If you're implementing homomorphic encryption for payment processing or tokenization systems, you've likely encountered noise flooding as a defense against decryption attacks. What you may not know is that the standard security proof for this defense has a critical composition problem that affects how confidently you can set your security parameters.
This checklist guides you through the verification requirements for adaptive decryption defenses when you're evaluating or implementing approximate homomorphic encryption schemes. It's designed for teams that need to validate cryptographic implementations against real-world attack scenarios where adversaries can adaptively query your system.
Prerequisites
Before using this checklist, confirm:
- Your encryption scheme provides approximate correctness (not exact computation).
- You've documented your IND-CPA security requirements.
- You understand your plaintext dimension (n) and can adjust flooding-width parameters.
- You have access to formal verification tools or can evaluate machine-checked proofs.
- Your threat model includes adaptive adversaries who can make multiple decryption queries.
Verification Checklist
1. Confirm your base encryption scheme meets IND-CPA security
Verify that your fully homomorphic encryption implementation is secure under chosen-plaintext attack conditions. This is your foundation.
Good looks like: Documentation showing formal proof or certification that your scheme resists adversaries who can encrypt arbitrary plaintexts and attempt to distinguish ciphertexts.
2. Document the number of adaptive queries (q) your system must resist
Count the maximum decryption queries an adversary could make in your threat model. This directly affects your security parameter calculations.
Good looks like: A written threat scenario stating "attacker can submit up to 1,000 adaptive decryption requests before detection" with justification based on your monitoring capabilities.
3. Calculate your security loss from composition
Determine whether your proof uses linear composition (loses linearly in q) or Pythagorean composition (loses by square root). The difference is critical for parameter sizing.
Good looks like: Clear documentation showing your security bound includes the composition method. For Pythagorean composition, you should see a term like √(qn)/2γ rather than qn/γ.
4. Verify conditional KL cost accumulation
If your proof uses Pythagorean composition, confirm it accumulates conditional Kullback-Leibler divergence costs throughout the reduction rather than converting to statistical distance at each step.
Good looks like: Proof documentation showing KL budgets tracked separately and converted to statistical distance once at the final step, not after each hybrid game transition.
5. Set your flooding-width multiplier (γ) appropriately
Calculate the γ parameter that gives you acceptable security loss given your q and plaintext dimension n. Remember this appears in the denominator of your security bound.
Good looks like: Written justification showing "with q=1000, n=4096, and γ=10, our security loss is √(4096000)/20 ≈ 101, which meets our 128-bit security target with margin."
6. Validate your relational program logic
If you're using machine-checked proofs, verify the relational program logic correctly models your adaptive adversary's capabilities.
Good looks like: Formal verification output from tools like Rocq showing your oracle interactions are correctly represented and your trace compiler lifts local rules to arbitrary adaptive programs.
7. Check statistical distance conversion happens once
Confirm your security reduction converts from KL divergence to statistical distance at the final step only, not after each decryption answer simulation.
Good looks like: Proof structure showing n simulation steps accumulating KL costs, followed by a single application of Pinsker's inequality at the end.
8. Test your implementation against the verified reduction
Run your actual code against the attack model described in your formal proof. Your real-world parameters should match or exceed those in the verified reduction.
Good looks like: Test results showing "our implementation with γ=12 maintains security bound of 2^-100 against 500 adaptive queries, matching formal proof predictions."
9. Document the security-parameter tradeoff
Write down how increasing q or n affects your required γ and therefore your noise overhead and performance.
Good looks like: A table showing "doubling q requires increasing γ by √2 to maintain security level, resulting in X% performance degradation."
10. Establish monitoring for query patterns
Set up detection for adversaries attempting to approach your q-query limit through adaptive probing.
Good looks like: Alerting rule that triggers when a single source makes more than 10% of your maximum q value within your detection window.
Common Mistakes
Using ordinary hybrid arguments for adaptive scenarios. The linear loss in q makes this approach impractical for realistic query counts. If your security bound includes a bare q term (not √q), you're likely using the wrong composition method.
Converting to statistical distance too early. Each intermediate conversion compounds your security loss. The Pythagorean approach's advantage comes from deferring this conversion.
Ignoring the plaintext dimension in parameter selection. Your security loss includes both q and n under the square root. A larger plaintext space requires proportionally larger γ.
Treating machine-checked proofs as optional. For cryptographic implementations protecting cardholder data or payment credentials, formal verification isn't academic overhead. It's how you know your security bound actually holds.
Mismatching threat model and implementation. Your formal proof assumes specific adversary capabilities. If your production system allows more queries or different interaction patterns than your verified model, your security guarantees don't transfer.
Next Steps
After completing this checklist:
- Review your parameter choices with your cryptography team quarterly as query patterns evolve.
- Document any deviations between your verified model and production implementation.
- Plan for parameter updates if you need to support higher query volumes.
- Consider whether your monitoring can detect adaptive attack patterns before they reach q queries.
- Evaluate whether your current formal verification tooling can accommodate scheme updates.
If you're implementing new homomorphic encryption systems for payment processing, start with schemes that provide machine-checked Pythagorean composition proofs. The tighter security bounds let you use smaller noise parameters, which directly improves performance without sacrificing the security margins you need for PCI DSS environments.
