elastic_elgamal/proofs/

ring.rs

//! Ring proofs.

use core::{fmt, mem};

use elliptic_curve::rand_core::{CryptoRng, RngCore};
use merlin::Transcript;
#[cfg(feature = "serde")]
use serde::{Deserialize, Serialize};

#[cfg(feature = "serde")]
use crate::serde::{ScalarHelper, VecHelper};
use crate::{
    Ciphertext, PublicKey, SecretKey,
    alloc::{Vec, vec},
    encryption::ExtendedCiphertext,
    group::Group,
    proofs::{TranscriptForGroup, VerificationError},
};

/// An incomplete ring proving that the encrypted value is in the a priori known set of
/// admissible values.
struct Ring<'a, G: Group> {
    // Public parameters of the ring.
    index: usize,
    admissible_values: &'a [G::Element],
    ciphertext: Ciphertext<G>,

    // ZKP-related public values.
    transcript: Transcript,
    responses: &'a mut [G::Scalar],
    terminal_commitments: (G::Element, G::Element),

    // Secret values.
    value_index: usize,
    discrete_log: SecretKey<G>,
    random_scalar: SecretKey<G>,
}

impl<G: Group> fmt::Debug for Ring<'_, G> {
    fn fmt(&self, formatter: &mut fmt::Formatter<'_>) -> fmt::Result {
        formatter
            .debug_struct("Ring")
            .field("index", &self.index)
            .field("admissible_values", &self.admissible_values)
            .field("ciphertext", &self.ciphertext)
            .field("responses", &self.responses)
            .field("terminal_commitments", &self.terminal_commitments)
            .finish()
    }
}

impl<'a, G: Group> Ring<'a, G> {
    #[allow(clippy::too_many_arguments)] // fine for a private function
    fn new<R: CryptoRng + RngCore>(
        index: usize,
        log_base: G::Element,
        ciphertext: ExtendedCiphertext<G>,
        admissible_values: &'a [G::Element],
        value_index: usize,
        transcript: &Transcript,
        responses: &'a mut [G::Scalar],
        rng: &mut R,
    ) -> Self {
        assert!(
            !admissible_values.is_empty(),
            "No admissible values supplied"
        );
        assert!(
            value_index < admissible_values.len(),
            "Specified value index is out of bounds"
        );
        debug_assert_eq!(
            responses.len(),
            admissible_values.len(),
            "Number of responses doesn't match number of admissible values"
        );

        let random_element = ciphertext.inner.random_element;
        let blinded_value = ciphertext.inner.blinded_element;
        debug_assert!(
            {
                let expected_blinded_value = log_base * ciphertext.random_scalar.expose_scalar()
                    + admissible_values[value_index];
                expected_blinded_value == blinded_value
            },
            "Specified ciphertext does not match the specified `value_index`"
        );

        let mut transcript = transcript.clone();
        transcript.start_proof(b"ring_enc");
        transcript.append_message(b"enc", &ciphertext.inner.to_bytes());
        // NB: we don't add `admissible_values` to the transcript since we assume that
        // they are fixed in the higher-level protocol.
        transcript.append_u64(b"i", index as u64);

        // Choose a random scalar to use in the equation matching the known discrete log.
        let random_scalar = SecretKey::<G>::generate(rng);
        let mut commitments = (
            G::mul_generator(random_scalar.expose_scalar()),
            log_base * random_scalar.expose_scalar(),
        );

        let it = admissible_values.iter().enumerate().skip(value_index + 1);
        for (eq_index, &admissible_value) in it {
            let mut eq_transcript = transcript.clone();
            eq_transcript.append_u64(b"j", eq_index as u64 - 1);
            eq_transcript.append_element::<G>(b"R_G", &commitments.0);
            eq_transcript.append_element::<G>(b"R_K", &commitments.1);
            let challenge = eq_transcript.challenge_scalar::<G>(b"c");

            let response = G::generate_scalar(rng);
            responses[eq_index] = response;
            let dh_element = blinded_value - admissible_value;
            commitments = (
                G::mul_generator(&response) - random_element * &challenge,
                G::multi_mul([&response, &-challenge], [log_base, dh_element]),
            );
        }

