# WIA-SEC-012: Homomorphic Encryption Standard
## PHASE 1 - CORE SPECIFICATION

**Standard ID:** WIA-SEC-012
**Version:** 1.0.0
**Status:** Draft
**Last Updated:** 2025-12-25
**Category:** Security (SEC)
**Emoji:** 🔐

---

## 1. INTRODUCTION

### 1.1 Purpose

The WIA-SEC-012 Homomorphic Encryption Standard defines a comprehensive framework for privacy-preserving computation on encrypted data. This standard enables organizations to perform computations on sensitive data without ever decrypting it, maintaining complete data confidentiality throughout the computational process.

### 1.2 Philosophy

**弘益人間 (홍익인간) - Benefit All Humanity**

This standard embodies the principle of benefiting all humanity by providing cryptographic tools that protect individual privacy while enabling collaborative computation and data sharing for the common good.

### 1.3 Scope

This specification covers:

- Fully Homomorphic Encryption (FHE) schemes
- Partially Homomorphic Encryption (PHE) schemes
- Secure Multi-Party Computation protocols
- Privacy-preserving machine learning
- Key management and distribution
- Performance optimization techniques
- Integration with existing systems

### 1.4 Target Audience

- Cryptography engineers
- Security architects
- Privacy officers
- Data scientists
- Cloud service providers
- Healthcare and financial institutions

---

## 2. TERMINOLOGY

### 2.1 Core Concepts

**Homomorphic Encryption (HE)**
A form of encryption that allows computation on ciphertexts, generating an encrypted result which, when decrypted, matches the result of operations performed on the plaintext.

**Fully Homomorphic Encryption (FHE)**
Encryption scheme that supports arbitrary computation (both addition and multiplication) on encrypted data without limitation.

**Partially Homomorphic Encryption (PHE)**
Encryption scheme that supports only specific operations (e.g., only addition or only multiplication) on encrypted data.

**Ciphertext**
Data in its encrypted form, unreadable without the decryption key.

**Plaintext**
Original, unencrypted data.

**Noise Budget**
The amount of computational operations that can be performed before the ciphertext becomes too noisy to decrypt correctly.

**Bootstrapping**
The process of refreshing the noise budget in FHE to enable unlimited computation depth.

---

## 3. ENCRYPTION SCHEMES

### 3.1 Fully Homomorphic Encryption (FHE)

#### 3.1.1 BFV (Brakerski-Fan-Vercauteren)

**Purpose:** Integer arithmetic with exact results

**Parameters:**
```
Polynomial Modulus Degree (n): 4096, 8192, 16384, 32768
Coefficient Modulus (q): Multi-prime chain
Plaintext Modulus (t): 1024, 2048, 4096
Security Level: 128-bit, 192-bit, 256-bit
```

**Supported Operations:**
- Addition: Enc(a) + Enc(b) = Enc(a + b)
- Multiplication: Enc(a) × Enc(b) = Enc(a × b)
- Scalar multiplication: k × Enc(a) = Enc(k × a)
- Rotation: Rotate(Enc([a₁, a₂, ...]), k)

**Use Cases:**
- Financial calculations
- Statistical analysis
- Database queries on encrypted data
- Voting systems

#### 3.1.2 CKKS (Cheon-Kim-Kim-Song)

**Purpose:** Approximate arithmetic on real/complex numbers

**Parameters:**
```
Polynomial Modulus Degree (n): 4096, 8192, 16384, 32768
Coefficient Modulus: Multi-level chain for rescaling
Scale: 2^40, 2^50, 2^60
Precision: 30-50 bits
Security Level: 128-bit, 192-bit, 256-bit
```

**Supported Operations:**
- Approximate addition: Enc(a) + Enc(b) ≈ Enc(a + b)
- Approximate multiplication: Enc(a) × Enc(b) ≈ Enc(a × b)
- Rotation and slot operations
- Rescaling for multiplicative depth management

