2021SC@SDUSC
2021-12-27
前言
本篇开始我将讨论SEAL中rotation.cpp的代码实现。
简介
BFV方案(使用BatchEncoder)和CKKS方案都支持对加密数字进行native计算。除了slot-wise计算外，还可以循环地旋转加密向量。
代码分析
void example_rotation_bfv()
先是和前文类似的设置一些参数。
print_example_banner("Example: Rotation / Rotation in BFV");
EncryptionParameters parms(scheme_type::bfv);
size_t poly_modulus_degree = 8192;
parms.set_poly_modulus_degree(poly_modulus_degree);
parms.set_coeff_modulus(CoeffModulus::BFVDefault(poly_modulus_degree));
parms.set_plain_modulus(PlainModulus::Batching(poly_modulus_degree, 20));
SEALContext context(parms);
print_parameters(context);
cout << endl;
KeyGenerator keygen(context);
SecretKey secret_key = keygen.secret_key();
PublicKey public_key;
keygen.create_public_key(public_key);
RelinKeys relin_keys;
keygen.create_relin_keys(relin_keys);
Encryptor encryptor(context, public_key);
Evaluator evaluator(context);
Decryptor decryptor(context, secret_key);
BatchEncoder batch_encoder(context);
size_t slot_count = batch_encoder.slot_count();
size_t row_size = slot_count / 2;
cout << "Plaintext matrix row size: " << row_size << endl;
vector<uint64_t> pod_matrix(slot_count, 0ULL);
pod_matrix[0] = 0ULL;
pod_matrix[1] = 1ULL;
pod_matrix[2] = 2ULL;
pod_matrix[3] = 3ULL;
pod_matrix[row_size] = 4ULL;
pod_matrix[row_size + 1] = 5ULL;
pod_matrix[row_size + 2] = 6ULL;
pod_matrix[row_size + 3] = 7ULL;
cout << "Input plaintext matrix:" << endl;
print_matrix(pod_matrix, row_size);
首先，我们使用BatchEncoder将矩阵编码成明文。我们像往常一样加密明文。
Plaintext plain_matrix;
print_line(__LINE__);
cout << "Encode and encrypt." << endl;
batch_encoder.encode(pod_matrix, plain_matrix);
Ciphertext encrypted_matrix;
encryptor.encrypt(plain_matrix, encrypted_matrix);//加密结果放在encrypted_matrix
cout << " + Noise budget in fresh encryption: "
<< decryptor.invariant_noise_budget(encrypted_matrix) << " bits" << endl;
cout << endl;
旋转需要另一种特殊的键，叫做“伽罗瓦键 Galois keys”。这些很容易从密钥生成器获得。
GaloisKeys galois_keys;
keygen.create_galois_keys(galois_keys);
现在将两个矩阵行向左旋转3步，解密、解码和打印。
print_line(__LINE__);
cout << "Rotate rows 3 steps left." << endl;
evaluator.rotate_rows_inplace(encrypted_matrix, 3, galois_keys);
Plaintext plain_result;
cout << " + Noise budget after rotation: " << decryptor.invariant_noise_budget(encrypted_matrix) << " bits"
<< endl;
cout << " + Decrypt and decode ...... Correct." << endl;
decryptor.decrypt(encrypted_matrix, plain_result);
batch_encoder.decode(plain_result, pod_matrix);
print_matrix(pod_matrix, row_size);
我们还可以旋转列，即交换行。
print_line(__LINE__);
cout << "Rotate columns." << endl;
evaluator.rotate_columns_inplace(encrypted_matrix, galois_keys);
cout << " + Noise budget after rotation: " << decryptor.invariant_noise_budget(encrypted_matrix) << " bits"
<< endl;
cout << " + Decrypt and decode ...... Correct." << endl;
decryptor.decrypt(encrypted_matrix, plain_result);
batch_encoder.decode(plain_result, pod_matrix);
print_matrix(pod_matrix, row_size);
最后，我们将行向右旋转4步，解密、解码和打印。
print_line(__LINE__);
cout << "Rotate rows 4 steps right." << endl;
evaluator.rotate_rows_inplace(encrypted_matrix, -4, galois_keys);
cout << " + Noise budget after rotation: " << decryptor.invariant_noise_budget(encrypted_matrix) << " bits"
<< endl;
cout << " + Decrypt and decode ...... Correct." << endl;
decryptor.decrypt(encrypted_matrix, plain_result);
batch_encoder.decode(plain_result, pod_matrix);
print_matrix(pod_matrix, row_size);
