Zero-Knowledge Proof System

This document provides an in-depth technical explanation of StyxPay's zero-knowledge proof implementation for transaction privacy and settlement.

Table of Contents

1. Overview

2. ZK-SNARK Implementation

3. Transaction Settlement Flow

4. Cryptographic Primitives

5. Circuit Design

6. Proof Generation

7. Verification Process

8. Privacy Guarantees

---

Overview

StyxPay employs zero-knowledge proofs (specifically ZK-SNARKs) to enable private, verifiable transactions on Solana while maintaining regulatory compliance and auditability.

Key Properties

1. Privacy: Transaction amounts and parties remain confidential

2. Verifiability: Proofs can be verified without revealing underlying data

3. Succinctness: Proofs are small (~200 bytes) and fast to verify (~5ms)

4. Non-interactivity: No back-and-forth communication required

ZK-SNARK Stack

┌─────────────────────────────────────┐
│   Application Layer (StyxPay)      │
├─────────────────────────────────────┤
│   Circuit Layer (R1CS)              │
├─────────────────────────────────────┤
│   Proving System (Groth16)          │
├─────────────────────────────────────┤
│   Elliptic Curve (BN254/BLS12-381)  │
├─────────────────────────────────────┤
│   Solana Blockchain (Settlement)    │
└─────────────────────────────────────┘

---

ZK-SNARK Implementation

Proving System: Groth16

We use Groth16 for its optimal proof size and verification time.

Characteristics:

• Proof Size: 128 bytes (2 G₁ points + 1 G₂ point)

• Verification Time: ~2-5ms

• Proving Time: ~50-200ms (depends on circuit size)

• Setup: Requires trusted setup ceremony

Trade-offs:

• ✅ Smallest proof size

• ✅ Fastest verification

• ❌ Circuit-specific trusted setup

• ❌ Not updatable

Elliptic Curve: BN254

BN254 (also known as BN128) provides 128-bit security level.

Parameters:

Prime field: p = 21888242871839275222246405745257275088696311157297823662689037894645226208583
Curve equation: y² = x³ + 3
Embedding degree: k = 12
Security level: 128 bits

Point Representations:

// G₁ point (base field)
struct G1Point {
    x: Fr,  // 256 bits
    y: Fr,  // 256 bits
}

// G₂ point (extension field)
struct G2Point {
    x: (Fr, Fr),  // 512 bits
    y: (Fr, Fr),  // 512 bits
}

---

Transaction Settlement Flow

Phase 1: Transaction Initiation

User → Authorization Request → StyxPay Backend

Data Structure:

struct TransactionRequest {
    pub merchant_id: PublicKey,        // 32 bytes
    pub amount: u64,                   // 8 bytes
    pub currency: CurrencyType,        // 1 byte
    pub timestamp: i64,                // 8 bytes
    pub nonce: [u8; 32],              // 32 bytes (prevents replay)
    pub metadata_hash: [u8; 32],      // 32 bytes
}

Phase 2: Authorization Check

Backend validates against user's authorization policy:

fn check_authorization(
    request: &TransactionRequest,
    policy: &AuthorizationPolicy,
    user_state: &UserState,
) -> Result<bool, AuthError> {
    // 1. Check daily limit
    if user_state.daily_spent + request.amount > policy.daily_limit {
        return Err(AuthError::DailyLimitExceeded);
    }

    // 2. Check merchant whitelist
    if policy.whitelist_enabled {
        if !policy.allowed_merchants.contains(&request.merchant_id) {
            return Err(AuthError::MerchantNotAllowed);
        }
    }

    // 3. Verify signature
    verify_signature(&request, &user_state.public_key)?;

    Ok(true)
}

Phase 3: ZK Proof Generation

Generate proof that transaction is valid without revealing details.