        Self {
            index,
            value_index,
            admissible_values,
            ciphertext: ciphertext.inner,
            transcript,
            responses,
            terminal_commitments: commitments,
            discrete_log: ciphertext.random_scalar,
            random_scalar,
        }
    }

    /// Completes the ring by calculating the common challenge and closing all rings using it.
    ///
    /// # Return value
    ///
    /// Returns the common challenge.
    fn aggregate<R: CryptoRng + RngCore>(
        rings: Vec<Self>,
        log_base: G::Element,
        transcript: &mut Transcript,
        rng: &mut R,
    ) -> G::Scalar {
        debug_assert!(
            rings.iter().enumerate().all(|(i, ring)| i == ring.index),
            "Rings have bogus indexes"
        );

        for ring in &rings {
            let commitments = &ring.terminal_commitments;
            transcript.append_element::<G>(b"R_G", &commitments.0);
            transcript.append_element::<G>(b"R_K", &commitments.1);
        }

        let common_challenge = transcript.challenge_scalar::<G>(b"c");
        for ring in rings {
            ring.finalize(log_base, common_challenge, rng);
        }
        common_challenge
    }

    fn finalize<R: CryptoRng + RngCore>(
        self,
        log_base: G::Element,
        common_challenge: G::Scalar,
        rng: &mut R,
    ) {
        // Compute remaining responses for non-reversible equations.
        let mut challenge = common_challenge;
        let it = self.admissible_values[..self.value_index]
            .iter()
            .enumerate();
        for (eq_index, &admissible_value) in it {
            let response = G::generate_scalar(rng);
            self.responses[eq_index] = response;
            let dh_element = self.ciphertext.blinded_element - admissible_value;
            let commitments = (
                G::mul_generator(&response) - self.ciphertext.random_element * &challenge,
                G::multi_mul([&response, &-challenge], [log_base, dh_element]),
            );

            let mut eq_transcript = self.transcript.clone();
            eq_transcript.append_u64(b"j", eq_index as u64);
            eq_transcript.append_element::<G>(b"R_G", &commitments.0);
            eq_transcript.append_element::<G>(b"R_K", &commitments.1);
            challenge = eq_transcript.challenge_scalar::<G>(b"c");
        }

        // Finally, compute the response for equation #`value_index`, using our knowledge
        // of the trapdoor.
        debug_assert_eq!(self.responses[self.value_index], G::Scalar::from(0_u64));
        self.responses[self.value_index] =
            challenge * self.discrete_log.expose_scalar() + self.random_scalar.expose_scalar();
    }
}

