregate verification,
/// performing repeated bisection to find invalid signatures (if any exist).
///
/// TODO (#903): parallelize this
fn partial_verify_multiple_public_keys_bisect<'a, V, VP>(
    pending: &[(VP, &'a PartialSignature<V>)],
    mut invalid: Vec<&'a PartialSignature<V>>,
    namespace: Option<&[u8]>,
    message: &[u8],
) -> Result<(), Vec<&'a PartialSignature<V>>>
where
    V: Variant,
    VP: AsRef<V::Public>,
{
    // Iteratively bisect to find invalid signatures
    let mut stack = vec![(0, pending.len())];
    while let Some((start, end)) = stack.pop() {
        // Skip if range is empty
        let slice = &pending[start..end];
        if slice.is_empty() {
            continue;
        }

        // Create aggregate public key and signature
        let mut agg_pk = V::Public::zero();
        let mut agg_sig = V::Signature::zero();
        for (pk, partial) in slice {
            agg_pk.add(pk.as_ref());
            agg_sig.add(&partial.value);
        }

        // If aggregate signature is invalid, bisect. Otherwise, continue.
        if verify_message::<V>(&agg_pk, namespace, message, &agg_sig).is_err() {
            if slice.len() == 1 {
                invalid.push(slice[0].1);
            } else {
                let mid = slice.len() / 2;
                stack.push((start + mid, end));
                stack.push((start, start + mid));
            }
        }
    }

    // Return invalid partial signatures, if any
    if !invalid.is_empty() {
        return Err(invalid);
    }
    Ok(())
}

/// Attempts to verify multiple [PartialSignature]s over the same message as a single
/// aggregate signature (or returns any invalid signature found).
///
/// Unlike `partial_verify_multiple_public_keys`, this function requires the public keys
/// of all partial signatures to be precomputed (avoids a significant amount of compute
/// evaluating each signer on the public polynomial).
pub fn partial_verify_multiple_public_keys_precomputed<'a, V, I>(
    polynomial: &[V::Public],
    namespace: Option<&[u8]>,
    message: &[u8],
    partials: I,
) -> Result<(), Vec<&'a PartialSignature<V>>>
where
    V: Variant,
    I: IntoIterator<Item = &'a PartialSignature<V>>,
{
    // Ensure all partial signatures are associated with a signer
    let partials = partials.into_iter();
    let mut pending = Vec::with_capacity(partials.size_hint().0);
    let mut invalid = Vec::new();
    for partial in partials {
        match polynomial.get(partial.index as usize) {
            Some(public_key) => pending.push((public_key, partial)),
            None => invalid.push(partial),
        }
    }

    // Find any invalid partial signatures
    partial_verify_multiple_public_keys_bisect::<V, _>(&pending, invalid, namespace, message)
}

/// Attempts to verify multiple [PartialSignature]s over the same message as a single
/// aggregate signature (or returns any invalid signature found).
///
/// # Warning
///
/// This function assumes a group check was already performed on each `signature`.
pub fn partial_verify_multiple_public_keys<'a, V, I>(
    public: &poly::Public<V>,
    namespace: Option<&[u8]>,
    message: &[u8],
    partials: I,
) -> Result<(), Vec<&'a PartialSignature<V>>>
where
    V: Variant,
    I: IntoIterator<Item = &'a PartialSignature<V>>,
{
    // Evaluate public polynomial to compute signer public keys
    let pending = partials
        .into_iter()
        .map(|partial| {
            let public_key = public.evaluate(partial.index).value;
            (public_key, partial)
        })
        .collect::<Vec<_>>();

    // Find any invalid partial signatures
    partial_verify_multiple_public_keys_bisect::<V, _>(&pending, Vec::new(), namespace, message)
}

/// Interpolate the value of some [Point] with precomputed Barycentric Weights
/// and multi-scalar multiplication (MSM).
pub fn msm_interpolate<'a, P, I>(weights: &BTreeMap<u32, Weight>, evals: I) -> Result<P, Error>
where
    P: Point + 'a,
    I: IntoIterator<Item = &'a Eval<P>>,
{
    // Populate points and scalars
    let mut points = Vec::with_capacity(weights.len());
    let mut scalars = Vec::with_capacity(weights.len());
    for e in evals {
        points.push(e.value.clone());
        scalars.push(
            weights
                .get(&e.index)
                .ok_or(Error::InvalidIndex)?
                .as_scalar()
                .clone(),
        );
    }

    // Perform multi-scalar multiplication
    Ok(P::msm(&points, &scalars))
}

/// Recovers a signature from `threshold` partial signatures.
///
/// # Determinism
///
/// Signatures recovered by this function are deterministic and are safe
/// to use in a consensus-critical context.
///
/// # Warning
///
/// This function assumes that each partial signature is unique and that
/// that there exists exactly one partial signature for each index in
/// the `weights` map.
pub fn threshold_signature_recover_with_weights<'a, V, I>(
    weights: &BTreeMap<u32, Weight>,
    partials: I,
) -> Result<V::Signature, Error>
where
    V: Variant,
    I: IntoIterator<Item = &'a PartialSignature<V>>,
    V::Signature: 'a,
{
    msm_interpolate(weights, partials)
}

/// Recovers a signature from at least `threshold` partial signatures.
///
/// # Determinism
///
/// Signatures recovered by this function are deterministic and are safe
/// to use in a consensus-critical context.
///
/// # Warning
///
/// This function assumes that each partial signature is unique.
pub fn threshold_signature_recover<'a, V, I>(
    threshold: u32,
    partials: I,
) -> Result<V::Signature, Error>
where
    V: Variant,
    I: IntoIterator<Item = &'a PartialSignature<V>>,
    V::Signature: 'a,
{
    // Prepare evaluations
    let evals = prepare_evaluations(threshold, partials)?;

    // Compute weights
    let indices = evals.iter().map(|e| e.index).collect::<Vec<_>>();
    let weights = compute_weights(indices)?;

    // Perform interpolation with the precomputed weights.
    //
    // We call this function instead of `poly::recover_with_weights` because
    // it will use multi-scalar multiplication (MSM) to recover the signature.
    threshold_signature_recover_with_weights::<V, _>(&weights, evals)
}

/// Recovers multiple signatures from multiple sets of at least `threshold`
/// partial signatures.
///
/// # Determinism
///
/// Signatures recovered by this function are deterministic and are safe
/// to use in a consensus-critical context.
///
/// # Warning
///
/// This function assumes that each partial signature is unique and that
/// each set of partial signatures has the same indices.
pub fn threshold_signature_recover_multiple<'a, V, I>(
    threshold: u32,
    mut many_evals: Vec<I>,
    #[cfg_attr(not(feature = "std"), allow(unused_variables))] concurrency: usize,
) -> Result<Vec<V::Signature>, Error>
where
    V: Variant,
    I: IntoIterator<Item = &'a PartialSignature<V>>,
    V::Signature: 'a,
{
    // Process first set of evaluations
    let evals = many_evals.swap_remove(0).into_iter().collect::<Vec<_>>();
    let evals = prepare_evaluations(threshold, evals)?;
    let mut prepared_evals = vec![evals];

    // Prepare other evaluations and ensure they have the same indices
    for evals in many_evals {
        let evals = evals.into_iter().collect::<Vec<_>>();
        let evals = prepare_evaluations(threshold, evals)?;
        for (i, e) in prepared_evals[0].iter().enumerate() {
            if e.index != evals[i].index {
                return Err(Error::InvalidIndex);
            }
        }
        prepared_evals.push(evals);
    }

    // Compute weights
    let indices = prepared_evals[0]
        .iter()
        .map(|e| e.index)
        .collect::<Vec<_>>();
    let weights = compute_weights(indices)?;

    #[cfg(not(feature = "std"))]
    return prepared_evals
        .into_iter()
        .map(|evals| {
            threshold_signature_recover_with_weights::<V, _>(&weights, evals.iter().cloned())
        })
        .collect();

    #[cfg(feature = "std")]
    {
        let concurrency = core::cmp::min(concurrency, prepared_evals.len());
        if concurrency == 1 {
            return prepared_evals
                .into_iter()
                .map(|evals| {
                    threshold_signature_recover_with_weights::<V, _>(
                        &weights,
                        evals.iter().cloned(),
                    )
                })
                .collect();
        }

        // Build a thread pool with the specified concurrency
        let pool = ThreadPoolBuilder::new()
            .num_threads(concurrency)
            .build()
            .expect("Unable to build thread pool");

        // Recover signatures
        pool.install(move || {
            prepared_evals
                .par_iter()
                .map(|evals| {
                    threshold_signature_recover_with_weights::<V, _>(
                        &weights,
                        evals.iter().cloned(),
                    )
                })
                .collect()
        })
    }
}