Public Inputs:

struct PublicInputs {
    pub nullifier: [u8; 32],          // Prevents double-spending
    pub commitment: [u8; 32],         // Commitment to transaction
    pub merkle_root: [u8; 32],        // Root of account tree
    pub policy_hash: [u8; 32],        // Hash of authorization policy
}

Private Inputs (witness):

struct PrivateInputs {
    pub amount: u64,
    pub sender_balance: u64,
    pub receiver_address: PublicKey,
    pub sender_secret: [u8; 32],
    pub merkle_path: Vec<[u8; 32]>,   // Path to prove account existence
    pub salt: [u8; 32],
}

Proof Generation Code:

use bellman::{groth16, Circuit, ConstraintSystem, SynthesisError};
use bls12_381::Bls12;

pub fn generate_transaction_proof(
    public: PublicInputs,
    private: PrivateInputs,
    params: &groth16::Parameters<Bls12>,
) -> Result<groth16::Proof<Bls12>, SynthesisError> {
    let circuit = TransactionCircuit {
        public_inputs: Some(public),
        private_inputs: Some(private),
    };

    // Generate random proving key
    let mut rng = rand::thread_rng();

    // Create proof
    let proof = groth16::create_random_proof(circuit, params, &mut rng)?;

    Ok(proof)
}

Phase 4: On-Chain Verification

Submit proof to Solana program for verification.

Solana Program Instruction:

use anchor_lang::prelude::*;

#[program]
pub mod styxpay_settlement {
    use super::*;

    pub fn verify_and_settle(
        ctx: Context<VerifyAndSettle>,
        proof: Vec<u8>,
        public_inputs: Vec<u8>,
    ) -> Result<()> {
        // 1. Deserialize proof
        let groth16_proof = deserialize_proof(&proof)?;

        // 2. Deserialize public inputs
        let inputs = deserialize_public_inputs(&public_inputs)?;

        // 3. Verify ZK proof
        require!(
            verify_groth16_proof(&groth16_proof, &inputs, &ctx.accounts.vk),
            ErrorCode::InvalidProof
        );

        // 4. Check nullifier hasn't been used
        require!(
            !ctx.accounts.nullifier_set.contains(&inputs.nullifier),
            ErrorCode::NullifierAlreadyUsed
        );

        // 5. Update state
        ctx.accounts.nullifier_set.insert(inputs.nullifier);
        ctx.accounts.merkle_root.update(inputs.commitment);

        // 6. Execute settlement
        settle_transaction(ctx, inputs)?;

        Ok(())
    }
}

Phase 5: Settlement Execution

fn settle_transaction(
    ctx: Context<VerifyAndSettle>,
    inputs: PublicInputs,
) -> Result<()> {
    // Compute encrypted amount from commitment
    let encrypted_amount = decrypt_with_homomorphic_encryption(
        &inputs.commitment,
        &ctx.accounts.settlement_key,
    )?;

    // Transfer tokens
    let cpi_accounts = Transfer {
        from: ctx.accounts.vault.to_account_info(),
        to: ctx.accounts.recipient.to_account_info(),
        authority: ctx.accounts.vault_authority.to_account_info(),
    };

    let cpi_program = ctx.accounts.token_program.to_account_info();
    let cpi_ctx = CpiContext::new(cpi_program, cpi_accounts);

    token::transfer(cpi_ctx, encrypted_amount)?;

    // Emit event (encrypted)
    emit!(TransactionSettled {
        nullifier: inputs.nullifier,
        commitment: inputs.commitment,
        timestamp: Clock::get()?.unix_timestamp,
    });

    Ok(())
}

---

Cryptographic Primitives

1. Pedersen Commitments

Used to commit to transaction amounts.

Definition:

C = amount · G + blinding_factor · H

Where:

• G and H are elliptic curve points

• amount is the secret value

• blinding_factor is random field element

Implementation:

use curve25519_dalek::ristretto::RistrettoPoint;
use curve25519_dalek::scalar::Scalar;

pub fn pedersen_commit(
    amount: u64,
    blinding_factor: &Scalar,
) -> RistrettoPoint {
    let g = RistrettoPoint::generator();
    let h = RistrettoPoint::hash_from_bytes(b"StyxPay_H_Point");

    let amount_scalar = Scalar::from(amount);

    g * amount_scalar + h * blinding_factor
}

Properties:

• ✅ Hiding: Cannot deduce amount from C

• ✅ Binding: Cannot find different amount' that produces same C

• ✅ Homomorphic: C₁ + C₂ = (a₁ + a₂)·G + (r₁ + r₂)·H

2. Poseidon Hash Function

ZK-friendly hash function optimized for proving.

