Zero-Knowledge Proofs for Homomorphically Encrypted Transactions: A Practical Guide

How to verify computations over encrypted transaction data without revealing the data itself using ZK proofs and Homomorphic Encryption.

Zero-Knowledge Proofs for Homomorphically Encrypted Transactions: A Practical Guide

In the evolving landscape of financial systems, blockchain networks, and privacy-preserving computation, one challenge stands out:

How can we verify computations over encrypted transaction data without revealing the data itself?

This is where Zero-Knowledge (ZK) proofs and Homomorphic Encryption (HE) meet. HE allows computation on encrypted data; ZK proofs let you prove those computations were done correctly — without exposing the plaintext.

In this blog, we take a deep dive into practical designs for such systems, covering:

Core cryptographic building blocks
Security models and goals
Three architecture patterns for ZK + HE
Concrete protocol sketches
Implementation guidance and use cases
Current limitations and open research problems

The Core Problem

Imagine:

A custodian aggregates transactions for compliance checks.
A DeFi aggregator computes risk metrics over encrypted balances.
A bank sums encrypted customer deposits for a regulator.

In all cases:

The data must stay encrypted for confidentiality.
The verifier (auditor, smart contract, regulator) needs assurance that the computation was correct and policy rules were followed.

Solution: Combine HE (for privacy) with ZK proofs (for verifiable correctness).

Building Blocks

Homomorphic Encryption (HE)

Additive HE (e.g., Paillier) – supports addition and scalar multiplication.
Somewhat / Leveled HE (e.g., BFV, CKKS) – supports bounded-depth circuits.
Fully Homomorphic Encryption (FHE) – supports arbitrary circuits, but costly.

Key property:

Eval(f, Enc(x1), ..., Enc(xn)) = Enc(f(x1, ..., xn))

Zero-Knowledge Proofs

Sigma Protocols – efficient, interactive, best for simple linear relations.
NIZKs / SNARKs – succinct, non-interactive proofs for arbitrary circuits.
Bulletproofs – logarithmic-size range and inner-product proofs.

Security Goals

Any architecture must ensure:

Correctness – Output ciphertext decrypts to the intended function of inputs.
Zero-Knowledge – Proof leaks no extra info about inputs.
Soundness – Cheating provers can't pass verification with false results.
Efficiency – Proof size and verification must be practical.

Three Practical Architectures

A. Sigma-Protocol over Additive HE (Efficient Linear Checks)

Best for:

Balance conservation checks
Simple sum/weighted sum verification

How it works:

Clients encrypt values with Paillier and also publish Pedersen commitments.
Prover computes the encrypted sum homomorphically.
Prover generates a ZK proof that the decrypted sum matches the committed sum.

✓ Pros:

Very fast, small proofs, low verification cost.

✗ Cons:

Works only for linear relations, needs trusted or threshold decryption.

B. zkSNARK over Circuit-Friendly HE (General-Purpose)

Best for:

Arbitrary policy checks
On-chain verification

How it works:

Prover runs homomorphic evaluation to get encrypted output.
Builds a SNARK circuit that proves the correctness of the homomorphic evaluation.
Verifier checks the SNARK — cheap, constant-time.

✓ Pros:

Succinct, verifier-friendly, fits blockchain verification.

✗ Cons:

High prover cost, complex circuits for HE encryption logic.

C. Homomorphic MAC + ZK Proof (Streaming & Batch-Friendly)

Best for:

Continuous aggregation
IoT/streaming data

How it works:

Each encrypted value has a homomorphic authentication tag.
Tags combine along with ciphertexts during computation.
Prover produces a ZK proof that the tag matches the encrypted result.

✓ Pros:

Lightweight aggregation, efficient batching.

✗ Cons:

Needs pre-distributed MAC keys, special setup.

Example — Sum Verification with Paillier HE

Encryption: C_i = g^m_i r_i^N mod N²

Commitment: Com(m_i) = g^m_i h^r'_i

Proof: Use Chaum–Pedersen to show decrypted sum matches committed sum.

Performance & Deployment Guidance

Approach trade-offs:

Sigma + Additive HE: Fast, minimal overhead, ideal for audits.
zkSNARK + HE: Succinct, works for any computation, best for blockchain.
MAC + ZK: Efficient in high-frequency streaming settings.

Implementation tips:

Use established libraries: libsnark, bellman, SEAL, TFHE.
Precompute where possible (especially for SNARK witnesses).
Store proofs & commitments immutably for audit trails.

Real-World Applications

Banking compliance: Encrypted aggregate balances with proofs for regulators.
Custody services: MPC wallets proving no double-spend or quota violations.
DeFi privacy pools: Prove liquidity balances without revealing addresses.
Federated learning: Verifiable encrypted model aggregation.

Challenges & Open Problems

Efficiency: SNARK + HE circuits still heavy — need optimized encodings.
Post-Quantum security: Current pairing-based SNARKs are not PQ-safe.
Composable security: Formal frameworks for HE + ZK in multi-party setups.
Metadata leakage: Side-channel protections for timing and traffic analysis.

Conclusion

Combining Zero-Knowledge Proofs with Homomorphic Encryption is a powerful way to ensure confidentiality + integrity in modern financial, blockchain, and analytic systems.

The right architecture depends on:

The type of relations (linear vs. arbitrary)
Performance constraints (prover/verifier resources)
Deployment context (off-chain, on-chain, streaming)

As cryptographic tooling matures, we expect more efficient, post-quantum-ready, and developer-friendly frameworks for building verifiable encrypted transaction systems.

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