An implementation of Schnorr signatures on the Ed448 curve for both single and threshold numbers
of signers (FROST).

## Example: key generation with trusted dealer and FROST signing

Creating a key with a trusted dealer and splitting into shares; then signing a message
and aggregating the signature. Note that the example just simulates a distributed
scenario in a single thread and it abstracts away any communication between peers.


```rust
# // ANCHOR: tkg_gen
use frost_ed448 as frost;
use rand::thread_rng;
use std::collections::BTreeMap;

let mut rng = thread_rng();
let max_signers = 5;
let min_signers = 3;
let (shares, pubkey_package) = frost::keys::generate_with_dealer(
    max_signers,
    min_signers,
    frost::keys::IdentifierList::Default,
    &mut rng,
)?;
# // ANCHOR_END: tkg_gen

// Verifies the secret shares from the dealer and store them in a BTreeMap.
// In practice, the KeyPackages must be sent to its respective participants
// through a confidential and authenticated channel.
let mut key_packages: BTreeMap<_, _> = BTreeMap::new();

for (identifier, secret_share) in shares {
    # // ANCHOR: tkg_verify
    let key_package = frost::keys::KeyPackage::try_from(secret_share)?;
    # // ANCHOR_END: tkg_verify
    key_packages.insert(identifier, key_package);
}

let mut nonces_map = BTreeMap::new();
let mut commitments_map = BTreeMap::new();

////////////////////////////////////////////////////////////////////////////
// Round 1: generating nonces and signing commitments for each participant
////////////////////////////////////////////////////////////////////////////

// In practice, each iteration of this loop will be executed by its respective participant.
for participant_index in 1..=min_signers {
    let participant_identifier = participant_index.try_into().expect("should be nonzero");
    let key_package = &key_packages[&participant_identifier];
    // Generate one (1) nonce and one SigningCommitments instance for each
    // participant, up to _threshold_.
    # // ANCHOR: round1_commit
    let (nonces, commitments) = frost::round1::commit(
        key_package.signing_share(),
        &mut rng,
    );
    # // ANCHOR_END: round1_commit
    // In practice, the nonces must be kept by the participant to use in the
    // next round, while the commitment must be sent to the coordinator
    // (or to every other participant if there is no coordinator) using
    // an authenticated channel.
    nonces_map.insert(participant_identifier, nonces);
    commitments_map.insert(participant_identifier, commitments);
}

// This is what the signature aggregator / coordinator needs to do:
// - decide what message to sign
// - take one (unused) commitment per signing participant
let mut signature_shares = BTreeMap::new();
# // ANCHOR: round2_package
let message = "message to sign".as_bytes();
# // In practice, the SigningPackage must be sent to all participants
# // involved in the current signing (at least min_signers participants),
# // using an authenticate channel (and confidential if the message is secret).
let signing_package = frost::SigningPackage::new(commitments_map, message);
# // ANCHOR_END: round2_package

////////////////////////////////////////////////////////////////////////////
// Round 2: each participant generates their signature share
////////////////////////////////////////////////////////////////////////////

// In practice, each iteration of this loop will be executed by its respective participant.
for participant_identifier in nonces_map.keys() {
    let key_package = &key_packages[participant_identifier];

    let nonces = &nonces_map[participant_identifier];

    // Each participant generates their signature share.
    # // ANCHOR: round2_sign
    let signature_share = frost::round2::sign(&signing_package, nonces, key_package)?;
    # // ANCHOR_END: round2_sign

    // In practice, the signature share must be sent to the Coordinator
    // using an authenticated channel.
    signature_shares.insert(*participant_identifier, signature_share);
}

////////////////////////////////////////////////////////////////////////////
// Aggregation: collects the signing shares from all participants,
// generates the final signature.
////////////////////////////////////////////////////////////////////////////

// Aggregate (also verifies the signature shares)
# // ANCHOR: aggregate
let group_signature = frost::aggregate(&signing_package, &signature_shares, &pubkey_package)?;
# // ANCHOR_END: aggregate


// Check that the threshold signature can be verified by the group public
// key (the verification key).
# // ANCHOR: verify
let is_signature_valid = pubkey_package
    .verifying_key()
    .verify(message, &group_signature)
    .is_ok();
# // ANCHOR_END: verify
assert!(is_signature_valid);

# Ok::<(), frost::Error>(())
```