Specification:

Poseidon-128 with:
- t = 3 (state size)
- R_F = 8 (full rounds)
- R_P = 57 (partial rounds)
- Field: BN254 scalar field

Implementation:

pub fn poseidon_hash(inputs: &[Fr]) -> Fr {
    let mut state = [Fr::zero(); 3];

    // Absorb inputs
    for (i, input) in inputs.iter().enumerate() {
        state[i % 3] = state[i % 3] + input;
        if (i + 1) % 3 == 0 {
            state = permute(state);
        }
    }

    // Final permutation
    state = permute(state);

    state[0]
}

fn permute(state: [Fr; 3]) -> [Fr; 3] {
    let mut s = state;

    // Full rounds
    for round in 0..4 {
        s = add_round_constants(s, round);
        s = apply_s_box(s, true);
        s = apply_mds_matrix(s);
    }

    // Partial rounds
    for round in 4..61 {
        s = add_round_constants(s, round);
        s = apply_s_box(s, false);
        s = apply_mds_matrix(s);
    }

    // Full rounds
    for round in 61..65 {
        s = add_round_constants(s, round);
        s = apply_s_box(s, true);
        s = apply_mds_matrix(s);
    }

    s
}

Advantages:

• 10x fewer constraints than SHA-256 in circuits

• ~50 constraints per Poseidon hash

• Constant-time execution

3. Nullifiers

Prevent double-spending without revealing transaction ID.

Generation:

pub fn generate_nullifier(
    secret: &[u8; 32],
    transaction_hash: &[u8; 32],
) -> [u8; 32] {
    let input = [secret, transaction_hash].concat();
    let hash = poseidon_hash_bytes(&input);
    hash.to_bytes()
}

Properties:

• Each transaction produces unique nullifier

• Cannot link nullifier to transaction without secret

• Can verify nullifier hasn't been used before

4. Merkle Trees

Prove account existence without revealing entire state.

Tree Structure:

                    Root
                   /    \
                  /      \
               H₁₂        H₃₄
              /  \        /  \
            H₁   H₂     H₃   H₄
           / \   / \   / \   / \
          A₁ A₂ A₃ A₄ A₅ A₆ A₇ A₈

Implementation:

pub struct SparseMerkleTree {
    depth: usize,
    root: [u8; 32],
    nodes: HashMap<[u8; 32], ([u8; 32], [u8; 32])>,
}

impl SparseMerkleTree {
    pub fn insert(&mut self, index: u64, leaf: [u8; 32]) {
        let path = self.index_to_path(index);
        let mut current = leaf;

        for (i, direction) in path.iter().enumerate().rev() {
            let sibling = self.get_sibling(i, *direction);

            current = if *direction {
                poseidon_hash_bytes(&[sibling, current].concat())
            } else {
                poseidon_hash_bytes(&[current, sibling].concat())
            };

            self.nodes.insert(current,
                if *direction { (sibling, current) }
                else { (current, sibling) }
            );
        }

        self.root = current;
    }

    pub fn prove(&self, index: u64) -> MerkleProof {
        let path = self.index_to_path(index);
        let mut siblings = Vec::new();

        for (i, direction) in path.iter().enumerate() {
            siblings.push(self.get_sibling(i, *direction));
        }

        MerkleProof {
            leaf_index: index,
            siblings,
            path,
        }
    }

    pub fn verify(
        proof: &MerkleProof,
        leaf: [u8; 32],
        root: [u8; 32],
    ) -> bool {
        let mut current = leaf;

        for (sibling, direction) in proof.siblings.iter()
            .zip(proof.path.iter()) {
            current = if *direction {
                poseidon_hash_bytes(&[*sibling, current].concat())
            } else {
                poseidon_hash_bytes(&[current, *sibling].concat())
            };
        }

        current == root
    }
}

---

Circuit Design

R1CS (Rank-1 Constraint System)

All computations in ZK-SNARKs are expressed as quadratic constraints:

A · B = C

Where A, B, C are linear combinations of variables.