/// Recovers a pair of signatures from two sets of at least `threshold` partial signatures.
///
/// This is just a wrapper around `threshold_signature_recover_multiple` with concurrency set to 2.
pub fn threshold_signature_recover_pair<'a, V, I>(
    threshold: u32,
    first: I,
    second: I,
) -> Result<(V::Signature, V::Signature), Error>
where
    V: Variant,
    I: IntoIterator<Item = &'a PartialSignature<V>>,
    V::Signature: 'a,
{
    let mut sigs = threshold_signature_recover_multiple::<V, _>(threshold, vec![first, second], 2)?;
    let second_sig = sigs.pop().unwrap();
    let first_sig = sigs.pop().unwrap();
    Ok((first_sig, second_sig))
}

/// Aggregates multiple public keys.
///
/// # Warning
///
/// This function assumes a group check was already performed on all `public_keys`,
/// that each `public_key` is unique, and that the caller has a Proof-of-Possession (PoP)
/// for each `public_key`. If any of these assumptions are violated, an attacker can
/// exploit this function to verify an incorrect aggregate signature.
pub fn aggregate_public_keys<'a, V, I>(public_keys: I) -> V::Public
where
    V: Variant,
    I: IntoIterator<Item = &'a V::Public>,
    V::Public: 'a,
{
    let mut p = V::Public::zero();
    for pk in public_keys {
        p.add(pk);
    }
    p
}

/// Aggregates multiple signatures.
///
/// # Warning
///
/// This function assumes a group check was already performed on each `signature` and
/// that each `signature` is unique. If any of these assumptions are violated, an attacker can
/// exploit this function to verify an incorrect aggregate signature.
pub fn aggregate_signatures<'a, V, I>(signatures: I) -> V::Signature
where
    V: Variant,
    I: IntoIterator<Item = &'a V::Signature>,
    V::Signature: 'a,
{
    let mut s = V::Signature::zero();
    for sig in signatures {
        s.add(sig);
    }
    s
}

/// Verifies the aggregate signature over a single message from multiple public keys.
///
/// # Warning
///
/// This function assumes the caller has performed a group check and collected a proof-of-possession
/// for all provided `public`. This function assumes a group check was already performed on the
/// `signature`. It is not safe to provide duplicate public keys.
pub fn aggregate_verify_multiple_public_keys<'a, V, I>(
    public: I,
    namespace: Option<&[u8]>,
    message: &[u8],
    signature: &V::Signature,
) -> Result<(), Error>
where
    V: Variant,
    I: IntoIterator<Item = &'a V::Public>,
    V::Public: 'a,
{
    // Aggregate public keys
    //
    // We can take advantage of the bilinearity property of pairings to aggregate public keys
    // that have all signed the same message (as long as all public keys are unique).
    let agg_public = aggregate_public_keys::<V, _>(public);

    // Verify the signature
    verify_message::<V>(&agg_public, namespace, message, signature)
}

/// Verifies the aggregate signature over multiple unique messages from a single public key.
///
/// # Warning
///
/// This function assumes a group check was already performed on `public` and `signature`. It is not
/// safe to provide an aggregate public key or to provide duplicate messages.
///
/// ## Why not aggregate public keys?
///
/// We rely on bilinearity to reduce pairing operations in this function, like in `aggregate_verify_multiple_public_keys`,
/// and sum hashed messages together before performing a single pairing operation (instead of summing `len(messages)` pairings of
/// hashed message and public key). If the public key itself is an aggregate of multiple public keys, an attacker can exploit
/// this optimization to cause this function to return that an aggregate signature is valid when it really isn't.
pub fn aggregate_verify_multiple_messages<'a, V, I>(
    public: &V::Public,
    messages: I,
    signature: &V::Signature,
    #[cfg_attr(not(feature = "std"), allow(unused_variables))] concurrency: usize,
) -> Result<(), Error>
where
    V: Variant,
    I: IntoIterator<Item = &'a (Option<&'a [u8]>, &'a [u8])> + Send + Sync,
    I::IntoIter: Send + Sync,
{
    #[cfg(not(feature = "std"))]
    let hm_sum = compute_hm_sum::<V, I>(messages);

    #[cfg(feature = "std")]
    let hm_sum = if concurrency == 1 {
        compute_hm_sum::<V, I>(messages)
    } else {
        let pool = ThreadPoolBuilder::new()
            .num_threads(concurrency)
            .build()
            .expect("Unable to build thread pool");

        // TODO(#1496): Revisit use of `.par_bridge()`
        pool.install(move || {
            messages
                .into_iter()
                .par_bridge()
                .map(|(namespace, msg)| match namespace {
                    Some(namespace) => hash_message_namespace::<V>(V::MESSAGE, namespace, msg),
                    None => hash_message::<V>(V::MESSAGE, msg),
                })
                .reduce(V::Signature::zero, |mut sum, hm| {
                    sum.add(&hm);
                    sum
                })
        })
    };

    V::verify(public, &hm_sum, signature)
}

/// Computes the sum over the hash of each message.
fn compute_hm_sum<'a, V, I>(messages: I) -> V::Signature
where
    V: Variant,
    I: IntoIterator<Item = &'a (Option<&'a [u8]>, &'a [u8])>,
{
    let mut hm_sum = V::Signature::zero();
    for (namespace, msg) in messages {
        let hm = match namespace {
            Some(namespace) => hash_message_namespace::<V>(V::MESSAGE, namespace, msg),
            None => hash_message::<V>(V::MESSAGE, msg),
        };
        hm_sum.add(&hm);
    }
    hm_sum
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::bls12381::{
        dkg::ops::{evaluate_all, generate_shares},
        primitives::variant::{MinPk, MinSig},
    };
    use blst::BLST_ERROR;
    use commonware_codec::{DecodeExt, ReadExt};
    use commonware_utils::{from_hex_formatted, quorum};
    use group::{Private, G1_MESSAGE, G2_MESSAGE};
    use poly::Poly;
    use rand::{prelude::*, rngs::OsRng};

    fn codec<V: Variant>() {
        // Encode private/public key
        let (private, public) = keypair::<_, V>(&mut thread_rng());
        let (private_bytes, public_bytes) = (private.encode(), public.encode());

        // Decode private/public key
        let (private_decoded, public_decoded) = (
            group::Private::decode(private_bytes.clone()).unwrap(),
            V::Public::decode(public_bytes.clone()).unwrap(),
        );

        // Ensure equal
        assert_eq!(private, private_decoded);
        assert_eq!(public, public_decoded);

        // Ensure blst compatibility
        match V::MESSAGE {
            G1_MESSAGE => {
                blst::min_sig::SecretKey::from_bytes(&private_bytes).unwrap();
                let blst_public_decoded =
                    blst::min_sig::PublicKey::from_bytes(&public_bytes).unwrap();
                blst_public_decoded.validate().unwrap();
                let blst_public_encoded = blst_public_decoded.compress().to_vec();
                assert_eq!(public_bytes, blst_public_encoded.as_slice());
            }
            G2_MESSAGE => {
                blst::min_pk::SecretKey::from_bytes(&private_bytes).unwrap();
                let blst_public_decoded =
                    blst::min_pk::PublicKey::from_bytes(&public_bytes).unwrap();
                blst_public_decoded.validate().unwrap();
                let blst_public_encoded = blst_public_decoded.compress().to_vec();
                assert_eq!(public_bytes, blst_public_encoded.as_slice());
            }
            _ => panic!("Unsupported Variant"),
        }
    }

    #[test]
    fn test_codec() {
        codec::<MinPk>();
        codec::<MinSig>();
    }

    /// Verify that a given proof-of-possession signature is valid according to `blst`.
    fn blst_verify_proof_of_possession<V: Variant>(
        public: &V::Public,
        signature: &V::Signature,
    ) -> Result<(), BLST_ERROR> {
        let msg = public.encode();
        match V::MESSAGE {
            G1_MESSAGE => {
                let public = blst::min_sig::PublicKey::from_bytes(&public.encode()).unwrap();
                let signature = blst::min_sig::Signature::from_bytes(&signature.encode()).unwrap();
                match signature.verify(true, &msg, V::PROOF_OF_POSSESSION, &[], &public, true) {
                    BLST_ERROR::BLST_SUCCESS => Ok(()),
                    e => Err(e),
                }
            }
            G2_MESSAGE => {
                let public = blst::min_pk::PublicKey::from_bytes(&public.encode()).unwrap();
                let signature = blst::min_pk::Signature::from_bytes(&signature.encode()).unwrap();
                match signature.verify(true, &msg, V::PROOF_OF_POSSESSION, &[], &public, true) {
                    BLST_ERROR::BLST_SUCCESS => Ok(()),
                    e => Err(e),
                }
            }
            _ => panic!("Unsupported Variant"),
        }
    }

    fn single_proof_of_possession<V: Variant>() {
        // Generate PoP
        let (private, public) = keypair::<_, V>(&mut thread_rng());
        let pop = sign_proof_of_possession::<V>(&private);

        // Verify PoP
        verify_proof_of_possession::<V>(&public, &pop).expect("PoP should be valid");

        // Verify PoP using blst
        blst_verify_proof_of_possession::<V>(&public, &pop).expect("PoP should be valid");
    }

