---
id: wiki-2026-0508-homomorphic-encryption
title: Homomorphic Encryption (HE)
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [HE, FHE, homomorphic encryption, fully homomorphic, CKKS, BFV, BGV, TFHE, privacy-preserving ML]
duplicate_of: none
source_trust_level: A
confidence_score: 0.93
verification_status: applied
tags: [cryptography, fhe, privacy, encryption, ml-privacy, secure-computation]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: C++ / Python
  framework: Microsoft SEAL / OpenFHE / Concrete / TFHE-rs
---

# Homomorphic Encryption

## 매 한 줄
> **"매 encrypted data 의 의 의 직접 의 compute"**. 매 plaintext 의 reveal X. 매 Gentry 2009 (FHE breakthrough). 매 modern: 매 CKKS (real / approximate), BFV/BGV (integer), TFHE (Boolean). 매 응용: 매 privacy-preserving ML, 매 cloud, 매 medical.

## 매 핵심

### 매 levels
- **Partial HE** (PHE): 매 single op (add or mul).
- **Somewhat HE** (SHE): 매 limited depth.
- **Leveled HE**: 매 known depth.
- **Fully HE** (FHE): 매 unlimited depth via bootstrapping.

### 매 schemes
- **CKKS**: 매 approximate arithmetic on real / complex (modern ML).
- **BFV / BGV**: 매 exact integer.
- **TFHE**: 매 fast bootstrapping, Boolean.
- **NTRU**, **GSW**: 매 historical / specialized.

### 매 modern library
- **Microsoft SEAL** (CKKS, BFV).
- **OpenFHE** (모든 schemes).
- **Concrete** (Zama, TFHE).
- **TFHE-rs** (Rust).
- **PySEAL**, **TenSEAL** (Python).

### 매 응용
1. **Privacy-preserving ML inference**.
2. **Medical** (encrypted genomic).
3. **Financial** (encrypted query).
4. **Cloud compute** (zero-knowledge).
5. **Federated learning** + HE.
6. **Smart contract** (FHEVM).

### 매 cost
- 매 100x-10000x slowdown vs plaintext.
- 매 modern hardware (GPU FHE) 의 의 의 reduce.
- 매 ciphertext 매 100-1000x larger.

## 💻 패턴

### TenSEAL (CKKS, Python)
```python
import tenseal as ts

context = ts.context(ts.SCHEME_TYPE.CKKS, poly_modulus_degree=8192,
                      coeff_mod_bit_sizes=[60, 40, 40, 60])
context.global_scale = 2**40
context.generate_galois_keys()

# 매 encrypt
v1 = ts.ckks_vector(context, [1.0, 2.0, 3.0])
v2 = ts.ckks_vector(context, [4.0, 5.0, 6.0])

# 매 compute on ciphertext
add = v1 + v2  # 매 [5, 7, 9]
mul = v1 * v2  # 매 [4, 10, 18]
dot = v1.dot(v2)  # 매 32

# 매 decrypt only when needed
print(add.decrypt())
```

### Microsoft SEAL (BFV, C++)
```cpp
#include "seal/seal.h"
using namespace seal;

EncryptionParameters parms(scheme_type::bfv);
parms.set_poly_modulus_degree(4096);
parms.set_coeff_modulus(CoeffModulus::BFVDefault(4096));
parms.set_plain_modulus(1024);

SEALContext context(parms);
KeyGenerator keygen(context);
PublicKey public_key; keygen.create_public_key(public_key);
SecretKey secret_key = keygen.secret_key();

Encryptor encryptor(context, public_key);
Evaluator evaluator(context);
Decryptor decryptor(context, secret_key);

Plaintext x_plain("5"); Ciphertext x_encrypted;
encryptor.encrypt(x_plain, x_encrypted);

Ciphertext result;
evaluator.square(x_encrypted, result);  // 매 5² = 25 (encrypted)
```

### Encrypted ML inference (logistic regression)
```python
def encrypted_logistic_regression(encrypted_x, weights_plaintext, bias_plaintext):
    """매 매 매 weights / bias 매 plaintext, 매 input 매 encrypted."""
    z = encrypted_x.dot(weights_plaintext) + bias_plaintext
    # 매 sigmoid approximation (poly degree 3)
    z2 = z * z
    z3 = z2 * z
    sigmoid_approx = 0.5 + 0.197 * z - 0.004 * z3  # 매 around 0
    return sigmoid_approx
```

### Sigmoid polynomial approx (CKKS)
```python
def sigmoid_poly_approx(z):
    """매 매 [-5, 5] 의 의 의 valid."""
    # 매 deg 3 Taylor-like
    z2 = z * z
    z3 = z2 * z
    z5 = z3 * z2
    return 0.5 + 0.197 * z - 0.004 * z3 + 0.0001 * z5
```