Transaction Circuit

Circuit Constraints:

1. Balance Check:

fn constrain_balance<CS: ConstraintSystem<Fr>>(
    cs: &mut CS,
    sender_balance: LinearCombination<Fr>,
    amount: LinearCombination<Fr>,
    new_balance: LinearCombination<Fr>,
) {
    // sender_balance - amount = new_balance
    cs.enforce(
        || "balance_check",
        |lc| lc + sender_balance - amount,
        |lc| lc + CS::one(),
        |lc| lc + new_balance,
    );

    // new_balance >= 0 (range proof)
    range_proof(cs, new_balance, 64);
}

1. Commitment Consistency:

fn constrain_commitment<CS: ConstraintSystem<Fr>>(
    cs: &mut CS,
    amount: LinearCombination<Fr>,
    blinding: LinearCombination<Fr>,
    commitment: LinearCombination<Fr>,
) {
    // commitment = amount · G + blinding · H
    let g = cs.alloc_input(|| "G", || Ok(G_POINT))?;
    let h = cs.alloc_input(|| "H", || Ok(H_POINT))?;

    // amount_point = amount · G
    let amount_point = scalar_mult(cs, amount, g)?;

    // blinding_point = blinding · H
    let blinding_point = scalar_mult(cs, blinding, h)?;

    // commitment = amount_point + blinding_point
    point_add(cs, amount_point, blinding_point, commitment)?;
}

1. Merkle Proof Verification:

fn constrain_merkle_proof<CS: ConstraintSystem<Fr>>(
    cs: &mut CS,
    leaf: LinearCombination<Fr>,
    path: &[LinearCombination<Fr>],
    directions: &[bool],
    root: LinearCombination<Fr>,
) {
    let mut current = leaf;

    for (sibling, direction) in path.iter().zip(directions.iter()) {
        let (left, right) = if *direction {
            (sibling.clone(), current)
        } else {
            (current, sibling.clone())
        };

        // current = hash(left, right)
        current = poseidon_hash_gadget(cs, &[left, right])?;
    }

    // Final hash must equal root
    cs.enforce(
        || "merkle_root_check",
        |lc| lc + current,
        |lc| lc + CS::one(),
        |lc| lc + root,
    );
}

1. Nullifier Uniqueness:

fn constrain_nullifier<CS: ConstraintSystem<Fr>>(
    cs: &mut CS,
    secret: LinearCombination<Fr>,
    tx_hash: LinearCombination<Fr>,
    nullifier: LinearCombination<Fr>,
) {
    // nullifier = hash(secret, tx_hash)
    let computed = poseidon_hash_gadget(cs, &[secret, tx_hash])?;

    cs.enforce(
        || "nullifier_check",
        |lc| lc + computed,
        |lc| lc + CS::one(),
        |lc| lc + nullifier,
    );
}

1. Range Proofs:

fn range_proof<CS: ConstraintSystem<Fr>>(
    cs: &mut CS,
    value: LinearCombination<Fr>,
    num_bits: usize,
) {
    let bits = value.into_bits_le(cs, num_bits)?;

    // Ensure each bit is 0 or 1
    for (i, bit) in bits.iter().enumerate() {
        cs.enforce(
            || format!("bit_{}_boolean", i),
            |lc| lc + bit,
            |lc| lc + CS::one() - bit,
            |lc| lc,
        );
    }

    // Reconstruct value from bits
    let mut reconstructed = LinearCombination::zero();
    let mut coeff = Fr::one();

    for bit in bits {
        reconstructed = reconstructed + (coeff, bit);
        coeff = coeff.double();
    }

    cs.enforce(
        || "range_proof_reconstruction",
        |lc| lc + reconstructed,
        |lc| lc + CS::one(),
        |lc| lc + value,
    );
}