    #[test]
    fn test_single_proof_of_possession() {
        single_proof_of_possession::<MinPk>();
        single_proof_of_possession::<MinSig>();
    }

    fn threshold_proof_of_possession<V: Variant>() {
        // Generate PoP
        let (n, t) = (5, 4);
        let mut rng = StdRng::seed_from_u64(0);
        let (public, shares) = generate_shares::<_, V>(&mut rng, None, n, t);
        let partials: Vec<_> = shares
            .iter()
            .map(|s| partial_sign_proof_of_possession::<V>(&public, s))
            .collect();
        for p in &partials {
            partial_verify_proof_of_possession::<V>(&public, p).expect("signature should be valid");
        }
        let threshold_sig = threshold_signature_recover::<V, _>(t, &partials).unwrap();
        let threshold_pub = poly::public::<V>(&public);

        // Verify PoP
        verify_proof_of_possession::<V>(threshold_pub, &threshold_sig)
            .expect("signature should be valid");

        // Verify PoP using blst
        blst_verify_proof_of_possession::<V>(threshold_pub, &threshold_sig)
            .expect("signature should be valid");
    }

    #[test]
    fn test_threshold_proof_of_possession() {
        threshold_proof_of_possession::<MinPk>();
        threshold_proof_of_possession::<MinSig>();
    }

    /// Verify that a given message signature is valid according to `blst`.
    fn blst_verify_message<V: Variant>(
        public: &V::Public,
        msg: &[u8],
        signature: &V::Signature,
    ) -> Result<(), BLST_ERROR> {
        match V::MESSAGE {
            G1_MESSAGE => {
                let public = blst::min_sig::PublicKey::from_bytes(&public.encode()).unwrap();
                let signature = blst::min_sig::Signature::from_bytes(&signature.encode()).unwrap();
                match signature.verify(true, msg, V::MESSAGE, &[], &public, true) {
                    BLST_ERROR::BLST_SUCCESS => Ok(()),
                    e => Err(e),
                }
            }
            G2_MESSAGE => {
                let public = blst::min_pk::PublicKey::from_bytes(&public.encode()).unwrap();
                let signature = blst::min_pk::Signature::from_bytes(&signature.encode()).unwrap();
                match signature.verify(true, msg, V::MESSAGE, &[], &public, true) {
                    BLST_ERROR::BLST_SUCCESS => Ok(()),
                    e => Err(e),
                }
            }
            _ => panic!("Unsupported Variant"),
        }
    }

    fn bad_namespace<V: Variant>() {
        let (private, public) = keypair::<_, V>(&mut thread_rng());
        let msg = &[1, 9, 6, 9];
        let sig = sign_message::<V>(&private, Some(b"good"), msg);
        assert!(matches!(
            verify_message::<V>(&public, Some(b"bad"), msg, &sig).unwrap_err(),
            Error::InvalidSignature
        ));
    }

    #[test]
    fn test_bad_namespace() {
        bad_namespace::<MinPk>();
        bad_namespace::<MinSig>();
    }

    fn single_message<V: Variant>() {
        let (private, public) = keypair::<_, V>(&mut thread_rng());
        let msg = &[1, 9, 6, 9];
        let namespace = b"test";
        let sig = sign_message::<V>(&private, Some(namespace), msg);
        verify_message::<V>(&public, Some(namespace), msg, &sig)
            .expect("signature should be valid");
        let payload = union_unique(namespace, msg);
        blst_verify_message::<V>(&public, &payload, &sig).expect("signature should be valid");
    }

    #[test]
    fn test_single_message() {
        single_message::<MinPk>();
        single_message::<MinSig>();
    }

    fn threshold_message<V: Variant>() {
        // Generate signature
        let (n, t) = (5, 4);
        let mut rng = StdRng::seed_from_u64(0);
        let (public, shares) = generate_shares::<_, V>(&mut rng, None, n, t);
        let msg = &[1, 9, 6, 9];
        let namespace = b"test";
        let partials: Vec<_> = shares
            .iter()
            .map(|s| partial_sign_message::<V>(s, Some(namespace), msg))
            .collect();
        for p in &partials {
            partial_verify_message::<V>(&public, Some(namespace), msg, p)
                .expect("signature should be valid");
        }
        let threshold_sig = threshold_signature_recover::<V, _>(t, &partials).unwrap();
        let threshold_pub = poly::public::<V>(&public);

        // Verify the signature
        verify_message::<V>(threshold_pub, Some(namespace), msg, &threshold_sig)
            .expect("signature should be valid");

        // Verify the signature using blst
        let payload = union_unique(namespace, msg);
        blst_verify_message::<V>(threshold_pub, &payload, &threshold_sig)
            .expect("signature should be valid");
    }

    #[test]
    fn test_threshold_message() {
        threshold_message::<MinPk>();
        threshold_message::<MinSig>();
    }

    fn blst_aggregate_verify_multiple_public_keys<'a, V, I>(
        public: I,
        message: &[u8],
        signature: &V::Signature,
    ) -> Result<(), BLST_ERROR>
    where
        V: Variant,
        I: IntoIterator<Item = &'a V::Public>,
        V::Public: 'a,
    {
        match V::MESSAGE {
            G1_MESSAGE => {
                let public = public
                    .into_iter()
                    .map(|pk| blst::min_sig::PublicKey::from_bytes(&pk.encode()).unwrap())
                    .collect::<Vec<_>>();
                let public = public.iter().collect::<Vec<_>>();
                let signature = blst::min_sig::Signature::from_bytes(&signature.encode()).unwrap();
                match signature.fast_aggregate_verify(true, message, V::MESSAGE, &public) {
                    BLST_ERROR::BLST_SUCCESS => Ok(()),
                    e => Err(e),
                }
            }
            G2_MESSAGE => {
                let public = public
                    .into_iter()
                    .map(|pk| blst::min_pk::PublicKey::from_bytes(&pk.encode()).unwrap())
                    .collect::<Vec<_>>();
                let public = public.iter().collect::<Vec<_>>();
                let signature = blst::min_pk::Signature::from_bytes(&signature.encode()).unwrap();
                match signature.fast_aggregate_verify(true, message, V::MESSAGE, &public) {
                    BLST_ERROR::BLST_SUCCESS => Ok(()),
                    e => Err(e),
                }
            }
            _ => panic!("Unsupported Variant"),
        }
    }

    fn aggregate_verify_multiple_public_keys_correct<V: Variant>() {
        // Generate signatures
        let (private1, public1) = keypair::<_, V>(&mut thread_rng());
        let (private2, public2) = keypair::<_, V>(&mut thread_rng());
        let (private3, public3) = keypair::<_, V>(&mut thread_rng());
        let namespace = b"test";
        let message = b"message";
        let sig1 = sign_message::<V>(&private1, Some(namespace), message);
        let sig2 = sign_message::<V>(&private2, Some(namespace), message);
        let sig3 = sign_message::<V>(&private3, Some(namespace), message);
        let pks = vec![public1, public2, public3];
        let signatures = vec![sig1, sig2, sig3];

        // Aggregate the signatures
        let aggregate_sig = aggregate_signatures::<V, _>(&signatures);

        // Verify the aggregated signature
        aggregate_verify_multiple_public_keys::<V, _>(
            &pks,
            Some(namespace),
            message,
            &aggregate_sig,
        )
        .expect("Aggregated signature should be valid");

        // Verify the aggregated signature using blst
        let payload = union_unique(namespace, message);
        blst_aggregate_verify_multiple_public_keys::<V, _>(&pks, &payload, &aggregate_sig)
            .expect("Aggregated signature should be valid");
    }

    #[test]
    fn test_aggregate_verify_multiple_public_keys() {
        aggregate_verify_multiple_public_keys_correct::<MinPk>();
        aggregate_verify_multiple_public_keys_correct::<MinSig>();
    }

    fn aggregate_verify_wrong_public_keys<V: Variant>() {
        // Generate signatures
        let (private1, public1) = keypair::<_, V>(&mut thread_rng());
        let (private2, public2) = keypair::<_, V>(&mut thread_rng());
        let (private3, _) = keypair::<_, V>(&mut thread_rng());
        let namespace = b"test";
        let message = b"message";
        let sig1 = sign_message::<V>(&private1, Some(namespace), message);
        let sig2 = sign_message::<V>(&private2, Some(namespace), message);
        let sig3 = sign_message::<V>(&private3, Some(namespace), message);
        let signatures = vec![sig1, sig2, sig3];

        // Aggregate the signatures
        let aggregate_sig = aggregate_signatures::<V, _>(&signatures);

        // Verify the aggregated signature
        let (_, public4) = keypair::<_, V>(&mut thread_rng());
        let wrong_pks = vec![public1, public2, public4];
        let result = aggregate_verify_multiple_public_keys::<V, _>(
            &wrong_pks,
            Some(namespace),
            message,
            &aggregate_sig,
        );
        assert!(matches!(result, Err(Error::InvalidSignature)));
    }

