Approximate Arithmetic • Encrypted ML

Compute on Encrypted Floats.
ML Without Decryption.

CKKS approximate arithmetic over encrypted floating-point vectors. Dot products, similarity search, bootstrapping — all on data that never leaves ciphertext. N=8192 turbo mode with auto depth management.

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8,192

Poly Degree

4,096

Slots

50+

Scale Bits

128-bit

Security

Capabilities

Full Encrypted Arithmetic

Every operation runs entirely on ciphertext. The server never sees your data — not the inputs, not the intermediates, not the results.

Encrypted Addition

Add ciphertexts element-wise. Constant-time across all input sizes. Level-preserving — no depth consumed.

Level-Preserving

Encrypted Multiplication

Multiply with auto-rescale and relinearization. Consumes one multiplicative level per operation. Scale management is automatic.

Level-Consuming

Dot Product

Encrypted vector dot product — the key operation for ML inference. Multiply + rotate + sum in a single API call.

Level-Consuming

Rotation

Slot rotation for encrypted SIMD operations. Enables sum-reduction patterns, matrix transpositions, and data movement across encrypted slots.

SIMD Operation

Similarity Search

Normalized dot product in the encrypted domain. Cosine similarity without decryption — compare embeddings while they stay encrypted.

Level-Consuming

Bootstrapping

Level refresh via Chebyshev polynomial approximation. Enables unlimited computation depth — reset noise budget and keep computing.

Level Refresh

Architecture

Turbo → Standard Auto-Switch

CKKS turbo mode (N=8192) delivers fast operations for shallow circuits. When depth is exhausted, the bridge transparently transitions to standard parameters (N=16384) for deeper computation — zero application code changes.

Turbo Mode

N = 8,192

4,096 slots
4 mul levels
Scale: 2⁴⁰
Chain: [58, 40, 40, 40, 40]

→

Bridge

Auto-Switch

Transparent
parameter
transition

→

Standard Mode

N = 16,384

8,192 slots
9 mul levels
Scale: 2⁴⁰
Chain: [60, 40×9]

Fast Start

Most workloads fit in 4 multiplicative levels. Turbo mode runs them at maximum speed with smaller polynomials and lower memory overhead.

Seamless Bridge

When turbo levels run out, ckks_bridge.rs handles the parameter transition. Your application code stays identical.

Deep Compute

Standard mode provides 9 multiplicative levels. Bootstrapping available for circuits that need unlimited depth. Precision mode (N=32768) for extreme depth.

API

Three Tiers of Cryptographic Trust

Start with raw CKKS computation. Layer Dilithium signatures for post-quantum attestation. Add H33-3-Key for full triple-layer verification.

CKKS Endpoints

POST/v1/fhe/h33-ckks/keygen

POST/v1/fhe/h33-ckks/encrypt

POST/v1/fhe/h33-ckks/decrypt

POST/v1/fhe/h33-ckks/add

POST/v1/fhe/h33-ckks/multiply

POST/v1/fhe/h33-ckks/rotate

POST/v1/fhe/h33-ckks/inner-product

POST/v1/fhe/h33-ckks/similarity

POST/v1/fhe/h33-ckks/bootstrap

GET/v1/fhe/h33-ckks/health

Example Request

curl -X POST https://api.h33.ai/v1/fhe/h33-ckks/encrypt \
  -H "Authorization: Bearer $H33_API_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "public_key_id": "pk_ckks_...",
    "values": [0.123, -0.456, 0.789, 1.234],
    "mode": "turbo"
  }'

CKKS + Dilithium Endpoints

POST/v1/fhe/h33-ckks/keygen

POST/v1/fhe/h33-ckks/encrypt

POST/v1/fhe/h33-ckks/decrypt

POST/v1/fhe/h33-ckks/add

POST/v1/fhe/h33-ckks/multiply

POST/v1/fhe/h33-ckks/rotate

POST/v1/fhe/h33-ckks/inner-product

POST/v1/fhe/h33-ckks/similarity

POST/v1/fhe/h33-ckks/bootstrap

POST/v1/fhe/h33-ckks/dilithium/keygen

POST/v1/fhe/h33-ckks/dilithium/sign

POST/v1/fhe/h33-ckks/dilithium/verify

GET/v1/fhe/h33-ckks/health

Example Request

curl -X POST https://api.h33.ai/v1/fhe/h33-ckks/inner-product \
  -H "Authorization: Bearer $H33_API_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "ciphertext_a": "ct_ckks_...",
    "ciphertext_b": "ct_ckks_...",
    "dilithium_key_id": "dk_...",
    "sign_result": true
  }'

Dilithium tier: All compute endpoints (add, multiply, rotate, inner-product, similarity, bootstrap) include a dilithium_signature field in the response, attesting that the computation was performed correctly.