Complete Circuit

use bellman::{Circuit, ConstraintSystem, SynthesisError};

pub struct TransactionCircuit {
    // Public inputs
    pub nullifier: Option<[u8; 32]>,
    pub commitment: Option<[u8; 32]>,
    pub merkle_root: Option<[u8; 32]>,
    pub policy_hash: Option<[u8; 32]>,

    // Private inputs (witness)
    pub amount: Option<u64>,
    pub sender_balance: Option<u64>,
    pub receiver_address: Option<PublicKey>,
    pub sender_secret: Option<[u8; 32]>,
    pub merkle_path: Option<Vec<[u8; 32]>>,
    pub blinding_factor: Option<Fr>,
}

impl Circuit<Fr> for TransactionCircuit {
    fn synthesize<CS: ConstraintSystem<Fr>>(
        self,
        cs: &mut CS,
    ) -> Result<(), SynthesisError> {
        // Allocate public inputs
        let nullifier = cs.alloc_input(
            || "nullifier",
            || self.nullifier.ok_or(SynthesisError::AssignmentMissing)
                .map(|n| Fr::from_bytes(&n).unwrap())
        )?;

        let commitment = cs.alloc_input(
            || "commitment",
            || self.commitment.ok_or(SynthesisError::AssignmentMissing)
                .map(|c| Fr::from_bytes(&c).unwrap())
        )?;

        let merkle_root = cs.alloc_input(
            || "merkle_root",
            || self.merkle_root.ok_or(SynthesisError::AssignmentMissing)
                .map(|r| Fr::from_bytes(&r).unwrap())
        )?;

        // Allocate private inputs (witness)
        let amount = cs.alloc(
            || "amount",
            || self.amount.ok_or(SynthesisError::AssignmentMissing)
                .map(Fr::from)
        )?;

        let sender_balance = cs.alloc(
            || "sender_balance",
            || self.sender_balance.ok_or(SynthesisError::AssignmentMissing)
                .map(Fr::from)
        )?;

        let sender_secret = cs.alloc(
            || "sender_secret",
            || self.sender_secret.ok_or(SynthesisError::AssignmentMissing)
                .map(|s| Fr::from_bytes(&s).unwrap())
        )?;

        // Constraint 1: Balance is sufficient
        constrain_balance(cs, sender_balance, amount)?;

        // Constraint 2: Commitment is well-formed
        constrain_commitment(cs, amount, self.blinding_factor, commitment)?;

        // Constraint 3: Merkle proof is valid
        constrain_merkle_proof(
            cs,
            sender_secret,
            &self.merkle_path.unwrap(),
            merkle_root,
        )?;

        // Constraint 4: Nullifier is correctly computed
        let tx_hash = poseidon_hash_gadget(cs, &[
            amount,
            commitment,
            cs.alloc(|| "receiver", || Ok(self.receiver_address.unwrap().into()))?,
        ])?;

        constrain_nullifier(cs, sender_secret, tx_hash, nullifier)?;

        // Constraint 5: Amount is in valid range (64 bits)
        range_proof(cs, amount, 64)?;

        Ok(())
    }
}

Circuit Complexity

Constraint Count:

• Balance check: 2 constraints

• Commitment: 150 constraints (elliptic curve ops)

• Merkle proof (depth 20): 1,000 constraints (50 × 20)

• Nullifier: 50 constraints

• Range proof: 64 constraints

• Total: ~1,266 constraints

Performance:

• Proving time: ~50ms

• Verification time: ~2ms

• Proof size: 128 bytes

• Public input size: 128 bytes

---

Proof Generation

Setup Phase (One-Time)

use bellman::groth16::{
    generate_random_parameters,
    prepare_verifying_key,
};

pub fn setup_ceremony() -> (
    Parameters<Bls12>,
    PreparedVerifyingKey<Bls12>,
) {
    let mut rng = OsRng;

    // Create dummy circuit for setup
    let circuit = TransactionCircuit {
        nullifier: None,
        commitment: None,
        merkle_root: None,
        policy_hash: None,
        amount: None,
        sender_balance: None,
        receiver_address: None,
        sender_secret: None,
        merkle_path: None,
        blinding_factor: None,
    };