    #[test]
    fn test_aggregate_verify_wrong_public_keys() {
        aggregate_verify_wrong_public_keys::<MinPk>();
        aggregate_verify_wrong_public_keys::<MinSig>();
    }

    fn aggregate_verify_wrong_public_key_count<V: Variant>() {
        // Generate signatures
        let (private1, public1) = keypair::<_, V>(&mut thread_rng());
        let (private2, public2) = keypair::<_, V>(&mut thread_rng());
        let (private3, _) = keypair::<_, V>(&mut thread_rng());
        let namespace = b"test";
        let message = b"message";
        let sig1 = sign_message::<V>(&private1, Some(namespace), message);
        let sig2 = sign_message::<V>(&private2, Some(namespace), message);
        let sig3 = sign_message::<V>(&private3, Some(namespace), message);
        let signatures = vec![sig1, sig2, sig3];

        // Aggregate the signatures
        let aggregate_sig = aggregate_signatures::<V, _>(&signatures);

        // Verify the aggregated signature
        let wrong_pks = vec![public1, public2];
        let result = aggregate_verify_multiple_public_keys::<V, _>(
            &wrong_pks,
            Some(namespace),
            message,
            &aggregate_sig,
        );
        assert!(matches!(result, Err(Error::InvalidSignature)));
    }

    #[test]
    fn test_aggregate_verify_wrong_public_key_count() {
        aggregate_verify_wrong_public_key_count::<MinPk>();
        aggregate_verify_wrong_public_key_count::<MinSig>();
    }

    fn blst_aggregate_verify_multiple_messages<'a, V, I>(
        public: &V::Public,
        msgs: I,
        signature: &V::Signature,
    ) -> Result<(), BLST_ERROR>
    where
        V: Variant,
        I: IntoIterator<Item = &'a [u8]>,
    {
        match V::MESSAGE {
            G1_MESSAGE => {
                let public = blst::min_sig::PublicKey::from_bytes(&public.encode()).unwrap();
                let msgs = msgs.into_iter().collect::<Vec<_>>();
                let pks = vec![&public; msgs.len()];
                let signature = blst::min_sig::Signature::from_bytes(&signature.encode()).unwrap();
                match signature.aggregate_verify(true, &msgs, V::MESSAGE, &pks, true) {
                    BLST_ERROR::BLST_SUCCESS => Ok(()),
                    e => Err(e),
                }
            }
            G2_MESSAGE => {
                let public = blst::min_pk::PublicKey::from_bytes(&public.encode()).unwrap();
                let msgs = msgs.into_iter().collect::<Vec<_>>();
                let pks = vec![&public; msgs.len()];
                let signature = blst::min_pk::Signature::from_bytes(&signature.encode()).unwrap();
                match signature.aggregate_verify(true, &msgs, V::MESSAGE, &pks, true) {
                    BLST_ERROR::BLST_SUCCESS => Ok(()),
                    e => Err(e),
                }
            }
            _ => panic!("Unsupported Variant"),
        }
    }

    fn aggregate_verify_multiple_messages_correct<V: Variant>() {
        // Generate signatures
        let (private, public) = keypair::<_, V>(&mut thread_rng());
        let namespace = Some(&b"test"[..]);
        let messages: Vec<(Option<&[u8]>, &[u8])> = vec![
            (namespace, b"Message 1"),
            (namespace, b"Message 2"),
            (namespace, b"Message 3"),
        ];
        let signatures: Vec<_> = messages
            .iter()
            .map(|(namespace, msg)| sign_message::<V>(&private, *namespace, msg))
            .collect();

        // Aggregate the signatures
        let aggregate_sig = aggregate_signatures::<V, _>(&signatures);

        // Verify the aggregated signature without parallelism
        aggregate_verify_multiple_messages::<V, _>(&public, &messages, &aggregate_sig, 1)
            .expect("Aggregated signature should be valid");

        // Verify the aggregated signature with parallelism
        aggregate_verify_multiple_messages::<V, _>(&public, &messages, &aggregate_sig, 4)
            .expect("Aggregated signature should be valid");

        // Verify the aggregated signature using blst
        let messages = messages
            .iter()
            .map(|(namespace, msg)| union_unique(namespace.unwrap(), msg))
            .collect::<Vec<_>>();
        let messages = messages
            .iter()
            .map(|msg| msg.as_slice())
            .collect::<Vec<_>>();
        blst_aggregate_verify_multiple_messages::<V, _>(&public, messages, &aggregate_sig)
            .expect("Aggregated signature should be valid");
    }

    #[test]
    fn test_aggregate_verify_multiple_messages() {
        aggregate_verify_multiple_messages_correct::<MinPk>();
        aggregate_verify_multiple_messages_correct::<MinSig>();
    }

    fn aggregate_verify_wrong_messages<V: Variant>() {
        // Generate signatures
        let (private, public) = keypair::<_, V>(&mut thread_rng());
        let namespace = Some(&b"test"[..]);
        let messages: Vec<(Option<&[u8]>, &[u8])> = vec![
            (namespace, b"Message 1"),
            (namespace, b"Message 2"),
            (namespace, b"Message 3"),
        ];
        let signatures: Vec<_> = messages
            .iter()
            .map(|(namespace, msg)| sign_message::<V>(&private, *namespace, msg))
            .collect();

        // Aggregate the signatures
        let aggregate_sig = aggregate_signatures::<V, _>(&signatures);

        // Construct wrong messages
        let wrong_messages: Vec<(Option<&[u8]>, &[u8])> = vec![
            (namespace, b"Message 1"),
            (namespace, b"Message 2"),
            (namespace, b"Message 4"),
        ];

        // Verify the aggregated signature without parallelism
        let result =
            aggregate_verify_multiple_messages::<V, _>(&public, &wrong_messages, &aggregate_sig, 1);
        assert!(matches!(result, Err(Error::InvalidSignature)));

        // Verify the aggregated signature with parallelism
        let result =
            aggregate_verify_multiple_messages::<V, _>(&public, &wrong_messages, &aggregate_sig, 4);
        assert!(matches!(result, Err(Error::InvalidSignature)));
    }

    #[test]
    fn test_aggregate_verify_wrong_messages() {
        aggregate_verify_wrong_messages::<MinPk>();
        aggregate_verify_wrong_messages::<MinSig>();
    }

    fn aggregate_verify_wrong_message_count<V: Variant>() {
        // Generate signatures
        let (private, public) = keypair::<_, V>(&mut thread_rng());
        let namespace = Some(&b"test"[..]);
        let messages: Vec<(Option<&[u8]>, &[u8])> = vec![
            (namespace, b"Message 1"),
            (namespace, b"Message 2"),
            (namespace, b"Message 3"),
        ];
        let signatures: Vec<_> = messages
            .iter()
            .map(|(namespace, msg)| sign_message::<V>(&private, *namespace, msg))
            .collect();

        // Aggregate the signatures
        let aggregate_sig = aggregate_signatures::<V, _>(&signatures);

        // Construct wrong messages
        let wrong_messages: Vec<(Option<&[u8]>, &[u8])> =
            vec![(namespace, b"Message 1"), (namespace, b"Message 2")];

        // Verify the aggregated signature without parallelism
        let result =
            aggregate_verify_multiple_messages::<V, _>(&public, &wrong_messages, &aggregate_sig, 1);
        assert!(matches!(result, Err(Error::InvalidSignature)));

        // Verify the aggregated signature with parallelism
        let result =
            aggregate_verify_multiple_messages::<V, _>(&public, &wrong_messages, &aggregate_sig, 4);
        assert!(matches!(result, Err(Error::InvalidSignature)));
    }

    #[test]
    fn test_aggregate_verify_wrong_message_count() {
        aggregate_verify_wrong_message_count::<MinPk>();
        aggregate_verify_wrong_message_count::<MinSig>();
    }

    fn partial_verify_multiple_messages_correct<V: Variant>() {
        // Generate polynomial and shares
        let n = 5;
        let t = quorum(n);
        let (public, shares) = generate_shares::<_, V>(&mut thread_rng(), None, n, t);

        // Select signer with index 0
        let signer = &shares[0];

        // Successful verification with namespaced messages
        let messages: Vec<(Option<&[u8]>, &[u8])> = vec![
            (Some(&b"ns"[..]), b"msg1"),
            (Some(&b"ns"[..]), b"msg2"),
            (Some(&b"ns"[..]), b"msg3"),
        ];
        let partials: Vec<PartialSignature<V>> = messages
            .iter()
            .map(|(ns, msg)| partial_sign_message::<V>(signer, *ns, msg))
            .collect();
        partial_verify_multiple_messages::<V, _, _>(&public, signer.index, &messages, &partials)
            .expect("Verification with namespaced messages should succeed");

        // Successful verification with non-namespaced messages
        let messages_no_ns: Vec<(Option<&[u8]>, &[u8])> =
            vec![(None, b"msg1"), (None, b"msg2"), (None, b"msg3")];
        let partials_no_ns: Vec<PartialSignature<V>> = messages_no_ns
            .iter()
            .map(|(ns, msg)| partial_sign_message::<V>(signer, *ns, msg))
            .collect();
        partial_verify_multiple_messages::<V, _, _>(
            &public,
            signer.index,
            &messages_no_ns,
            &partials_no_ns,
        )
        .expect("Verification with non-namespaced messages should succeed");