CKKS + H33-3-Key Endpoints

POST/v1/fhe/h33-ckks/keygen

POST/v1/fhe/h33-ckks/encrypt

POST/v1/fhe/h33-ckks/decrypt

POST/v1/fhe/h33-ckks/add

POST/v1/fhe/h33-ckks/multiply

POST/v1/fhe/h33-ckks/rotate

POST/v1/fhe/h33-ckks/inner-product

POST/v1/fhe/h33-ckks/similarity

POST/v1/fhe/h33-ckks/bootstrap

POST/v1/fhe/h33-ckks/dilithium/keygen

POST/v1/fhe/h33-ckks/dilithium/sign

POST/v1/fhe/h33-ckks/dilithium/verify

POST/v1/fhe/h33-ckks/3key/keygen

POST/v1/fhe/h33-ckks/3key/sign

POST/v1/fhe/h33-ckks/3key/verify

GET/v1/fhe/h33-ckks/health

Example Request

curl -X POST https://api.h33.ai/v1/fhe/h33-ckks/similarity \
  -H "Authorization: Bearer $H33_API_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "query_ct": "ct_ckks_...",
    "candidates": ["ct_ckks_...", "ct_ckks_..."],
    "three_key_id": "3k_...",
    "full_attestation": true
  }'

H33-3-Key tier: Every response includes a full 3-layer signature bundle — Dilithium (ML-DSA-65) attestation, SHA3-256 computation digest, and the H33 triple-key verification proof. Provides the highest level of cryptographic trust available.

Included with every API call

H33 Noise Pilot

CKKS Noise Management on Autopilot

CKKS is powerful but fragile — precision degrades with every multiplication. Noise Pilot manages rescaling, level tracking, and bootstrap scheduling automatically so your encrypted floats stay accurate.

Auto-Rescale

After every multiplication, Noise Pilot automatically rescales to maintain precision. No manual level management required.

Bootstrap Scheduling

Noise Pilot monitors remaining levels and triggers bootstrapping before depth exhaustion — enabling unlimited computation depth.

Turbo→Standard Switch

Starts with Turbo parameters for speed, seamlessly switches to Standard when deeper circuits are detected. Zero developer config.

Scale Precision Tracking

Tracks floating-point scale across operations. Warns before precision loss exceeds your tolerance threshold.

Use Cases

Built for Real-World Encrypted Computation

From healthcare to finance, CKKS enables computation on the most sensitive floating-point data without ever exposing it.

ML Inference

Run neural network layers on encrypted inputs. Dot products, activations, and weighted sums — all in ciphertext. The model never sees raw user data.

Encrypted Analytics

Aggregate statistics, compute means, variances, and correlations across encrypted datasets. Multi-party analytics without data sharing agreements.

Healthcare

Process patient vitals, lab results, and diagnostic scores without exposing PHI. HIPAA compliance built into the math — data is never decrypted on server.

Recommendations

Compute similarity scores between encrypted user preference vectors. Serve personalized recommendations without ever learning user preferences.

Financial Modeling

Risk scoring, portfolio optimization, and credit modeling on encrypted financial data. Regulatory compliance and trade secret protection in one API call.

Secure Search

Vector similarity search over encrypted embeddings. The search engine computes relevance scores without learning query content or document contents.

New Capability

Encrypted Activation Functions

Evaluate sigmoid, ReLU, tanh, and GELU directly on CKKS ciphertexts. Chebyshev polynomial approximation with automatic degree selection — no plaintext exposure, no manual polynomial tuning.

Sigmoid

1 / (1 + e^-x)

The standard logistic function for binary classification layers. Chebyshev approximation over [-8, 8] with configurable degree for precision vs. depth trade-off.

Degree 7–15 Classification

ReLU

max(0, x)

Smooth polynomial approximation of rectified linear unit. Enables deep encrypted neural networks with the most widely used activation function in modern architectures.

