Homomorphic Encryption: Computing on Encrypted Data 2026

class BasicHomomorphic:
    """
    Simple examples of homomorphic properties.
    """
    
    def paillier_addition(self, c1, c2, n, g):
        """
        Paillier: Homomorphic addition.
        Enc(E(m1) * Enc(m2)) = Enc(m1 + m2)
        """
        # Given two ciphertexts
        # c1 = g^m1 * r1^n mod n^2
        # c2 = g^m2 * r2^n mod n^2
        
        # Multiply ciphertexts
        result = (c1 * c2) % (n ** 2)
        
        # Decrypting gives m1 + m2
        return result
    
    def multiplicative_homomorphism(self, m1, m2, n, e):
        """
        RSA: Homomorphic multiplication.
        Enc(m1) * Enc(m2) = Enc(m1 * m2)
        """
        # RSA encryption: c = m^e mod n
        c1 = pow(m1, e, n)
        c2 = pow(m2, e, n)
        
        # Multiply ciphertexts
        c_result = (c1 * c2) % n
        
        # Decrypt: m_result = c_result^d mod n
        # = (m1^e * m2^e)^d = (m1 * m2)^ed = m1 * m2 mod n
        return c_result

class BFVScheme:
    """
    Brakerski-Fan-Vercauteren scheme.
    """
    
    def __init__(self, poly_degree=4096, plaintext_modulus=256):
        self.n = poly_degree  # Ring dimension
        self.t = plaintext_modulus
        self.q = 2**40  # Ciphertext modulus
    
    def key_generation(self):
        """
        Generate FHE keys.
        """
        # Secret key (small polynomial)
        sk = self.generate_secret_key()
        
        # Public key
        pk = self.generate_public_key(sk)
        
        # Evaluation keys for relinearization
        evk = self.generate_evaluation_key(sk)
        
        return {'sk': sk, 'pk': pk, 'evk': evk}
    
    def encode(self, message):
        """
        Encode plaintext into polynomial.
        """
        # Map integers to polynomial coefficients
        poly = self.polynomial_from_ints(message)
        return poly
    
    def encrypt(self, plaintext, public_key):
        """
        Encrypt plaintext.
        """
        # Sample random polynomial
        u = self.sample_random()
        
        # Sample error
        e = self.sample_error()
        
        # c0 = pk[0] * u + e
        # c1 = pk[1] * u + e
        c0 = public_key[0] * u + e
        c1 = public_key[1] * u + e
        
        return (c0, c1)
    
    def add(self, ct1, ct2):
        """
        Homomorphic addition.
        """
        return (ct1[0] + ct2[0], ct1[1] + ct2[1])
    
    def multiply(self, ct1, ct2, eval_key):
        """
        Homomorphic multiplication (with relinearization).
        """
        # Raw multiplication
        ct_raw = (ct1[0] * ct2[0], 
                  ct1[0] * ct2[1] + ct1[1] * ct2[0],
                  ct1[1] * ct2[1])
        
        # Relinearize using evaluation key
        ct = self.relinearize(ct_raw, eval_key)
        
        return ct

class CKKSScheme:
    """
    Cheon-Kim-Kim-Song scheme for approximate arithmetic.
    Ideal for ML inference.
    """
    
    def __init__(self, poly_degree=16384, scale=2**40):
        self.n = poly_degree
        self.scale = scale
    
    def encode_vector(self, values):
        """
        Encode vector of complex/real numbers.
        """
        # Scale and round to integers
        scaled = [int(v * self.scale) for v in values]
        
        # Interpolate polynomial through points
        poly = self.lagrange_interpolate(scaled)
        
        return poly
    
    def decrypt_and_decode(self, ciphertext, secret_key):
        """
        Decrypt and rescale.
        """
        poly = self.decrypt(ciphertext, secret_key)
        
        # Extract values
        values = [p / self.scale for p in poly.coeffs]
        
        return values

// Microsoft SEAL FHE implementation
#include "seal/seal.h"

using namespace seal;

int main() {
    // Parameters
    EncryptionParameters parms(scheme_type::bfv);
    parms.set_poly_modulus_degree(4096);
    parms.set_coeff_modulus(CoeffModulus::BFVDefault(4096));
    parms.set_plain_modulus(256);
    