        // Successful verification with mixed namespaces
        let messages_mixed: Vec<(Option<&[u8]>, &[u8])> = vec![
            (Some(&b"ns1"[..]), b"msg1"),
            (None, b"msg2"),
            (Some(&b"ns2"[..]), b"msg3"),
        ];
        let partials_mixed: Vec<PartialSignature<V>> = messages_mixed
            .iter()
            .map(|(ns, msg)| partial_sign_message::<V>(signer, *ns, msg))
            .collect();
        partial_verify_multiple_messages::<V, _, _>(
            &public,
            signer.index,
            &messages_mixed,
            &partials_mixed,
        )
        .expect("Verification with mixed namespaces should succeed");

        // Failure with wrong signer index
        assert!(matches!(
            partial_verify_multiple_messages::<V, _, _>(&public, 1, &messages, &partials),
            Err(Error::InvalidSignature)
        ));

        // Success with swapped partial signatures
        let mut partials_swapped = partials.clone();
        partials_swapped.swap(0, 1);
        partial_verify_multiple_messages::<V, _, _>(
            &public,
            signer.index,
            &messages,
            &partials_swapped,
        )
        .expect("Verification with swapped partials should succeed");

        // Failure with fewer signatures than messages
        let partials_fewer = partials[..2].to_vec();
        assert!(matches!(
            partial_verify_multiple_messages::<V, _, _>(
                &public,
                signer.index,
                &messages,
                &partials_fewer
            ),
            Err(Error::InvalidSignature)
        ));

        // Failure with more signatures than messages
        let extra_message = (Some(&b"ns"[..]), b"msg4");
        let extra_partial = partial_sign_message::<V>(signer, extra_message.0, extra_message.1);
        let mut partials_more = partials.clone();
        partials_more.push(extra_partial);
        assert!(matches!(
            partial_verify_multiple_messages::<V, _, _>(
                &public,
                signer.index,
                &messages,
                &partials_more
            ),
            Err(Error::InvalidSignature)
        ));

        // Failure with signatures from different public_keys
        let signer2 = &shares[1];
        let partial2 = partial_sign_message::<V>(signer2, messages[0].0, messages[0].1);
        let mut partials_mixed_public_keys = partials.clone();
        partials_mixed_public_keys[0] = partial2;
        assert!(matches!(
            partial_verify_multiple_messages::<V, _, _>(
                &public,
                signer.index,
                &messages,
                &partials_mixed_public_keys
            ),
            Err(Error::InvalidSignature)
        ));
    }

    #[test]
    fn test_partial_verify_multiple_messages() {
        partial_verify_multiple_messages_correct::<MinPk>();
        partial_verify_multiple_messages_correct::<MinSig>();
    }

    fn threshold_signature_recover_with_weights_correct<V: Variant>() {
        let mut rng = StdRng::seed_from_u64(3333);
        let (n, t) = (6, quorum(6));
        let (group_poly, shares) = generate_shares::<_, V>(&mut rng, None, n, t);

        // Produce partial signatures for the first `t` shares.
        let partials: Vec<_> = shares
            .iter()
            .take(t as usize)
            .map(|s| partial_sign_message::<V>(s, None, b"payload"))
            .collect();

        // Compute barycentric weights once.
        let indices = partials.iter().map(|e| e.index).collect::<Vec<_>>();
        let weights = compute_weights(indices).unwrap();

        // Path-1: generic recover
        let sig1 = threshold_signature_recover::<V, _>(t, &partials).unwrap();

        // Path-2: recover with *pre-computed* weights
        let sig2 = threshold_signature_recover_with_weights::<V, _>(&weights, &partials).unwrap();

        assert_eq!(sig1, sig2);

        // Verify with the aggregated public key.
        let pk = poly::public::<V>(&group_poly);
        verify_message::<V>(pk, None, b"payload", &sig1).unwrap();
    }

    #[test]
    fn test_threshold_signature_recover_with_weights() {
        threshold_signature_recover_with_weights_correct::<MinPk>();
        threshold_signature_recover_with_weights_correct::<MinSig>();
    }

    fn threshold_signature_recover_multiple<V: Variant>() {
        let mut rng = StdRng::seed_from_u64(3333);
        let (n, t) = (6, quorum(6));
        let (group_poly, shares) = generate_shares::<_, V>(&mut rng, None, n, t);

        // Produce partial signatures for the first `t` shares.
        let partials_1: Vec<_> = shares
            .iter()
            .take(t as usize)
            .map(|s| partial_sign_message::<V>(s, None, b"payload1"))
            .collect();
        let partials_2: Vec<_> = shares
            .iter()
            .take(t as usize)
            .map(|s| partial_sign_message::<V>(s, None, b"payload2"))
            .collect();

        // Recover signatures
        let (sig_1, sig_2) =
            threshold_signature_recover_pair::<V, _>(t, &partials_1, &partials_2).unwrap();

        // Verify with the aggregated public key.
        let pk = poly::public::<V>(&group_poly);
        verify_message::<V>(pk, None, b"payload1", &sig_1).unwrap();
        verify_message::<V>(pk, None, b"payload2", &sig_2).unwrap();
    }

    #[test]
    fn test_threshold_signature_recover_multiple() {
        threshold_signature_recover_multiple::<MinPk>();
        threshold_signature_recover_multiple::<MinSig>();
    }

    fn msm_interpolate_vs_poly_recover<V: Variant>() {
        let mut rng = StdRng::seed_from_u64(4242);
        let degree = 5;
        let threshold = degree + 1;
        let poly_scalar = poly::new_from(degree, &mut rng);

        // Commit to Signature group
        let poly_g1 = Poly::<V::Signature>::commit(poly_scalar);

        // Generate evaluations (enough to meet threshold)
        let evals: Vec<_> = (0..threshold).map(|i| poly_g1.evaluate(i)).collect();
        let eval_refs: Vec<_> = evals.iter().collect(); // Get references

        // Compute weights
        let indices: Vec<u32> = eval_refs.iter().map(|e| e.index).collect();
        let weights = poly::compute_weights(indices).expect("Failed to compute weights");

        // Calculate using original polynomial recovery (naive interpolation)
        let expected_result =
            poly::Signature::<V>::recover_with_weights(&weights, eval_refs.clone())
                .expect("poly::recover_with_weights failed");

        // Calculate using MSM interpolation
        let msm_result = msm_interpolate(&weights, eval_refs).expect("msm_interpolate failed");

        // Compare results
        assert_eq!(
            expected_result, msm_result,
            "MSM interpolation result differs from polynomial recovery"
        );

        // Also check against the known constant term
        assert_eq!(
            expected_result,
            *poly_g1.constant(),
            "Recovered value does not match original constant term"
        );
    }

    #[test]
    fn test_msm_interpolate_vs_poly_recover() {
        msm_interpolate_vs_poly_recover::<MinPk>();
        msm_interpolate_vs_poly_recover::<MinSig>();
    }

    fn msm_interpolate_invalid_index<V: Variant>() {
        let mut rng = StdRng::seed_from_u64(5555);
        let degree = 2;
        let threshold = degree + 1;
        let poly_scalar = poly::new_from(degree, &mut rng);
        let poly_g2 = Poly::<V::Public>::commit(poly_scalar);

        // Generate threshold evaluations
        let evals: Vec<_> = (0..threshold).map(|i| poly_g2.evaluate(i)).collect();
        let eval_refs: Vec<_> = evals.iter().collect();

        // Compute weights for *different* indices
        let wrong_indices: Vec<u32> = (threshold..threshold * 2).collect();
        let weights = poly::compute_weights(wrong_indices).expect("Failed to compute weights");

        // Try to interpolate with mismatched weights/evals
        let result = msm_interpolate::<V::Public, _>(&weights, eval_refs);

        // Expect InvalidIndex error
        assert!(
            matches!(result, Err(Error::InvalidIndex)),
            "Expected InvalidIndex error for mismatched weights"
        );
    }

    #[test]
    fn test_msm_interpolate_invalid_index() {
        msm_interpolate_invalid_index::<MinPk>();
        msm_interpolate_invalid_index::<MinSig>();
    }

    fn msm_interpolate_empty<V: Variant>() {
        let weights: BTreeMap<u32, Weight> = BTreeMap::new();
        let evals: Vec<Eval<V::Public>> = Vec::new();
        let eval_refs: Vec<&Eval<V::Public>> = evals.iter().collect();

        // Interpolate with empty inputs
        let result = msm_interpolate(&weights, eval_refs).expect("msm_interpolate failed on empty");

        // Expect identity element
        assert_eq!(
            result,
            V::Public::zero(),
            "Expected G2 identity for empty interpolation"
        );
    }

    #[test]
    fn test_msm_interpolate_empty() {
        msm_interpolate_empty::<MinPk>();
        msm_interpolate_empty::<MinSig>();
    }

    fn partial_aggregate_signature_correct<V: Variant>() {
        let (n, t) = (5, 4);
        let mut rng = StdRng::seed_from_u64(0);

        // Create the private key polynomial and evaluate it at `n`
        // points to generate the shares.
        //
        // If receiving a share from an untrusted party, the recipient
        // should verify the share is on the public polynomial.
        let (group, shares) = generate_shares::<_, V>(&mut rng, None, n, t);