Degree 5–13 Deep Networks

Tanh

(e^x - e^-x) / (e^x + e^-x)

Hyperbolic tangent for recurrent and normalization layers. Naturally bounded output [-1, 1] makes it well-suited for Chebyshev approximation with high accuracy.

Degree 7–15 RNNs / LSTMs

GELU

x · Φ(x)

Gaussian Error Linear Unit — the activation behind transformer architectures. Smooth approximation enables encrypted inference on BERT, GPT, and ViT-style models.

Degree 9–19 Transformers

Auto-Degree Selection

You specify the activation function and a precision target. H33-CKKS automatically selects the minimum Chebyshev polynomial degree that meets your accuracy requirement while consuming the fewest multiplicative levels. Higher degrees yield better approximation but cost more depth — the auto-selector finds the optimal balance for your remaining noise budget.

Function	Degree Range	Levels Consumed
Sigmoid	7 – 15	3 – 4
ReLU	5 – 13	3 – 4
Tanh	7 – 15	3 – 4
GELU	9 – 19	4 – 5

Comparison

H33-CKKS vs. the Field

How H33-CKKS compares to Microsoft SEAL, OpenFHE, and Zama Concrete on the features that matter for production CKKS workloads.

Feature	H33-CKKS	MS SEAL CKKS	OpenFHE CKKS	Zama Concrete
Language	Rust (native)	C++	C++	Rust/Python
Turbo mode (N=8192)	Yes (auto-switch)	Manual only	Manual only	No
Bootstrapping	Built-in, auto	Manual setup	Manual setup	Compiler-driven
ML inference support	Native dot product	Manual encoding	Manual encoding	Via Concrete-ML
Post-quantum attestation	ML-DSA-65	None	None	None
ZKP integration	Built-in STARK	None	None	None
API model	One REST call	Library	Library	Compiler
Noise management	Automatic	Manual	Manual	Automatic
ARM NEON optimized	Yes (Graviton4)	x86 only	x86 focus	x86 focus
Encrypted activations	Sigmoid, ReLU, tanh, GELU	Manual polynomial	Manual polynomial	Compiler-generated

FAQ

Frequently Asked Questions

Common questions about CKKS, turbo mode, noise management, and encrypted ML inference.

What is CKKS and how does it differ from BFV?

CKKS supports approximate arithmetic on encrypted floating-point numbers. BFV is exact integer arithmetic. CKKS is ideal for ML inference, similarity search, and statistical analysis.

What does “turbo mode” mean?

N=8192 parameters for fast shallow circuits — roughly 2× faster. When computation depth is exhausted, H33-CKKS auto-bridges to N=16384 standard mode. Zero code changes needed.

What precision does CKKS provide?

40-bit mantissa precision in turbo mode. Sufficient for ML inference, cosine similarity, and most float operations. Not suitable for exact financial arithmetic — use BFV for that.

How does CKKS handle noise growth?

Each operation adds noise to ciphertexts. H33-CKKS automatically tracks noise budgets and applies rescaling. Bootstrapping resets noise for deeper circuits.

Can I run ML models on encrypted data?

Yes. H33-CKKS supports encrypted dot products, matrix multiplication, and activation function approximations. Encode model weights as plaintext, run inference on encrypted input.

What is the latency for an encrypted dot product?

Approximately 180µs for 128-dimensional vectors in turbo mode (N=8192). Standard mode (N=16384): approximately 420µs. Both include automatic rescaling.

Is CKKS post-quantum secure?

The lattice-based encryption is inherently quantum-resistant. H33 adds ML-DSA-65 attestation and STARK proofs. The full pipeline is post-quantum by construction.

Can I mix CKKS and BFV operations?

Not within a single ciphertext. Use FHE-IQ to automatically route workloads: floats to CKKS, integers to BFV. Same API call.

What vector sizes does CKKS support?

Turbo: up to 4096 slots per ciphertext. Standard: up to 8192 slots. For 128-dim biometric vectors: 32 users per ciphertext in turbo mode.

How do I choose between CKKS and BFV?

CKKS for: ML, similarity, statistics, any float math. BFV for: exact integer matching, biometric templates, database lookups. When unsure, use FHE-IQ auto-routing.

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