    SEALContext context(parms);
    
    // Key generation
    KeyGenerator keygen(context);
    SecretKey secret_key = keygen.secret_key();
    PublicKey public_key;
    keygen.create_public_key(public_key);
    
    // Encryptor/Decryptor
    Encryptor encryptor(context, public_key);
    Decryptor decryptor(context, secret_key);
    
    // Encoder
    BatchEncoder encoder(context);
    
    // Encode plaintext
    vector<uint64_t> plaintext_data = {1, 2, 3, 4};
    Plaintext plaintext;
    encoder.encode(plaintext_data, plaintext);
    
    // Encrypt
    Ciphertext ciphertext;
    encryptor.encrypt(plaintext, ciphertext);
    
    // Homomorphic operations
    Evaluator evaluator(context);
    
    // Square
    Ciphertext ciphertext_sq;
    evaluator.square(ciphertext, ciphertext_sq);
    
    // Add
    Ciphertext ciphertext_add;
    evaluator.add(ciphertext, ciphertext_sq, ciphertext_add);
    
    // Decrypt and decode
    Plaintext result;
    decryptor.decrypt(ciphertext_add, result);
    
    vector<uint64_t> result_data;
    encoder.decode(result, result_data);
    
    return 0;
}

// PALISADE FHE
#include "palisade.h"

using namespace lbcrypto;

int main() {
    // Set parameters
    CryptoContext<DCRTPoly> cc = GenCryptoContext(
        scheme::BFV,
        4096,  // poly degree
        256,   // plaintext modulus
        1024   // ciphertext modulus
    );
    
    // Enable operations
    cc->Enable(ENCRYPTION);
    cc->Enable(ADD);
    cc->Enable(MULTIPLY);
    
    // Key generation
    auto keyPair = cc->KeyGen();
    
    // Encryption
    Plaintext pt1 = cc->MakePackedPlaintext({1, 2, 3, 4});
    Ciphertext ct1 = cc->Encrypt(keyPair.publicKey, pt1);
    
    Plaintext pt2 = cc->MakePackedPlaintext({5, 6, 7, 8});
    Ciphertext ct2 = cc->Encrypt(keyPair.publicKey, pt2);
    
    // Homomorphic addition
    auto ctResult = cc->Add(ct1, ct2);
    
    // Decryption
    Plaintext result;
    cc->Decrypt(keyPair.secretKey, ctResult, &result);
    
    return 0;
}

import tenseal as ts

class TenSEALExample:
    """
    TenSEAL: Python FHE library.
    """
    
    def context_create(self):
        """
        Create TenSEAL context.
        """
        # BFV scheme
        context = ts.context(
            ts.SCHEME.BFV,
            poly_modulus_degree=4096,
            plain_modulus=256
        )
        
        # Generate keys
        context.generate_galois_keys()
        
        return context
    
    def encrypt_vector(self, context, vector):
        """
        Encrypt vector.
        """
        # Create encrypted tensor
        enc_vector = ts.bfv_vector(context, vector)
        
        return enc_vector
    
    def compute(self, enc_vector):
        """
        Compute on encrypted data.
        """
        # Addition
        result = enc_vector + 5
        
        # Multiplication
        result = result * 3
        
        # Polynomial
        result = result + result * result
        
        return result
    
    def decrypt(self, enc_vector):
        """
        Decrypt result.
        """
        return enc_vector.decrypt()

class PrivacyPreservingML:
    """
    FHE for ML inference on encrypted data.
    """
    
    def __init__(self, model):
        self.model = model
    
    def encrypt_model(self, fhe_context):
        """
        Encrypt model weights.
        """
        encrypted_weights = []
        
        for layer in self.model.layers:
            encrypted_layer = []
            for weight in layer.get_weights():
                # Encrypt each weight
                enc_weight = fhe_context.encrypt(weight)
                encrypted_layer.append(enc_weight)
            
            encrypted_weights.append(encrypted_layer)
        
        return encrypted_weights
    
    def predict_on_encrypted(self, encrypted_input, encrypted_model):
        """
        Run inference on encrypted data with encrypted model.
        """
        current = encrypted_input
        
        for layer_weights in encrypted_model:
            # Encrypted matrix multiplication
            current = self.encrypted_matmul(current, layer_weights)
            