        // Generate the partial signatures
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";
        let partials = shares
            .iter()
            .map(|s| partial_sign_message::<V>(s, namespace, msg))
            .collect::<Vec<_>>();

        // Each partial sig can be partially verified against the public polynomial
        partials.iter().for_each(|partial| {
            partial_verify_message::<V>(&group, namespace, msg, partial).unwrap();
        });

        // Generate and verify the threshold sig
        let threshold_sig = threshold_signature_recover::<V, _>(t, &partials).unwrap();
        let threshold_pub = poly::public::<V>(&group);
        verify_message::<V>(threshold_pub, namespace, msg, &threshold_sig).unwrap();
    }

    #[test]
    fn test_partial_aggregate_signature_correct() {
        partial_aggregate_signature_correct::<MinPk>();
        partial_aggregate_signature_correct::<MinSig>();
    }

    fn partial_aggregate_signature_bad_namespace<V: Variant>() {
        let (n, t) = (5, 4);
        let mut rng = StdRng::seed_from_u64(0);

        // Create the private key polynomial and evaluate it at `n`
        // points to generate the shares.
        //
        // If receiving a share from an untrusted party, the recipient
        // should verify the share is on the public polynomial.
        let (group, shares) = generate_shares::<_, V>(&mut rng, None, n, t);

        // Generate the partial signatures
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";
        let partials = shares
            .iter()
            .map(|s| partial_sign_message::<V>(s, namespace, msg))
            .collect::<Vec<_>>();

        // Each partial sig can be partially verified against the public polynomial
        let namespace = Some(&b"bad"[..]);
        partials.iter().for_each(|partial| {
            assert!(matches!(
                partial_verify_message::<V>(&group, namespace, msg, partial).unwrap_err(),
                Error::InvalidSignature
            ));
        });

        // Generate and verify the threshold sig
        let threshold_sig = threshold_signature_recover::<V, _>(t, &partials).unwrap();
        let threshold_pub = poly::public::<V>(&group);
        assert!(matches!(
            verify_message::<V>(threshold_pub, namespace, msg, &threshold_sig).unwrap_err(),
            Error::InvalidSignature
        ));
    }

    #[test]
    fn test_partial_aggregate_signature_bad_namespace() {
        partial_aggregate_signature_bad_namespace::<MinPk>();
        partial_aggregate_signature_bad_namespace::<MinSig>();
    }

    fn partial_aggregate_signature_insufficient<V: Variant>() {
        let (n, t) = (5, 4);
        let mut rng = StdRng::seed_from_u64(0);

        // Create the private key polynomial and evaluate it at `n`
        // points to generate the shares
        let (group, shares) = generate_shares::<_, V>(&mut rng, None, n, t);

        // Only take t-1 shares
        let shares = shares.into_iter().take(t as usize - 1).collect::<Vec<_>>();

        // Generate the partial signatures
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";
        let partials = shares
            .iter()
            .map(|s| partial_sign_message::<V>(s, namespace, msg))
            .collect::<Vec<_>>();

        // Each partial sig can be partially verified against the public polynomial
        partials.iter().for_each(|partial| {
            partial_verify_message::<V>(&group, namespace, msg, partial).unwrap();
        });

        // Generate the threshold sig
        assert!(matches!(
            threshold_signature_recover::<V, _>(t, &partials).unwrap_err(),
            Error::NotEnoughPartialSignatures(4, 3)
        ));
    }

    #[test]
    fn test_partial_aggregate_signature_insufficient() {
        partial_aggregate_signature_insufficient::<MinPk>();
        partial_aggregate_signature_insufficient::<MinSig>();
    }

    fn partial_aggregate_signature_bad_share<V: Variant>() {
        let (n, t) = (5, 4);
        let mut rng = StdRng::seed_from_u64(0);

        // Create the private key polynomial and evaluate it at `n`
        // points to generate the shares
        let (group, mut shares) = generate_shares::<_, V>(&mut rng, None, n, t);

        // Corrupt a share
        let share = shares.get_mut(3).unwrap();
        share.private = Private::from_rand(&mut rand::thread_rng());

        // Generate the partial signatures
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";
        let partials = shares
            .iter()
            .map(|s| partial_sign_message::<V>(s, namespace, msg))
            .collect::<Vec<_>>();

        // Each partial sig can be partially verified against the public polynomial
        partials.iter().for_each(|partial| {
            partial_verify_message::<V>(&group, namespace, msg, partial).unwrap();
        });

        // Generate and verify the threshold sig
        let threshold_sig = threshold_signature_recover::<V, _>(t, &partials).unwrap();
        let threshold_pub = poly::public::<V>(&group);
        verify_message::<V>(threshold_pub, namespace, msg, &threshold_sig).unwrap();
    }

    #[test]
    #[should_panic(expected = "InvalidSignature")]
    fn test_partial_aggregate_signature_bad_share() {
        partial_aggregate_signature_bad_share::<MinPk>();
        partial_aggregate_signature_bad_share::<MinSig>();
    }

    #[test]
    fn test_partial_verify_multiple_public_keys() {
        let mut rng = StdRng::seed_from_u64(0);
        let (n, t) = (5, 4);
        let (public, shares) = generate_shares::<_, MinSig>(&mut rng, None, n, t);
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";

        // Generate partial signatures
        let partials: Vec<_> = shares
            .iter()
            .map(|s| partial_sign_message::<MinSig>(s, namespace, msg))
            .collect();

        // Verify all signatures
        partial_verify_multiple_public_keys::<MinSig, _>(&public, namespace, msg, &partials)
            .expect("all signatures should be valid");
        let polynomial = evaluate_all::<MinSig>(&public, n);
        partial_verify_multiple_public_keys_precomputed::<MinSig, _>(
            &polynomial,
            namespace,
            msg,
            &partials,
        )
        .expect("all signatures should be valid");
    }

    #[test]
    fn test_partial_verify_multiple_public_keys_one_invalid() {
        let mut rng = StdRng::seed_from_u64(0);
        let (n, t) = (5, 4);
        let (public, mut shares) = generate_shares::<_, MinSig>(&mut rng, None, n, t);
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";

        // Corrupt the second share's private key
        let corrupted_index = 1;
        shares[corrupted_index].private = Private::from_rand(&mut rng);

        // Generate partial signatures
        let partials: Vec<_> = shares
            .iter()
            .map(|s| partial_sign_message::<MinSig>(s, namespace, msg))
            .collect();

        // Attempt verification and expect failure with bisection identifying the invalid signature
        let result_1 =
            partial_verify_multiple_public_keys::<MinSig, _>(&public, namespace, msg, &partials);
        let polynomial = evaluate_all::<MinSig>(&public, n);
        let result_2 = partial_verify_multiple_public_keys_precomputed::<MinSig, _>(
            &polynomial,
            namespace,
            msg,
            &partials,
        );
        for result in [result_1, result_2] {
            match result {
                Err(invalid_sigs) => {
                    assert_eq!(
                        invalid_sigs.len(),
                        1,
                        "Exactly one signature should be invalid"
                    );
                    assert_eq!(
                        invalid_sigs[0].index, corrupted_index as u32,
                        "The invalid signature should match the corrupted share's index"
                    );
                }
                _ => panic!("Expected an error with invalid signatures"),
            }
        }
    }

    #[test]
    fn test_partial_verify_multiple_public_keys_many_invalid() {
        let mut rng = StdRng::seed_from_u64(0);
        let (n, t) = (6, 5);
        let (public, mut shares) = generate_shares::<_, MinSig>(&mut rng, None, n, t);
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";

        // Corrupt shares at indices 1 and 3
        let corrupted_indices = vec![1, 3];
        for &idx in &corrupted_indices {
            shares[idx].private = Private::from_rand(&mut rng);
        }

        // Generate partial signatures
        let partials: Vec<_> = shares
            .iter()
            .map(|s| partial_sign_message::<MinSig>(s, namespace, msg))
            .collect();

        // Attempt verification and expect failure with bisection identifying invalid signatures
        let result_1 =
            partial_verify_multiple_public_keys::<MinSig, _>(&public, namespace, msg, &partials);
        let polynomial = evaluate_all::<MinSig>(&public, n);
        let result_2 = partial_verify_multiple_public_keys_precomputed::<MinSig, _>(
            &polynomial,
            namespace,
            msg,
            &partials,
        );
        for result in [result_1, result_2] {
            match result {
                Err(invalid_sigs) => {
                    assert_eq!(
                        invalid_sigs.len(),
                        corrupted_indices.len(),
                        "Number of invalid signatures should match number of corrupted shares"
                    );
                    let invalid_indices: Vec<u32> =
                        invalid_sigs.iter().map(|sig| sig.index).collect();
                    let expected_indices: Vec<u32> =
                        corrupted_indices.iter().map(|&i| i as u32).collect();
                    assert_eq!(
                        invalid_indices, expected_indices,
                        "Invalid signature indices should match corrupted share indices"
                    );
                }
                _ => panic!("Expected an error with invalid signatures"),
            }
        }
    }