**Use Cases:**
- Machine learning inference
- Signal processing
- Scientific computing
- Real-time analytics

### 3.2 Partially Homomorphic Encryption (PHE)

#### 3.2.1 Paillier Cryptosystem

**Purpose:** Additive homomorphism for secure aggregation

**Key Size:** 2048, 3072, 4096 bits

**Operations:**
- Enc(a) × Enc(b) = Enc(a + b)
- Enc(a)^k = Enc(k × a)

**Use Cases:**
- Secure voting and auctions
- Privacy-preserving data aggregation
- Financial portfolio analysis

#### 3.2.2 RSA Homomorphic Properties

**Purpose:** Multiplicative homomorphism

**Key Size:** 2048, 3072, 4096 bits

**Operations:**
- Enc(a) × Enc(b) = Enc(a × b)

**Use Cases:**
- Digital signatures
- Zero-knowledge proofs
- Secure product calculations

#### 3.2.3 ElGamal Encryption

**Purpose:** Multiplicative homomorphism with semantic security

**Operations:**
- Enc(a) × Enc(b) = Enc(a × b)

**Use Cases:**
- Threshold cryptography
- Mix networks
- Anonymous credentials

---

## 4. KEY MANAGEMENT

### 4.1 Key Generation

#### 4.1.1 Key Types

**Public Key (pk)**
- Used for encryption
- Can be distributed freely
- Size: Dependent on security parameter

**Secret Key (sk)**
- Used for decryption
- Must be kept private
- Requires secure storage

**Relinearization Keys (rlk)**
- Used to reduce ciphertext size after multiplication
- Public component
- Generated from secret key

**Galois Keys (gk)**
- Enable rotation operations
- Public component
- Generated for specific rotation amounts

#### 4.1.2 Key Generation Parameters

```typescript
interface KeyGenParams {
  scheme: 'BFV' | 'CKKS' | 'Paillier' | 'RSA';
  polyModulusDegree: 4096 | 8192 | 16384 | 32768;
  coefficientModulus: number[];
  plainModulus?: number;  // For BFV
  scale?: number;         // For CKKS
  securityLevel: 128 | 192 | 256;
}
```

### 4.2 Key Distribution

#### 4.2.1 Public Key Distribution

**Methods:**
- Certificate-based PKI
- Direct distribution via secure channels
- Blockchain-based public key infrastructure
- WIA-SEC-012 compliant key servers

**Format:**
```json
{
  "standardId": "WIA-SEC-012",
  "version": "1.0.0",
  "keyType": "public",
  "scheme": "BFV",
  "parameters": {
    "polyModulusDegree": 8192,
    "coefficientModulus": [60, 40, 40, 60],
    "plainModulus": 1024
  },
  "publicKey": "base64-encoded-key-data",
  "createdAt": "2025-12-25T00:00:00Z",
  "expiresAt": "2026-12-25T00:00:00Z",
  "issuer": "did:wia:issuer-id"
}
```

#### 4.2.2 Secret Key Storage

**Requirements:**
- Hardware Security Module (HSM) storage preferred
- Encrypted at rest using AES-256-GCM
- Multi-factor authentication for access
- Audit logging for all access attempts
- Regular key rotation schedule

**Storage Format:**
```json
{
  "encryptedSecretKey": "base64-encrypted-key",
  "keyDerivationFunction": "PBKDF2",
  "iterations": 100000,
  "salt": "base64-salt",
  "version": "1.0.0"
}
```

### 4.3 Key Rotation

**Rotation Period:** Recommended every 90-180 days

**Process:**
1. Generate new key pair
2. Re-encrypt all stored ciphertexts with new public key
3. Update all distributed public keys
4. Securely destroy old secret key
5. Update audit logs

---

## 5. ENCRYPTION AND DECRYPTION

### 5.1 Encryption Process

#### 5.1.1 Data Preparation

**Input Validation:**
- Verify data type compatibility
- Check data size limits
- Validate encoding scheme

**Encoding:**
```typescript
interface EncodingParams {
  dataType: 'integer' | 'float' | 'complex' | 'vector' | 'matrix';
  encoding: 'polynomial' | 'batch' | 'simd';
  scale?: number;  // For CKKS
}

function encode(
  data: number | number[] | number[][],
  params: EncodingParams
): Plaintext {
  // Encode data into polynomial representation
  // Return plaintext object ready for encryption
}
```

#### 5.1.2 Encryption Algorithm

**BFV Encryption:**
```
Input: plaintext m, public key (pk₀, pk₁)
Output: ciphertext (c₀, c₁)

1. Sample error polynomial e from error distribution
2. Sample random polynomial r
3. c₀ ← pk₀ × r + e + m
4. c₁ ← pk₁ × r
5. Return (c₀, c₁)
```

**CKKS Encryption:**
```
Input: plaintext vector m, public key pk, scale Δ
Output: ciphertext ct

1. Encode m into polynomial using CKKS encoding
2. Scale by Δ and round to nearest integer
3. Add encryption noise
4. Return ciphertext
```