### Encrypted neural net (CKKS)
```python
def encrypted_dense_layer(enc_x, weights, bias):
    """매 매 layer 의 linear part 는 가능, 매 ReLU 는 polynomial approx."""
    z = enc_x.mm(weights) + bias  # 매 matrix-vector
    return polynomial_relu_approx(z)

def polynomial_relu_approx(z):
    """매 ReLU(x) ≈ a + b*x + c*x²."""
    z2 = z * z
    return 0.375 * z + 0.5 + 0.125 * z2  # 매 매 small range
```

### Bootstrapping (FHE refresh)
```python
# 매 SEAL CKKS 의 native bootstrap
# 매 OpenFHE
import openfhe
parameters = openfhe.CCParamsCKKSRNS()
parameters.SetMultiplicativeDepth(20)
cc = openfhe.GenCryptoContext(parameters)
cc.Enable(openfhe.PKESchemeFeature.PKE)
cc.Enable(openfhe.PKESchemeFeature.LEVELEDSHE)
cc.Enable(openfhe.PKESchemeFeature.ADVANCEDSHE)
cc.Enable(openfhe.PKESchemeFeature.FHE)

# 매 bootstrap
ct_refreshed = cc.EvalBootstrap(ct)
```

### TFHE (Boolean fast)
```rust
use tfhe::boolean::prelude::*;

let (client_key, server_key) = gen_keys();
let ct1 = client_key.encrypt(true);
let ct2 = client_key.encrypt(false);

// 매 server side
let result = server_key.and(&ct1, &ct2);

// 매 client side
let dec = client_key.decrypt(&result);  // 매 false
```

### Concrete-ML (Zama)
```python
from concrete.ml.sklearn import LogisticRegression

model = LogisticRegression(n_bits=8)
model.fit(X_train, y_train)

# 매 compile to FHE
model.compile(X_train)

# 매 encrypted inference
y_pred = model.predict(X_test_encrypted)
```

### Multi-party (HE + secret share)
```python
def hybrid_compute(encrypted_data_party_a, encrypted_data_party_b, secure_aggregator):
    # 매 매 party 매 encrypt own data
    # 매 aggregator 의 의 의 sum (HE)
    sum_encrypted = encrypted_data_party_a + encrypted_data_party_b
    # 매 only authorized 매 decrypt
    return sum_encrypted
```

### Performance benchmark
```python
import time
def benchmark_he(scheme='ckks'):
    setup_t = time.perf_counter()
    context = create_context(scheme)
    keys = setup_keys(context)
    setup_t = time.perf_counter() - setup_t
    
    enc_t = time.perf_counter()
    ct = encrypt([1, 2, 3], context, keys.pub)
    enc_t = time.perf_counter() - enc_t
    
    op_t = time.perf_counter()
    ct2 = ct * ct
    op_t = time.perf_counter() - op_t
    
    return {'setup': setup_t, 'encrypt': enc_t, 'op': op_t}
```

### Threshold HE (multi-key)
```python
# 매 multi-party with shared secret
# 매 매 party 매 partial decryption share
def threshold_decrypt(ciphertext, partial_shares, threshold):
    if len(partial_shares) >= threshold:
        return combine_shares(partial_shares, ciphertext)
    raise ThresholdNotMet()
```

### FHEVM (smart contract on encrypted)
```solidity
// 매 fhEVM (Zama)
import "fhevm/lib/TFHE.sol";

contract EncryptedAuction {
    euint64 highestBid;  // 매 encrypted
    
    function bid(bytes calldata encryptedAmount) public {
        euint64 amount = TFHE.asEuint64(encryptedAmount);
        highestBid = TFHE.max(highestBid, amount);
    }
}
```

## 매 결정 기준
| 상황 | Scheme |
|---|---|
| ML inference (real) | CKKS |
| Exact integer | BFV / BGV |
| Boolean / lookup | TFHE |
| Smart contract | FHEVM (TFHE-based) |
| Multi-party | Threshold HE |
| Best library | OpenFHE / SEAL / Concrete |

**기본값**: 매 ML = CKKS + 매 polynomial activation. 매 boolean = TFHE. 매 always benchmark before commit (100-1000x cost).

## 🔗 Graph
- 부모: [[Practical-Cryptography|Cryptography]]
- 응용: [[Privacy-Preserving-ML]] · [[Federated-Learning]]
- Adjacent: [[Differential-Privacy]] · [[Secure Multi-party Computation]]

## 🤖 LLM 활용
**언제**: 매 medical / financial. 매 cloud trust. 매 multi-party compute.
**언제 X**: 매 latency-critical. 매 plaintext 의 가능.

## ❌ 안티패턴
- **FHE for everything**: 매 cost overkill.
- **No polynomial approx of activation**: 매 deep ReLU 의 hard.
- **Wrong scheme**: 매 BFV for real-valued ML.
- **No bootstrap when deep**: 매 noise overflow.

## 🧪 검증 / 중복
- Verified (Gentry 2009, CKKS 2017, OpenFHE / SEAL / Concrete docs).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-04-26 | Auto |
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — schemes + 매 TenSEAL / SEAL / TFHE / Concrete / FHEVM code |