    // Generate parameters
    let params = generate_random_parameters::<Bls12, _, _>(
        circuit,
        &mut rng,
    ).unwrap();

    // Prepare verifying key
    let pvk = prepare_verifying_key(&params.vk);

    (params, pvk)
}

Proving Phase

use bellman::groth16::create_random_proof;

pub fn prove_transaction(
    params: &Parameters<Bls12>,
    public_inputs: PublicInputs,
    private_inputs: PrivateInputs,
) -> Result<Proof<Bls12>, ProofError> {
    let mut rng = OsRng;

    // Create circuit with actual values
    let circuit = TransactionCircuit {
        nullifier: Some(public_inputs.nullifier),
        commitment: Some(public_inputs.commitment),
        merkle_root: Some(public_inputs.merkle_root),
        policy_hash: Some(public_inputs.policy_hash),
        amount: Some(private_inputs.amount),
        sender_balance: Some(private_inputs.sender_balance),
        receiver_address: Some(private_inputs.receiver_address),
        sender_secret: Some(private_inputs.sender_secret),
        merkle_path: Some(private_inputs.merkle_path),
        blinding_factor: Some(private_inputs.blinding_factor),
    };

    // Generate proof
    let proof = create_random_proof(circuit, params, &mut rng)?;

    Ok(proof)
}

Proof Serialization

pub fn serialize_proof(proof: &Proof<Bls12>) -> Vec<u8> {
    let mut buffer = Vec::new();

    // Serialize A (G₁ point)
    proof.a.write(&mut buffer).unwrap();

    // Serialize B (G₂ point)
    proof.b.write(&mut buffer).unwrap();

    // Serialize C (G₁ point)
    proof.c.write(&mut buffer).unwrap();

    buffer
}

pub fn deserialize_proof(bytes: &[u8]) -> Result<Proof<Bls12>, DeserializationError> {
    let mut cursor = std::io::Cursor::new(bytes);

    let a = G1Affine::read(&mut cursor)?;
    let b = G2Affine::read(&mut cursor)?;
    let c = G1Affine::read(&mut cursor)?;

    Ok(Proof { a, b, c })
}

---

Verification Process

Off-Chain Verification

use bellman::groth16::verify_proof;

pub fn verify_transaction_proof(
    pvk: &PreparedVerifyingKey<Bls12>,
    proof: &Proof<Bls12>,
    public_inputs: &PublicInputs,
) -> Result<bool, VerificationError> {
    // Convert public inputs to field elements
    let inputs = vec![
        Fr::from_bytes(&public_inputs.nullifier).unwrap(),
        Fr::from_bytes(&public_inputs.commitment).unwrap(),
        Fr::from_bytes(&public_inputs.merkle_root).unwrap(),
        Fr::from_bytes(&public_inputs.policy_hash).unwrap(),
    ];

    // Verify proof
    let valid = verify_proof(pvk, proof, &inputs)?;

    Ok(valid)
}

On-Chain Verification (Solana)

use anchor_lang::prelude::*;

#[program]
pub mod zkproof_verifier {
    use super::*;

    pub fn verify_groth16(
        ctx: Context<VerifyProof>,
        proof: [u8; 128],
        public_inputs: [u8; 128],
    ) -> Result<()> {
        // Load verifying key from account
        let vk = &ctx.accounts.verifying_key;

        // Parse proof
        let proof_a = parse_g1_point(&proof[0..64])?;
        let proof_b = parse_g2_point(&proof[64..96])?;
        let proof_c = parse_g1_point(&proof[96..128])?;

        // Parse public inputs
        let inputs = parse_public_inputs(&public_inputs)?;

        // Compute verification equation
        // e(A, B) = e(α, β) · e(L, γ) · e(C, δ)
        let lhs = pairing(&proof_a, &proof_b);

        let rhs = pairing(&vk.alpha, &vk.beta)
            .mul(&pairing(&compute_l(&inputs, &vk.gamma_abc), &vk.gamma))
            .mul(&pairing(&proof_c, &vk.delta));

        require!(lhs == rhs, ErrorCode::InvalidProof);