            # Encrypted activation
            current = self.encrypted_relu(current)
        
        return current
    
    def use_case(self):
        """
        Real-world use cases.
        """
        return {
            'healthcare': 'Diagnose patients without seeing their data',
            'finance': 'Credit scoring without exposing financials',
            'ad_targeting': 'Target ads without profiling users'
        }

class SecureAggregation:
    """
    FHE for secure aggregation.
    """
    
    def __init__(self, fhe_context):
        self.context = fhe_context
    
    def sum_encrypted_values(self, encrypted_values):
        """
        Sum encrypted values without decryption.
        """
        total = encrypted_values[0]
        
        for val in encrypted_values[1:]:
            total = total + val
        
        return total
    
    def average_encrypted(self, encrypted_values):
        """
        Compute average of encrypted values.
        """
        total = self.sum_encrypted_values(encrypted_values)
        count = len(encrypted_values)
        
        # Divide by count
        avg = total / count
        
        return avg
    
    def federated_learning_aggregation(self, client_updates):
        """
        Aggregate model updates in federated learning.
        """
        # Each client sends encrypted update
        # Server aggregates without seeing individual updates
        aggregated = self.sum_encrypted_values(client_updates)
        
        # Return average to clients
        return aggregated / len(client_updates)

class EncryptedDatabase:
    """
    Query encrypted database.
    """
    
    def __init__(self, fhe_context, data):
        self.context = fhe_context
        # Data is stored encrypted
        self.encrypted_data = self.encrypt_database(data)
    
    def encrypt_database(self, data):
        """
        Encrypt entire database.
        """
        return {k: self.context.encrypt(v) for k, v in data.items()}
    
    def encrypted_sum_query(self, column):
        """
        SUM(column) on encrypted data.
        """
        values = [row[column] for row in self.encrypted_data]
        
        # Sum encrypted values
        total = values[0]
        for v in values[1:]:
            total = total + v
        
        return total.decrypt()
    
    def encrypted_average_query(self, column):
        """
        AVG(column) on encrypted data.
        """
        total = self.encrypted_sum_query(column)
        count = len(self.encrypted_data)
        
        return total / count

class FHEOptimizations:
    """
    Techniques to improve FHE performance.
    """
    
    def batching(self):
        """
        Single Instruction Multiple Data (SIMD) via CRT.
        """
        return {
            'technique': 'Pack multiple values into one ciphertext',
            'speedup': 'Up to n times (n = poly degree)',
            'example': '4096 values in one ciphertext'
        }
    
    def noise_budget(self):
        """
        Manage noise growth.
        """
        return {
            'key': 'Balance modulus size and depth',
            'technique': 'Refresh (re-encryption) periodically',
            'bootstrapping': 'Reset noise in FHEW/TFHE'
        }
    
    def hardware_acceleration(self):
        """
        Specialized hardware.
        """
        return {
            'gpu': 'NVIDIA GPU acceleration',
            'fpga': 'FPGA implementations',
            'asic': 'Dedicated FHE chips',
            'examples': 'Intel FHE accelerator, fabriccrypt'
        }

class HealthcareFHE:
    """
    FHE for privacy-preserving healthcare.
    """
    
    def __init__(self):
        self.fhe = BFVScheme()
    
    def train_on_encrypted(self, encrypted_patient_records):
        """
        Train model without seeing patient data.
        """
        # Model trains on encrypted records
        # Never decrypts individual patient data
        model = train_model(encrypted_patient_records)
        
        return model
    
    def predict_on_encrypted(self, encrypted_symptoms, encrypted_model):
        """
        Diagnose without decrypting symptoms.
        """
        # Symptoms never decrypted
        diagnosis = predict(encrypted_symptoms, encrypted_model)
        
        # Only patient sees diagnosis
        return diagnosis
    
    def collaborative_research(self, hospital_data_list):
        """
        Multiple hospitals collaborate without sharing data.
        """
        # Each hospital encrypts their data locally
        encrypted_hospitals = [
            hospital.encrypt_local() for hospital in hospital_data_list
        ]
        
        # Aggregate insights from encrypted data
        insights = self.aggregate_insights(encrypted_hospitals)
        
        return insights