/// Zero-knowledge proof that the one or more encrypted values is each in the a priori known set of
/// admissible values. (Admissible values may differ among encrypted values.)
///
/// # Construction
///
/// In short, a proof is constructed almost identically to [Borromean ring signatures] by
/// Maxwell and Poelstra, with the only major difference being that we work on ElGamal ciphertexts
/// instead of group elements (= public keys).
///
/// A proof consists of one or more *rings*. Each ring proves than a certain
/// ElGamal ciphertext `E = (R, B)` for public key `K` in a group with generator `G`
/// encrypts one of distinct admissible values `x_0`, `x_1`, ..., `x_n`.
/// `K` and `G` are shared among rings, admissible values are generally not.
/// Different rings may have different number of admissible values.
///
/// ## Single ring
///
/// A ring is a challenge `e_0` and a set of responses `s_0`, `s_1`, ..., `s_n`, which
/// must satisfy the following verification procedure:
///
/// For each `j` in `0..=n`, compute
///
/// ```text
/// R_G(j) = [s_j]G - [e_j]R;
/// R_K(j) = [s_j]K - [e_j](B - [x_j]G);
/// e_{j+1} = H(j, R_G(j), R_K(j));
/// ```
///
/// Here, `H` is a cryptographic hash function. The ring is valid if `e_0 = e_{n+1}`.
///
/// This construction is almost identical to [Abe–Ohkubo–Suzuki ring signatures][ring],
/// with the only difference that two group elements are hashed on each iteration instead of one.
/// If admissible values consist of a single value, this protocol reduces to
/// [`LogEqualityProof`] / Chaum–Pedersen protocol.
///
/// As with "ordinary" ring signatures, constructing a ring is only feasible when knowing
/// additional *trapdoor information*. Namely, the prover must know
///
/// ```text
/// r = dlog_G(R) = dlog_K(B - [x_j]G)
/// ```
///
/// for a certain `j`. (This discrete log `r` is the random scalar used in ElGamal encryption.)
/// With this info, the prover constructs the ring as follows:
///
/// 1. Select random scalar `x` and compute `R_G(j) = [x]G`, `R_K(j) = [x]K`.
/// 2. Compute `e_{j+1}`, ... `e_n`, ..., `e_j` ("wrapping" around `e_0 = e_{n+1}`)
///    as per verification formulas. `s_*` scalars are selected uniformly at random.
/// 3. Compute `s_j` using the trapdoor information: `s_j = x + e_j * r`.
///
/// ## Multiple rings
///
/// Transformation to multiple rings is analogous to one in [Borromean ring signatures].
/// Namely, challenge `e_0` is shared among all rings and is computed by hashing
/// values of `R_G` and `R_K` with the maximum index for each of the rings.
///
/// # Applications
///
/// ## Voting protocols
///
/// [`EncryptedChoice`](crate::app::EncryptedChoice) uses `RingProof` to prove that all encrypted
/// values are Boolean (0 or 1). Using a common challenge allows to reduce proof size by ~33%.
///
/// ## Range proofs
///
/// See [`RangeProof`](crate::RangeProof).
///
/// # Implementation details
///
/// - The proof is serialized as the common challenge `e_0` followed by `s_i` scalars for
///   all the rings.
/// - Standalone proof generation and verification are not exposed in public crate APIs.
///   Rather, proofs are part of large protocols, such as [`PublicKey::encrypt_bool()`] /
///   [`PublicKey::verify_bool()`].
/// - The context of the proof is set using [`Transcript`] APIs, which provides hash functions
///   in the protocol described above. Importantly, the proof itself commits to encrypted values
///   and ring indexes, but not to the admissible values across the rings. This must be taken
///   care of in a higher-level protocol, and this is the case for protocols exposed by the crate.
///
/// [`LogEqualityProof`]: crate::LogEqualityProof
/// [Borromean ring signatures]: https://raw.githubusercontent.com/Blockstream/borromean_paper/master/borromean_draft_0.01_34241bb.pdf
/// [ring]: https://link.springer.com/content/pdf/10.1007/3-540-36178-2_26.pdf
#[derive(Debug, Clone)]
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
#[cfg_attr(feature = "serde", serde(bound = ""))]
pub struct RingProof<G: Group> {
    #[cfg_attr(feature = "serde", serde(with = "ScalarHelper::<G>"))]
    common_challenge: G::Scalar,
    #[cfg_attr(feature = "serde", serde(with = "VecHelper::<ScalarHelper<G>, 2>"))]
    ring_responses: Vec<G::Scalar>,
}

impl<G: Group> RingProof<G> {
    fn initialize_transcript(transcript: &mut Transcript, receiver: &PublicKey<G>) {
        transcript.start_proof(b"multi_ring_enc");
        transcript.append_element_bytes(b"K", receiver.as_bytes());
    }

    pub(crate) fn new(common_challenge: G::Scalar, ring_responses: Vec<G::Scalar>) -> Self {
        Self {
            common_challenge,
            ring_responses,
        }
    }

    pub(crate) fn verify<'a>(
        &self,
        receiver: &PublicKey<G>,
        admissible_values: impl Iterator<Item = &'a [G::Element]> + Clone,
        ciphertexts: impl Iterator<Item = Ciphertext<G>>,
        transcript: &mut Transcript,
    ) -> Result<(), VerificationError> {
        // Do quick preliminary checks.
        let total_rings_size: usize = admissible_values.clone().map(<[_]>::len).sum();
        VerificationError::check_lengths(
            "items in all rings",
            self.total_rings_size(),
            total_rings_size,
        )?;