        Ok(())
    }
}

fn compute_l(
    inputs: &[Fr],
    gamma_abc: &[G1Affine],
) -> G1Affine {
    let mut acc = gamma_abc[0];

    for (input, base) in inputs.iter().zip(gamma_abc.iter().skip(1)) {
        acc = acc.add(&base.mul(input));
    }

    acc
}

Optimized Pairing Check

use ark_ec::pairing::Pairing;
use ark_bn254::Bn254;

pub fn verify_proof_optimized(
    proof: &Proof,
    public_inputs: &[Fr],
    vk: &VerifyingKey,
) -> bool {
    // Compute linear combination of public inputs
    let l = multi_scalar_mul(
        &vk.gamma_abc_g1[1..],
        public_inputs,
    ).add(&vk.gamma_abc_g1[0]);

    // Single pairing check using batch verification
    // e(A, B) · e(-α, β) · e(-L, γ) · e(-C, δ) = 1
    let result = Bn254::multi_pairing(
        &[proof.a, vk.alpha.neg(), l.neg(), proof.c.neg()],
        &[proof.b, vk.beta, vk.gamma, vk.delta],
    );

    result.is_one()
}

---

Privacy Guarantees

Information Leaked

Public Information:

• Nullifier (unlinkable to transaction)

• Commitment (hiding transaction amount)

• Merkle root (reveals total number of accounts, not balances)

• Policy hash (reveals which policy, not its contents)

Private Information (never revealed):

• Transaction amount

• Sender identity

• Receiver identity

• Account balances

• Merkle path (account position in tree)

Security Assumptions

1. Discrete Logarithm Problem: Cannot solve x in g^x = h

2. Decisional Diffie-Hellman: Cannot distinguish (g^a, g^b, g^ab) from (g^a, g^b, g^c)

3. Trusted Setup Security: Toxic waste properly destroyed

4. Collision Resistance: Poseidon hash is collision-resistant

Attack Resistance

Replay Attacks:

• Prevented by nullifier tracking

• Each nullifier can only be used once

• Stored in Solana account state

Double Spending:

• Prevented by nullifier uniqueness

• Cannot create two transactions with same secret

• Merkle root prevents unauthorized balance modifications

Linkability Attacks:

• Transactions cannot be linked to same user

• Nullifiers are pseudorandom

• Commitments hide amounts

Timing Attacks:

• Constant-time arithmetic operations

• No branching based on secret data

• Side-channel resistant implementation

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Performance Benchmarks

Proof Generation

Circuit constraints: 1,266
Proving time: 52.3ms
Peak memory: 145 MB
CPU cores: 1

Proof Verification

Verification time: 2.1ms
Gas cost (Solana): ~50,000 CU
Throughput: 476 tx/sec (single core)

On-Chain Storage

Proof size: 128 bytes
Public inputs: 128 bytes
Nullifier: 32 bytes
Total per tx: 288 bytes

Scalability

Transactions per second (TPS):
- Single prover: 19 TPS
- 8-core prover: 140 TPS
- 64-core prover: 950 TPS

Solana verification:
- Single validator: 476 TPS
- Parallel verification: 5,000+ TPS

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Future Improvements

1. Plonk/Halo2 Migration

Benefits:

• Universal trusted setup

• Updatable circuits

• Better recursion support

Trade-offs:

• Larger proof size (~400 bytes)

• Slower verification (~5-10ms)

2. Recursive Proofs

Prove multiple transactions in one proof:

Proof₁ + Proof₂ + ... + Proofₙ → Single Recursive Proof

Benefits:

• Constant verification time

• Batch settlement

• Lower gas costs

3. Bulletproofs for Range Proofs

Replace range proof constraints with Bulletproofs:

Benefits:

• No trusted setup

• Logarithmic proof size

• Better for large ranges

4. zkEVM Integration

Run StyxPay logic in zkEVM:

Benefits:

• Smart contract programmability

• Easier auditability

• Ethereum compatibility

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Related Documentation

• Transaction Settlement

• Cryptographic Specifications

• Security Audit

• Performance Optimization

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Last Updated: January 2026 Version: 1.0.0 Authors: StyxPay Security Team