### 5.2 Decryption Process

#### 5.2.1 Decryption Algorithm

**BFV Decryption:**
```
Input: ciphertext (c₀, c₁), secret key sk
Output: plaintext m

1. Compute intermediate = c₀ + c₁ × sk
2. Reduce modulo plaintext modulus
3. Return decoded plaintext m
```

**CKKS Decryption:**
```
Input: ciphertext ct, secret key sk
Output: approximate plaintext m

1. Compute polynomial result using secret key
2. Decode using CKKS decoding
3. Return approximate result within precision
```

#### 5.2.2 Noise Management

**Noise Budget Tracking:**
```typescript
interface NoiseBudget {
  initial: number;      // bits
  current: number;      // bits
  threshold: number;    // minimum safe bits
  operations: number;   // operations performed
}

function checkNoiseBudget(ciphertext: Ciphertext): NoiseBudget {
  // Calculate remaining noise budget
  // Return current noise status
}
```

**Bootstrapping Trigger:**
- Automatic when noise budget < threshold
- Manual trigger available
- Performance overhead: ~100-1000x slower than regular operations

---

## 6. HOMOMORPHIC OPERATIONS

### 6.1 Addition

**Operation:** `Enc(a) + Enc(b) = Enc(a + b)`

**Noise Growth:** Minimal (logarithmic)

**Implementation:**
```typescript
function add(
  ciphertext1: Ciphertext,
  ciphertext2: Ciphertext
): Ciphertext {
  // Component-wise polynomial addition
  // Return result ciphertext
}
```

**Cost:** O(n) where n is polynomial degree

### 6.2 Multiplication

**Operation:** `Enc(a) × Enc(b) = Enc(a × b)`

**Noise Growth:** Significant (exponential)

**Implementation:**
```typescript
function multiply(
  ciphertext1: Ciphertext,
  ciphertext2: Ciphertext,
  relinKeys: RelinearizationKeys
): Ciphertext {
  // Polynomial multiplication
  // Relinearization to reduce size
  // Return result ciphertext
}
```

**Cost:** O(n log n) with NTT optimization

**Note:** Requires relinearization to maintain ciphertext structure

### 6.3 Scalar Operations

**Scalar Addition:** `Enc(a) + k = Enc(a + k)`

**Scalar Multiplication:** `k × Enc(a) = Enc(k × a)`

**Implementation:**
```typescript
function scalarMultiply(
  ciphertext: Ciphertext,
  scalar: number
): Ciphertext {
  // Multiply each component by scalar
  // Return result ciphertext
}
```

**Cost:** O(n)

**Note:** No noise growth increase

### 6.4 Rotation

**Purpose:** Cyclic rotation of encrypted vectors

**Operation:** `Rotate(Enc([a₁, a₂, ..., aₙ]), k) = Enc([aₖ₊₁, ..., aₙ, a₁, ..., aₖ])`

**Implementation:**
```typescript
function rotate(
  ciphertext: Ciphertext,
  steps: number,
  galoisKeys: GaloisKeys
): Ciphertext {
  // Apply Galois automorphism
  // Return rotated ciphertext
}
```

**Requirements:**
- Pre-generated Galois keys for rotation amount
- Noise growth similar to multiplication

### 6.5 Comparison Operations

**Challenge:** Direct comparison not supported in standard HE

**Solutions:**

**Approximate Comparison (CKKS):**
```typescript
function approximateCompare(
  ciphertext1: Ciphertext,
  ciphertext2: Ciphertext,
  iterations: number = 3
): Ciphertext {
  // Use polynomial approximation of sign function
  // Return encrypted comparison result (≈0 or ≈1)
}
```

**Secure Comparison Protocol:**
- Use secure multi-party computation
- Garbled circuits for exact comparison
- Zero-knowledge proofs for specific comparisons

---

## 7. PERFORMANCE OPTIMIZATION

### 7.1 Batching (SIMD)

**Concept:** Encrypt multiple values in a single ciphertext

**Capacity:**
- BFV: n/2 slots (e.g., 4096 integers in one ciphertext)
- CKKS: n/2 slots (e.g., 4096 complex numbers)

**Benefits:**
- Reduced encryption/decryption overhead
- Parallel computation on vectors
- Improved throughput

**Implementation:**
```typescript
function batchEncode(
  values: number[],
  params: EncodingParams
): Plaintext {
  // Pack values into polynomial slots
  // Return plaintext with batched data
}
```