        Self::initialize_transcript(transcript, receiver);
        // We add common commitments to the `transcript` as we cycle through rings,
        // so we need a separate transcript copy to initialize ring transcripts.
        let initial_ring_transcript = transcript.clone();

        let it = admissible_values.zip(ciphertexts).enumerate();
        let mut starting_response = 0;
        for (ring_index, (values, ciphertext)) in it {
            let mut challenge = self.common_challenge;
            let mut commitments = (G::generator(), G::generator());

            let mut ring_transcript = initial_ring_transcript.clone();
            ring_transcript.start_proof(b"ring_enc");
            ring_transcript.append_message(b"enc", &ciphertext.to_bytes());
            ring_transcript.append_u64(b"i", ring_index as u64);

            for (eq_index, (&admissible_value, response)) in values
                .iter()
                .zip(&self.ring_responses[starting_response..])
                .enumerate()
            {
                let dh_element = ciphertext.blinded_element - admissible_value;
                let neg_challenge = -challenge;

                commitments = (
                    G::vartime_double_mul_generator(
                        &neg_challenge,
                        ciphertext.random_element,
                        response,
                    ),
                    G::vartime_multi_mul(
                        [response, &neg_challenge],
                        [receiver.as_element(), dh_element],
                    ),
                );

                // We can skip deriving the challenge for the last equation; it's not used anyway.
                if eq_index + 1 < values.len() {
                    let mut eq_transcript = ring_transcript.clone();
                    eq_transcript.append_u64(b"j", eq_index as u64);
                    eq_transcript.append_element::<G>(b"R_G", &commitments.0);
                    eq_transcript.append_element::<G>(b"R_K", &commitments.1);
                    challenge = eq_transcript.challenge_scalar::<G>(b"c");
                }
            }

            starting_response += values.len();
            transcript.append_element::<G>(b"R_G", &commitments.0);
            transcript.append_element::<G>(b"R_K", &commitments.1);
        }

        let expected_challenge = transcript.challenge_scalar::<G>(b"c");
        if expected_challenge == self.common_challenge {
            Ok(())
        } else {
            Err(VerificationError::ChallengeMismatch)
        }
    }

    pub(crate) fn total_rings_size(&self) -> usize {
        self.ring_responses.len()
    }

    /// Serializes this proof into bytes. As described [above](#implementation-details),
    /// the proof is serialized as the common challenge `e_0` followed by response scalars `s_*`
    /// corresponding successively to each admissible value in each ring.
    pub fn to_bytes(&self) -> Vec<u8> {
        let mut bytes = vec![0_u8; G::SCALAR_SIZE * (1 + self.total_rings_size())];
        G::serialize_scalar(&self.common_challenge, &mut bytes[..G::SCALAR_SIZE]);

        let chunks = bytes[G::SCALAR_SIZE..].chunks_mut(G::SCALAR_SIZE);
        for (response, buffer) in self.ring_responses.iter().zip(chunks) {
            G::serialize_scalar(response, buffer);
        }
        bytes
    }

    /// Attempts to deserialize a proof from bytes. Returns `None` if `bytes` do not represent
    /// a well-formed proof.
    #[allow(clippy::missing_panics_doc)] // triggered by `debug_assert`
    pub fn from_bytes(bytes: &[u8]) -> Option<Self> {
        if bytes.len() % G::SCALAR_SIZE != 0 || bytes.len() < 3 * G::SCALAR_SIZE {
            return None;
        }
        let common_challenge = G::deserialize_scalar(&bytes[..G::SCALAR_SIZE])?;

        let ring_responses: Option<Vec<_>> = bytes[G::SCALAR_SIZE..]
            .chunks(G::SCALAR_SIZE)
            .map(G::deserialize_scalar)
            .collect();
        let ring_responses = ring_responses?;
        debug_assert!(ring_responses.len() >= 2);

        Some(Self {
            common_challenge,
            ring_responses,
        })
    }
}