    #[test]
    fn test_partial_verify_multiple_public_keys_precomputed_out_of_range() {
        let mut rng = StdRng::seed_from_u64(0);
        let (n, t) = (5, 4);
        let (public, shares) = generate_shares::<_, MinSig>(&mut rng, None, n, t);
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";

        // Generate partial signatures
        let mut partials: Vec<_> = shares
            .iter()
            .map(|s| partial_sign_message::<MinSig>(s, namespace, msg))
            .collect();

        // Corrupt partial signature index
        partials[0].index = 100;

        // Attempt verification and expect failure with bisection identifying the invalid signature
        let polynomial = evaluate_all::<MinSig>(&public, n);
        let result = partial_verify_multiple_public_keys_precomputed::<MinSig, _>(
            &polynomial,
            namespace,
            msg,
            &partials,
        );
        match result {
            Err(invalid_sigs) => {
                assert_eq!(
                    invalid_sigs.len(),
                    1,
                    "Exactly one signature should be invalid"
                );
                assert_eq!(
                    invalid_sigs[0].index, 100,
                    "The invalid signature should match the corrupted index"
                );
            }
            _ => panic!("Expected an error with invalid signatures"),
        }
    }

    #[test]
    fn test_partial_verify_multiple_public_keys_single() {
        let mut rng = StdRng::seed_from_u64(0);
        let (public, shares) = generate_shares::<_, MinSig>(&mut rng, None, 1, 1);
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";

        let partials: Vec<_> = shares
            .iter()
            .map(|s| partial_sign_message::<MinSig>(s, namespace, msg))
            .collect();

        partial_verify_multiple_public_keys::<MinSig, _>(&public, namespace, msg, &partials)
            .expect("signature should be valid");
        let polynomial = evaluate_all::<MinSig>(&public, 1);
        partial_verify_multiple_public_keys_precomputed::<MinSig, _>(
            &polynomial,
            namespace,
            msg,
            &partials,
        )
        .expect("signature should be valid");
    }

    #[test]
    fn test_partial_verify_multiple_public_keys_single_invalid() {
        let mut rng = StdRng::seed_from_u64(0);
        let (public, mut shares) = generate_shares::<_, MinSig>(&mut rng, None, 1, 1);
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";

        shares[0].private = Private::from_rand(&mut rng);

        let partials: Vec<_> = shares
            .iter()
            .map(|s| partial_sign_message::<MinSig>(s, namespace, msg))
            .collect();

        let result1 =
            partial_verify_multiple_public_keys::<MinSig, _>(&public, namespace, msg, &partials);
        let polynomial = evaluate_all::<MinSig>(&public, 1);
        let result2 = partial_verify_multiple_public_keys_precomputed::<MinSig, _>(
            &polynomial,
            namespace,
            msg,
            &partials,
        );
        for result in [result1, result2] {
            match result {
                Err(invalid_sigs) => {
                    assert_eq!(invalid_sigs.len(), 1);
                    assert_eq!(invalid_sigs[0].index, 0);
                }
                _ => panic!("Expected an error with invalid signatures"),
            }
        }
    }

    #[test]
    fn test_partial_verify_multiple_public_keys_last_invalid() {
        let mut rng = StdRng::seed_from_u64(0);
        let (n, t) = (5, 4);
        let (public, mut shares) = generate_shares::<_, MinSig>(&mut rng, None, n, t);
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";

        let corrupted_index = n - 1;
        shares[corrupted_index as usize].private = Private::from_rand(&mut rng);

        let partials: Vec<_> = shares
            .iter()
            .map(|s| partial_sign_message::<MinSig>(s, namespace, msg))
            .collect();

        let result1 =
            partial_verify_multiple_public_keys::<MinSig, _>(&public, namespace, msg, &partials);
        let polynomial = evaluate_all::<MinSig>(&public, n);
        let result2 = partial_verify_multiple_public_keys_precomputed::<MinSig, _>(
            &polynomial,
            namespace,
            msg,
            &partials,
        );

        for result in [result1, result2] {
            match result {
                Err(invalid_sigs) => {
                    assert_eq!(invalid_sigs.len(), 1);
                    assert_eq!(invalid_sigs[0].index, corrupted_index);
                }
                _ => panic!("Expected an error with invalid signatures"),
            }
        }
    }

    // Source: https://github.com/paulmillr/noble-curves/blob/bee1ffe0000095f95b982a969d06baaa3dd8ce73/test/bls12-381/bls12-381-g1-test-vectors.txt
    //
    // The test vectors are in the format: `<private>:<message>:<signature>`.
    const MIN_SIG_TESTS: &str = "25d8cef413ba263e8d5732d3fca51fd369db74712655a5fd7b0b3a58d8095be8::800134e27aacc74dc91153a6bd65f96a5f8c8365c722da2f1e12eb048e0aed6987fa4168a51241ce41434fd05fd4bdd9
611810ebd8f5a7faad47b2249f9d13be0506131db987b6948f1ca3194fa6b643:68:94250c0cc62ae9041c6f6e5042202b3c327991ce4b2841a4145d270f6c8311bc95673c826ada72a6d69e92a833d649e6
419bb1de76e11a476f8d5cc5d85a648ec04f24bf75f6cf1f3fae43e57bf9a491:c8d0:898f660c5b26e8c9461ab3f42eb394465d5a115702c05d2a2bc761a8873ac0f33d21f9ea9cf4c435cd31391f5c8c0a91
0d1bd9077705325666408124339dca98c0c842b35a90bc3cea8e0c36f2d35583:c43623:94f60dc44a4dbb2505befe346c0c143190fc877ded5e877418f0f890b8ae357a40e8fcc189139aaa509d2b6500f623a5
50ff7bd9b21916e55debbd0757e945386b6159ef481d9d774ee67d9b07d0e4ed:7e846556:8d8d9e84012c9c0958018202fe944b4517b618cb7df0b61b1f1ce40b43c2da6330ee0c30a37ac6c7ba0f16aeaa5b99db
29a8af03f8c73c64e14807cdabae877cb0f273169bc5ebf17f3e4ef334690656:ce8e5953d7:b37528df8825b94349cfe90a8c8665915cc49e6c41d78e28f8f1a05c5956ee9af850b82e9be756f024e396fd85d9b1ca
732fbcaef0e216eae6420eff93c68e3547267b69ca48c7ae9d79d481a466fab9:ae9eb5f425ee:b0adf372fe871a5f7efd30ba8f4ea563460a14651b903789324b78fe12c06b23569766c2d7eecdfb734de4485fee2436
4a8135a8847019dad5c1f1b609b50ee72bf5e6459f9c4206ce43de04c2a7103a:01a6d7836c68ad:914a38d1fa13ffdff56cbadd1bd77a3108aae19f76ff2a99d18784cf5c7620d44543045d757f61bdd4fa66780b25eb46
0681339753344b5a346aeec93a9b3b9d1282d620a3cdfc4fb4f0e7a075a99fa0:c335129a7fa17398:a01d6f24c038ebc110d742babcc9dd0a32eb518e1e52fac73a3e0a3395012e708112e86a314649aa1edf90dc51007042
67a4cad01442be6649e8f3de5b14d126baee62c7525ac61e0b2fe3387e7681b5:bb8a6f6e15fce1f262:82aa70654ca48c6ef55ce3edc88ee77922e1064c763aa50fb0d4a2e8b206d4e14ed849b4d175b096481a6afac232c588
1e36e3af518a276dedb69eb0e9df882721116cdb336f692eb691a6d2c7f2ec15:5fbb696cc48ea826a789:8a8f9a764916f3fd5b6cc882f9869ebd1d6a24a057a6e436509c916a9a1e9308e5f891e8e49f39afa0e9afbd3d209cb5
5984b05cfa8100d150b3a9a0a0c1e2be149a09e2ff6218b0648651f82b4e773e:5d1bf1b69e2774fdb03500:91703b32c962a8bda991561258c29cb726fea6300742cfe37ed929f68087638169750a423b5c4b465f5498b64ec660ce
4d84a172794eeeda6217cf4d10fa36f1b21103742926d4948845a8a0e417d13f:8ec1a5032dfff9289fffefaa:92142c1955d234c700373b823ab4b4b308897218096a88ea504267b26c9330b939191c72e770aaed0af3281b418af173
4de6bd9f522e6edf20d0e54cc17cc22f558f115b58478ae6155291e67c28e096:3a5e5a964d6acedcae7b23189a:838d07c7d28c62b1e5aadf8c621c4f360407b3124ac7f7ae3a40a56b1b848b8104f59b4d74e278639e35f4ffa64a3767
64fe9caf26b773198b6700a23c2618d36c7382440339a60236e210fc7f61ade3:8873768b317b84b32fca283a4082:b9fb8bc5cfda68938437c17e9f5cb448ca4bb79be278d8f1eac42b9f9b03039673c3170af211c24d7006d1af522f805e
2b5bf5af15c13c167173ac0b4750a27cd36ebb90cb0ee90d6168fc81eea0c30a:0a93cd89817651705b4fd414054a44:a2216d350329553d3adc81d0d79d000c0edf634443dafe7c292e8f7193d09facf7a6361e02e5df957021429ae306d879
507bf8c00a3364d9f297b3df5f523fc786806b3fc60123d8b231831af5dabce5:691cec082f50711675043ed04233437d:aabdfee018464bba85027210a8a8322f9de64b452d296f31bc8c507c05289b70e8642fb2b6aaa33759d857ece7735231
46b4cd59574a6c1f845d2a57d41c42096db5d53ffc9ec8d4b080c1542e24f30b:9a877c2bef8fd2ef71b2852e35afde4912:88eaa4631bb68b510601e6b099376adfe05c1eed52757611897935974d82bcc2751036723bcbfaf29c7f8c09bfd93b1f
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6bbc2807b27b635285670a68a42d8fafb461fab581e4e2773c199b29088bc554:0c149764b95e6461e820206e5e2b7fa4acf62a3a132db955d5c4ff1cafbfef3b3816aab1bc9bbcf14af47a4e7753f0243842d9b53b3c3c26b9a2e244f2eb06461e2128949e9b437c96cddbddb52d9d6062d4692d05dc624f94fac39401ce51389576f0fd52e670841602645b4ed6a76cc845ec8454538782b3b179e44cd5ec:aaaa82c3fd810924f41b74bdf484460810f5f3274f8102520099f9ca384a5965e50202a36edfa5b0999c7b3c2548ae74";