### 7.2 Parallelization

**Thread-Level Parallelism:**
- Parallel polynomial operations
- Independent ciphertext operations
- Multi-threaded key generation

**Data Parallelism:**
- Batch processing of multiple ciphertexts
- GPU acceleration for polynomial arithmetic
- SIMD instructions for vector operations

**Recommended Configuration:**
```typescript
interface ParallelConfig {
  threads: number;           // CPU threads
  useGPU: boolean;          // GPU acceleration
  batchSize: number;        // Ciphertexts per batch
  pipelineDepth: number;    // Operation pipeline
}
```

### 7.3 Caching Strategies

**Key Material Caching:**
- Relinearization keys
- Galois keys
- NTT tables

**Intermediate Results:**
- Common subexpression results
- Frequently used encrypted constants
- Pre-computed rotation results

**Cache Configuration:**
```typescript
interface CacheConfig {
  maxSize: number;          // MB
  ttl: number;              // seconds
  evictionPolicy: 'LRU' | 'LFU' | 'FIFO';
  warmupKeys: string[];     // Pre-load keys
}
```

### 7.4 Number Theoretic Transform (NTT)

**Purpose:** Fast polynomial multiplication

**Complexity Reduction:** O(n²) → O(n log n)

**Implementation Requirements:**
- Special prime moduli
- Pre-computed roots of unity
- Butterfly network computation

**Performance Gain:** 100-1000x faster polynomial operations

---

## 8. SECURITY CONSIDERATIONS

### 8.1 Security Parameters

**Minimum Requirements:**

| Security Level | Poly Degree | Coeff Modulus (bits) | Estimated Security |
|---------------|-------------|----------------------|-------------------|
| Low           | 4096        | 109                  | ~128 bits         |
| Medium        | 8192        | 218                  | ~128 bits         |
| High          | 16384       | 438                  | ~192 bits         |
| Very High     | 32768       | 881                  | ~256 bits         |

**Security Model:** Ring-Learning with Errors (RLWE) hardness assumption

### 8.2 Side-Channel Protection

**Timing Attacks:**
- Constant-time polynomial operations
- Avoid data-dependent branches
- Uniform memory access patterns

**Power Analysis:**
- Hardware countermeasures in HSM
- Randomized computation order
- Noise injection techniques

**Cache Attacks:**
- Cache-oblivious algorithms
- Memory access randomization
- Secure memory allocation

### 8.3 Quantum Resistance

**Status:** Post-quantum secure under RLWE assumption

**NIST PQC Alignment:**
- Similar to CRYSTALS-Kyber (lattice-based)
- Parameters selected for quantum resistance
- Regular security assessment updates

### 8.4 Access Control

**Requirements:**
- Role-based access control (RBAC)
- Attribute-based access control (ABAC)
- Multi-factor authentication
- Audit logging

**Example Policy:**
```json
{
  "policy": {
    "resource": "encrypted-data",
    "actions": ["encrypt", "compute", "decrypt"],
    "conditions": {
      "roles": ["data-scientist", "admin"],
      "mfa": true,
      "ipWhitelist": ["10.0.0.0/8"],
      "timeWindow": "business-hours"
    }
  }
}
```

---

## 9. COMPLIANCE AND STANDARDS

### 9.1 Regulatory Compliance

**GDPR (General Data Protection Regulation):**
- Privacy by design through encryption
- Data minimization via homomorphic computation
- Right to be forgotten compatibility

**HIPAA (Health Insurance Portability and Accountability Act):**
- Protected health information (PHI) encryption
- Secure computation on medical data
- Audit trail requirements

**PCI DSS (Payment Card Industry Data Security Standard):**
- Cardholder data encryption
- Secure payment processing
- Key management requirements

### 9.2 Standards Alignment

**ISO/IEC 18033:** Encryption algorithms
**ISO/IEC 19790:** Security requirements for cryptographic modules
**NIST SP 800-57:** Key management recommendations
**FIPS 140-3:** Cryptographic module validation

### 9.3 WIA Standards Integration

**Related WIA Standards:**
- WIA-SEC-001: Foundation Security Framework
- WIA-SEC-005: Zero-Knowledge Proofs
- WIA-SEC-008: Blockchain Security
- WIA-IDENTITY-002: Decentralized Identity

**Integration Points:**
- Encrypted credential storage (WIA-IDENTITY)
- Zero-knowledge verification (WIA-SEC-005)
- Blockchain data privacy (WIA-SEC-008)