/// **NB.** Separate method calls of the builder depend on the position of the encrypted values
/// within admissible ones. This means that if a proof is constructed with interruptions between
/// method calls, there is a chance for an adversary to perform a timing attack.
#[doc(hidden)] // only public for benchmarking
pub struct RingProofBuilder<'a, G: Group, R> {
    receiver: &'a PublicKey<G>,
    transcript: &'a mut Transcript,
    rings: Vec<Ring<'a, G>>,
    ring_responses: &'a mut [G::Scalar],
    rng: &'a mut R,
}

impl<G: Group, R: fmt::Debug> fmt::Debug for RingProofBuilder<'_, G, R> {
    fn fmt(&self, formatter: &mut fmt::Formatter<'_>) -> fmt::Result {
        formatter
            .debug_struct("RingProofBuilder")
            .field("receiver", self.receiver)
            .field("rings", &self.rings)
            .field("rng", self.rng)
            .finish()
    }
}

impl<'a, G: Group, R: RngCore + CryptoRng> RingProofBuilder<'a, G, R> {
    /// Starts building a [`RingProof`].
    pub fn new(
        receiver: &'a PublicKey<G>,
        ring_count: usize,
        ring_responses: &'a mut [G::Scalar],
        transcript: &'a mut Transcript,
        rng: &'a mut R,
    ) -> Self {
        RingProof::<G>::initialize_transcript(transcript, receiver);
        Self {
            receiver,
            transcript,
            rings: Vec::with_capacity(ring_count),
            ring_responses,
            rng,
        }
    }

    /// Adds a value among `admissible_values` as a new ring to this proof.
    pub fn add_value(
        &mut self,
        admissible_values: &'a [G::Element],
        value_index: usize,
    ) -> ExtendedCiphertext<G> {
        let ext_ciphertext =
            ExtendedCiphertext::new(admissible_values[value_index], self.receiver, self.rng);
        self.add_precomputed_value(ext_ciphertext.clone(), admissible_values, value_index);
        ext_ciphertext
    }

    pub(crate) fn add_precomputed_value(
        &mut self,
        ciphertext: ExtendedCiphertext<G>,
        admissible_values: &'a [G::Element],
        value_index: usize,
    ) {
        let ring_responses = mem::take(&mut self.ring_responses);
        let (responses_for_ring, rest) = ring_responses.split_at_mut(admissible_values.len());
        self.ring_responses = rest;

        let ring = Ring::new(
            self.rings.len(),
            self.receiver.as_element(),
            ciphertext,
            admissible_values,
            value_index,
            &*self.transcript,
            responses_for_ring,
            self.rng,
        );
        self.rings.push(ring);
    }

    /// Finishes building all rings and returns a common challenge.
    pub fn build(self) -> G::Scalar {
        debug_assert!(
            self.ring_responses.is_empty(),
            "Not all ring_responses were used"
        );
        Ring::aggregate(
            self.rings,
            self.receiver.as_element(),
            self.transcript,
            self.rng,
        )
    }
}

#[cfg(test)]
mod tests {
    use core::iter;

    use rand::Rng;
    use test_casing::test_casing;

    use super::*;
    use crate::{
        curve25519::{ristretto::RistrettoPoint, scalar::Scalar as Scalar25519, traits::Identity},
        group::{ElementOps, Ristretto},
    };

    type Keypair = crate::Keypair<Ristretto>;

    #[test]
    fn single_ring_with_2_elements_works() {
        let mut rng = rand::rng();
        let keypair = Keypair::generate(&mut rng);
        let log_base = keypair.public().as_element();
        let admissible_values = [RistrettoPoint::identity(), Ristretto::generator()];

        let value = RistrettoPoint::identity();
        let ext_ciphertext = ExtendedCiphertext::new(value, keypair.public(), &mut rng);
        let ciphertext = ext_ciphertext.inner;

        let mut transcript = Transcript::new(b"test_ring_encryption");
        RingProof::initialize_transcript(&mut transcript, keypair.public());

        let mut ring_responses = vec![Scalar25519::default(); 2];
        let signature_ring = Ring::new(
            0,
            log_base,
            ext_ciphertext,
            &admissible_values,
            0,
            &transcript,
            &mut ring_responses,
            &mut rng,
        );
        let common_challenge =
            Ring::aggregate(vec![signature_ring], log_base, &mut transcript, &mut rng);