    #[test]
    fn test_min_sig() {
        // Source: https://github.com/paulmillr/noble-curves/blob/bee1ffe0000095f95b982a969d06baaa3dd8ce73/src/bls12-381.ts#L358
        const DST: &[u8] = b"BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_NUL_";

        // Parse lines
        let mut publics = Vec::new();
        let mut hms = Vec::new();
        let mut signatures = Vec::new();
        for line in MIN_SIG_TESTS.lines() {
            // Extract parts
            let parts: Vec<_> = line.split(':').collect();
            let private = from_hex_formatted(parts[0]).unwrap();
            let private = Scalar::read(&mut private.as_ref()).unwrap();
            let message = from_hex_formatted(parts[1]).unwrap();
            let signature = from_hex_formatted(parts[2]).unwrap();
            let mut signature =
                <MinSig as Variant>::Signature::read(&mut signature.as_ref()).unwrap();

            // Sign message
            let computed = sign::<MinSig>(&private, DST, &message);
            assert_eq!(signature, computed);

            // Verify signature
            let public = compute_public::<MinSig>(&private);
            verify::<MinSig>(&public, DST, &message, &signature).unwrap();

            // Add to batch
            publics.push(public);
            hms.push(hash_message::<MinSig>(DST, &message));
            signatures.push(signature);

            // Fail verification with a manipulated signature
            signature.add(&<MinSig as Variant>::Signature::one());
            assert!(verify::<MinSig>(&public, DST, &message, &signature).is_err());
        }

        // Batch verification
        assert!(MinSig::batch_verify(&mut OsRng, &publics, &hms, &signatures).is_ok());

        // Fail batch verification with a manipulated signature
        signatures[0].add(&<MinSig as Variant>::Signature::one());
        assert!(MinSig::batch_verify(&mut OsRng, &publics, &hms, &signatures).is_err());
    }

    /// Source: https://github.com/paulmillr/noble-curves/blob/bee1ffe0000095f95b982a969d06baaa3dd8ce73/test/bls12-381/bls12-381-g2-test-vectors.txt
    ///
    /// The test vectors are in the format: `<private>:<message>:<signature>`.
    const MIN_PK_TESTS: &str =
    "28b90deaf189015d3a325908c5e0e4bf00f84f7e639b056ff82d7e70b6eede4c:09:8647aa9680cd0cdf065b94e818ff2bb948cc97838bcee987b9bc1b76d0a0a6e0d85db4e9d75aaedfc79d4ea2733a21ae0579014de7636dd2943d45b87c82b1c66a289006b0b9767921bb8edd3f6c5c5dec0d54cd65f61513113c50cc977849e5
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    #[test]
    fn test_min_pk() {
        // Source: https://github.com/paulmillr/noble-curves/blob/bee1ffe0000095f95b982a969d06baaa3dd8ce73/src/bls12-381.ts#L358
        const DST: &[u8] = b"BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_NUL_";

        // Parse lines
        let mut publics = Vec::new();
        let mut hms = Vec::new();
        let mut signatures = Vec::new();
        for line in MIN_PK_TESTS.lines() {
            // Extract parts
            let parts: Vec<_> = line.split(':').collect();
            let private = from_hex_formatted(parts[0]).unwrap();
            let private = Scalar::read(&mut private.as_ref()).unwrap();
            let message = from_hex_formatted(parts[1]).unwrap();
            let signature = from_hex_formatted(parts[2]).unwrap();
            let mut signature =
                <MinPk as Variant>::Signature::read(&mut signature.as_ref()).unwrap();

            // Sign message
            let computed = sign::<MinPk>(&private, DST, &message);
            assert_eq!(signature, computed);

            // Verify signature
            let public = compute_public::<MinPk>(&private);
            verify::<MinPk>(&public, DST, &message, &signature).unwrap();

            // Add to batch
            publics.push(public);
            hms.push(hash_message::<MinPk>(DST, &message));
            signatures.push(signature);

            // Fail verification with a manipulated signature
            signature.add(&<MinPk as Variant>::Signature::one());
            assert!(verify::<MinPk>(&public, DST, &message, &signature).is_err());
        }

        // Batch verification
        assert!(MinPk::batch_verify(&mut OsRng, &publics, &hms, &signatures).is_ok());

        // Fail batch verification with a manipulated signature
        signatures[0].add(&<MinPk as Variant>::Signature::one());
        assert!(MinPk::batch_verify(&mut OsRng, &publics, &hms, &signatures).is_err());
    }

    fn threshold_derive_missing_partials<V: Variant>() {
        // Helper to compute the Lagrange basis polynomial l_i(x) evaluated at a specific point `eval_at_x`.
        fn lagrange_coeff(eval_x: u32, i_x: u32, x_coords: &[u32]) -> Scalar {
            // Initialize the numerator and denominator.
            let mut num = Scalar::one();
            let mut den = Scalar::one();

            // Initialize the evaluation point and the index.
            let eval_x = Scalar::from_index(eval_x);
            let xi = Scalar::from_index(i_x);

            // Compute the Lagrange coefficients.
            for &j_x in x_coords {
                // Skip if the index is the same.
                if i_x == j_x {
                    continue;
                }

                // Initialize the other index.
                let xj = Scalar::from_index(j_x);

                // Numerator: product over j!=i of (eval_x - x_j)
                let mut term = eval_x.clone();
                term.sub(&xj);
                num.mul(&term);

                // Denominator: product over j!=i of (x_i - x_j)
                let mut diff = xi.clone();
                diff.sub(&xj);
                den.mul(&diff);
            }

            // The result is num / den
            num.mul(&den.inverse().expect("should not have duplicate indices"));
            num
        }

        // Generate the public polynomial and the private shares for n participants.
        let mut rng = StdRng::seed_from_u64(0);
        let (n, t) = (5, quorum(5));
        let (public, shares) = generate_shares::<_, V>(&mut rng, None, n, t);

        // Produce partial signatures for every participant.
        let namespace = Some(&b"test"[..]);
        let msg = b"hello";
        let all_partials: Vec<_> = shares
            .iter()
            .map(|s| partial_sign_message::<V>(s, namespace, msg))
            .collect();

        // Take the first `t` partials to use for deriving the others.
        let recovery_partials: Vec<_> = all_partials.iter().take(t as usize).collect();
        let recovery_indices: Vec<u32> = recovery_partials.iter().map(|p| p.index).collect();

        // For each participant, derive their partial signature from the recovery set.
        //
        // The derived signature is a linear combination of the recovery signatures:
        // s_target = sum_{i in recovery_set} s_i * l_i(target_x)
        for target in &shares {
            // Get the target index.
            let target = target.index;

            // Compute the Lagrange coefficients (the scalars) for this combination.
            let scalars: Vec<Scalar> = recovery_indices
                .iter()
                .map(|&recovery_index| lagrange_coeff(target, recovery_index, &recovery_indices))
                .collect();

            // We then use MSM (Multi-Scalar Multiplication) to compute the sum efficiently.
            let points: Vec<_> = recovery_partials.iter().map(|p| p.value).collect();
            let derived = <V as Variant>::Signature::msm(&points, &scalars);
            let derived = Eval {
                index: target,
                value: derived,
            };

            // Verify that the derived partial signature is cryptographically valid.
            partial_verify_message::<V>(&public, namespace, msg, &derived)
                .expect("derived signature should be valid");

            // Verify that the derived signature matches the one originally created.
            let original = all_partials.iter().find(|p| p.index == target).unwrap();
            assert_eq!(derived.value, original.value);
        }
    }

    #[test]
    fn test_threshold_derive_missing_partials() {
        threshold_derive_missing_partials::<MinPk>();
        threshold_derive_missing_partials::<MinSig>();
    }
}