---

## 10. TESTING AND VALIDATION

### 10.1 Functional Testing

**Correctness Tests:**
```typescript
describe('Homomorphic Operations', () => {
  test('Addition correctness', async () => {
    const a = 15, b = 7;
    const encA = await encrypt(a, publicKey);
    const encB = await encrypt(b, publicKey);
    const encSum = await add(encA, encB);
    const result = await decrypt(encSum, secretKey);
    expect(result).toBe(a + b);
  });

  test('Multiplication correctness', async () => {
    const a = 15, b = 7;
    const encA = await encrypt(a, publicKey);
    const encB = await encrypt(b, publicKey);
    const encProduct = await multiply(encA, encB, relinKeys);
    const result = await decrypt(encProduct, secretKey);
    expect(result).toBe(a * b);
  });
});
```

### 10.2 Performance Benchmarks

**Key Metrics:**
- Encryption throughput (ops/sec)
- Decryption throughput (ops/sec)
- Homomorphic operation latency
- Memory consumption
- Noise budget consumption rate

**Benchmark Suite:**
```typescript
interface BenchmarkResults {
  operation: string;
  iterations: number;
  avgTime: number;      // milliseconds
  throughput: number;   // ops/sec
  memoryPeak: number;   // MB
}
```

### 10.3 Security Testing

**Required Tests:**
- Known-plaintext attack resistance
- Chosen-ciphertext attack resistance
- Side-channel attack resistance
- Parameter validation
- Key strength verification

**Security Audit:**
- Third-party cryptographic review
- Penetration testing
- Formal verification (optional)

---

## 11. REFERENCE IMPLEMENTATION

### 11.1 Core Interface

```typescript
interface WIAHomomorphicEncryption {
  // Key Management
  generateKeys(params: KeyGenParams): Promise<KeyPair>;
  exportPublicKey(format: 'pem' | 'json'): string;
  importPublicKey(key: string): PublicKey;

  // Encryption/Decryption
  encrypt(data: number | number[], publicKey: PublicKey): Promise<Ciphertext>;
  decrypt(ciphertext: Ciphertext, secretKey: SecretKey): Promise<number | number[]>;

  // Homomorphic Operations
  add(ct1: Ciphertext, ct2: Ciphertext): Promise<Ciphertext>;
  multiply(ct1: Ciphertext, ct2: Ciphertext): Promise<Ciphertext>;
  scalarAdd(ct: Ciphertext, scalar: number): Promise<Ciphertext>;
  scalarMultiply(ct: Ciphertext, scalar: number): Promise<Ciphertext>;
  rotate(ct: Ciphertext, steps: number): Promise<Ciphertext>;

  // Utility
  getNoiseBudget(ciphertext: Ciphertext): number;
  bootstrap(ciphertext: Ciphertext): Promise<Ciphertext>;
}
```

### 11.2 Example Usage

```typescript
import { WIAHomomorphic } from '@wia/sec-012';

// Initialize
const client = new WIAHomomorphic({
  scheme: 'BFV',
  polyModulusDegree: 8192,
  plainModulus: 1024,
  securityLevel: 128
});

// Generate keys
const { publicKey, secretKey, relinKeys, galoisKeys } =
  await client.generateKeys();

// Encrypt data
const plaintext1 = 42;
const plaintext2 = 58;
const ciphertext1 = await client.encrypt(plaintext1, publicKey);
const ciphertext2 = await client.encrypt(plaintext2, publicKey);

// Perform homomorphic operations
const sum = await client.add(ciphertext1, ciphertext2);
const product = await client.multiply(ciphertext1, ciphertext2);

// Decrypt results
const sumResult = await client.decrypt(sum, secretKey);
console.log(`Sum: ${sumResult}`); // 100

const productResult = await client.decrypt(product, secretKey);
console.log(`Product: ${productResult}`); // 2436
```

---

## 12. NEXT STEPS

Continue to **PHASE-2-&-3-&-4.md** for:
- Advanced use cases and applications
- Secure multi-party computation protocols
- Privacy-preserving machine learning
- Cloud deployment architectures
- Integration patterns and best practices

---

**Document Control**

| Version | Date       | Author          | Changes                    |
|---------|------------|-----------------|----------------------------|
| 1.0.0   | 2025-12-25 | WIA Standards   | Initial PHASE 1 release    |

---

弘益人間 (홍익인간) · Benefit All Humanity

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