        RingProof::new(common_challenge, ring_responses)
            .verify(
                keypair.public(),
                iter::once(&admissible_values as &[_]),
                iter::once(ciphertext),
                &mut Transcript::new(b"test_ring_encryption"),
            )
            .unwrap();

        // Check a proof for encryption of 1.
        let value = Ristretto::generator();
        let ext_ciphertext = ExtendedCiphertext::new(value, keypair.public(), &mut rng);
        let ciphertext = ext_ciphertext.inner;

        let mut transcript = Transcript::new(b"test_ring_encryption");
        RingProof::initialize_transcript(&mut transcript, keypair.public());
        let mut ring_responses = vec![Scalar25519::default(); 2];
        let signature_ring = Ring::new(
            0,
            log_base,
            ext_ciphertext,
            &admissible_values,
            1,
            &transcript,
            &mut ring_responses,
            &mut rng,
        );
        let common_challenge =
            Ring::aggregate(vec![signature_ring], log_base, &mut transcript, &mut rng);

        RingProof::new(common_challenge, ring_responses)
            .verify(
                keypair.public(),
                iter::once(&admissible_values as &[_]),
                iter::once(ciphertext),
                &mut Transcript::new(b"test_ring_encryption"),
            )
            .unwrap();
    }

    #[test]
    fn single_ring_with_4_elements_works() {
        let mut rng = rand::rng();
        let keypair = Keypair::generate(&mut rng);
        let log_base = keypair.public().as_element();
        let admissible_values: Vec<_> = (0_u32..4)
            .map(|i| Ristretto::mul_generator(&Scalar25519::from(i)))
            .collect();

        for _ in 0..100 {
            let val: u32 = rng.random_range(0..4);
            let element_val = Ristretto::mul_generator(&Scalar25519::from(val));
            let ext_ciphertext = ExtendedCiphertext::new(element_val, keypair.public(), &mut rng);
            let ciphertext = ext_ciphertext.inner;

            let mut transcript = Transcript::new(b"test_ring_encryption");
            RingProof::initialize_transcript(&mut transcript, keypair.public());

            let mut ring_responses = vec![Scalar25519::default(); 4];
            let signature_ring = Ring::new(
                0,
                log_base,
                ext_ciphertext,
                &admissible_values,
                val as usize,
                &transcript,
                &mut ring_responses,
                &mut rng,
            );
            let common_challenge =
                Ring::aggregate(vec![signature_ring], log_base, &mut transcript, &mut rng);

            RingProof::new(common_challenge, ring_responses)
                .verify(
                    keypair.public(),
                    iter::once(admissible_values.as_slice()),
                    iter::once(ciphertext),
                    &mut Transcript::new(b"test_ring_encryption"),
                )
                .unwrap();
        }
    }

    #[test_casing(5, 3..=7)]
    fn multiple_rings_with_boolean_flags_work(ring_count: usize) {
        let mut rng = rand::rng();
        let keypair = Keypair::generate(&mut rng);
        let log_base = keypair.public().as_element();
        let admissible_values = [RistrettoPoint::identity(), Ristretto::generator()];

        for _ in 0..20 {
            let mut transcript = Transcript::new(b"test_ring_encryption");
            RingProof::initialize_transcript(&mut transcript, keypair.public());

            let mut ring_responses = vec![Scalar25519::default(); ring_count * 2];

            let (ciphertexts, rings): (Vec<_>, Vec<_>) = ring_responses
                .chunks_mut(2)
                .enumerate()
                .map(|(ring_index, ring_responses)| {
                    let val: u32 = rng.random_range(0..=1);
                    let element_val = Ristretto::mul_generator(&Scalar25519::from(val));
                    let ext_ciphertext =
                        ExtendedCiphertext::new(element_val, keypair.public(), &mut rng);
                    let ciphertext = ext_ciphertext.inner;

                    let signature_ring = Ring::new(
                        ring_index,
                        log_base,
                        ext_ciphertext,
                        &admissible_values,
                        val as usize,
                        &transcript,
                        ring_responses,
                        &mut rng,
                    );

                    (ciphertext, signature_ring)
                })
                .unzip();

            let common_challenge = Ring::aggregate(rings, log_base, &mut transcript, &mut rng);

            RingProof::new(common_challenge, ring_responses)
                .verify(
                    keypair.public(),
                    iter::repeat_n(&admissible_values as &[_], ring_count),
                    ciphertexts.into_iter(),
                    &mut Transcript::new(b"test_ring_encryption"),
                )
                .unwrap();
        }
    }

    #[test]
    fn multiple_rings_with_base4_value_encoding_work() {
        // We're testing ciphertexts of `u8` integers, hence 4 rings with 4 elements (=2 bits) each.
        const RING_COUNT: u8 = 4;

        // Admissible values are `[O, G, [2]G, [3]G]` for the first ring,
        // `[O, [4]G, [8]G, [12]G]` for the second ring, etc.
        let admissible_values: Vec<_> = (0..RING_COUNT)
            .map(|ring_index| {
                let power: u32 = 1 << (2 * u32::from(ring_index));
                [
                    RistrettoPoint::identity(),
                    Ristretto::mul_generator(&Scalar25519::from(power)),
                    Ristretto::mul_generator(&Scalar25519::from(power * 2)),
                    Ristretto::mul_generator(&Scalar25519::from(power * 3)),
                ]
            })
            .collect();

        let mut rng = rand::rng();
        let keypair = Keypair::generate(&mut rng);
        let log_base = keypair.public().as_element();

        for _ in 0..20 {
            let overall_value: u8 = rng.random();
            let mut transcript = Transcript::new(b"test_ring_encryption");
            RingProof::initialize_transcript(&mut transcript, keypair.public());

            let mut ring_responses = vec![Scalar25519::default(); RING_COUNT as usize * 4];

            let (ciphertexts, rings): (Vec<_>, Vec<_>) = ring_responses
                .chunks_mut(4)
                .enumerate()
                .map(|(ring_index, ring_responses)| {
                    let mask = 3 << (2 * ring_index);
                    let val = overall_value & mask;
                    let val_index = (val >> (2 * ring_index)) as usize;
                    assert!(val_index < 4);

                    let element_val = Ristretto::mul_generator(&Scalar25519::from(val));
                    let ext_ciphertext =
                        ExtendedCiphertext::new(element_val, keypair.public(), &mut rng);
                    let ciphertext = ext_ciphertext.inner;

                    let signature_ring = Ring::new(
                        ring_index,
                        log_base,
                        ext_ciphertext,
                        &admissible_values[ring_index],
                        val_index,
                        &transcript,
                        ring_responses,
                        &mut rng,
                    );

                    (ciphertext, signature_ring)
                })
                .unzip();

            let common_challenge = Ring::aggregate(rings, log_base, &mut transcript, &mut rng);
            let admissible_values = admissible_values.iter().map(|values| values as &[_]);

            RingProof::new(common_challenge, ring_responses)
                .verify(
                    keypair.public(),
                    admissible_values,
                    ciphertexts.into_iter(),
                    &mut Transcript::new(b"test_ring_encryption"),
                )
                .unwrap();
        }
    }

    #[test_casing(5, 3..=7)]
    #[allow(clippy::needless_collect)]
    // ^-- false positive; `builder` is captured by the iterator and moved by creating a `proof`
    fn proof_builder_works(ring_count: usize) {
        let mut rng = rand::rng();
        let keypair = Keypair::generate(&mut rng);
        let mut transcript = Transcript::new(b"test_ring_encryption");
        let admissible_values = [RistrettoPoint::identity(), Ristretto::generator()];
        let mut ring_responses = vec![Scalar25519::default(); ring_count * 2];

        let mut builder = RingProofBuilder::new(
            keypair.public(),
            ring_count,
            &mut ring_responses,
            &mut transcript,
            &mut rng,
        );
        let ciphertexts: Vec<_> = (0..ring_count)
            .map(|i| builder.add_value(&admissible_values, i & 1).inner)
            .collect();

        RingProof::new(builder.build(), ring_responses)
            .verify(
                keypair.public(),
                iter::repeat_n(&admissible_values as &[_], ring_count),
                ciphertexts.into_iter(),
                &mut Transcript::new(b"test_ring_encryption"),
            )
            .unwrap();
    }
}