**注意，旋转不消耗任何噪音预算。**然而，只有当特殊素数至少与其他素数一样大时，才会出现这种情况。relinearization也是如此。Microsoft SEAL不要求特殊的prime具有任何特定的大小，因此确保这种情况是由用户来做的。
void example_rotation_ckks()
需要注意的是，CKKS方案中的旋转与BFV中的旋转非常相似，过程相似，我们不再作具体的分析。
EncryptionParameters parms(scheme_type::ckks);
size_t poly_modulus_degree = 8192;
parms.set_poly_modulus_degree(poly_modulus_degree);
parms.set_coeff_modulus(CoeffModulus::Create(poly_modulus_degree, {
40, 40, 40, 40, 40 }));
SEALContext context(parms);
print_parameters(context);
cout << endl;
KeyGenerator keygen(context);
SecretKey secret_key = keygen.secret_key();
PublicKey public_key;
keygen.create_public_key(public_key);
RelinKeys relin_keys;
keygen.create_relin_keys(relin_keys);
GaloisKeys galois_keys;
keygen.create_galois_keys(galois_keys);
Encryptor encryptor(context, public_key);
Evaluator evaluator(context);
Decryptor decryptor(context, secret_key);
CKKSEncoder ckks_encoder(context);
size_t slot_count = ckks_encoder.slot_count();
cout << "Number of slots: " << slot_count << endl;
vector<double> input;
input.reserve(slot_count);
double curr_point = 0;
double step_size = 1.0 / (static_cast<double>(slot_count) - 1);
for (size_t i = 0; i < slot_count; i++, curr_point += step_size)
{
input.push_back(curr_point);
}
cout << "Input vector:" << endl;
print_vector(input, 3, 7);
auto scale = pow(2.0, 50);
print_line(__LINE__);
cout << "Encode and encrypt." << endl;
Plaintext plain;
ckks_encoder.encode(input, scale, plain);
Ciphertext encrypted;
encryptor.encrypt(plain, encrypted);
Ciphertext rotated;
print_line(__LINE__);
cout << "Rotate 2 steps left." << endl;
evaluator.rotate_vector(encrypted, 2, galois_keys, rotated);
cout << " + Decrypt and decode ...... Correct." << endl;
decryptor.decrypt(rotated, plain);
vector<double> result;
ckks_encoder.decode(plain, result);
print_vector(result, 3, 7);
使用CKKS方案，还可以使用Evaluator:: complex_共轭来计算加密复数向量上的复共轭。这其实是一种旋转，也需要伽罗瓦键。
运行
即将上述两种方案依次运行即可。
void example_rotation()
{
print_example_banner("Example: Rotation");
/*
Run all rotation examples.
*/
example_rotation_bfv();
example_rotation_ckks();
}
运行效果展示
+-----------------------------------+
| Example: Rotation |
+-----------------------------------+
+-----------------------------------------------------+
| Example: Rotation / Rotation in BFV |
+-----------------------------------------------------+
/
| Encryption parameters :
| scheme: BFV
| poly_modulus_degree: 8192
| coeff_modulus size: 218 (43 + 43 + 44 + 44 + 44) bits
| plain_modulus: 1032193
\
Plaintext matrix row size: 4096
Input plaintext matrix:
[ 0, 1, 2, 3, 0, ..., 0, 0, 0, 0, 0 ]
[ 4, 5, 6, 7, 0, ..., 0, 0, 0, 0, 0 ]
Line 62 --> Encode and encrypt.
+ Noise budget in fresh encryption: 146 bits
Line 81 --> Rotate rows 3 steps left.
+ Noise budget after rotation: 142 bits
+ Decrypt and decode ...... Correct.
[ 3, 0, 0, 0, 0, ..., 0, 0, 0, 1, 2 ]
[ 7, 0, 0, 0, 0, ..., 0, 0, 4, 5, 6 ]
Line 95 --> Rotate columns.
+ Noise budget after rotation: 142 bits
+ Decrypt and decode ...... Correct.
[ 7, 0, 0, 0, 0, ..., 0, 0, 4, 5, 6 ]
[ 3, 0, 0, 0, 0, ..., 0, 0, 0, 1, 2 ]
Line 108 --> Rotate rows 4 steps right.
+ Noise budget after rotation: 142 bits
+ Decrypt and decode ...... Correct.
[ 0, 4, 5, 6, 7, ..., 0, 0, 0, 0, 0 ]
[ 0, 0, 1, 2, 3, ..., 0, 0, 0, 0, 0 ]
+------------------------------------------------------+
| Example: Rotation / Rotation in CKKS |
+------------------------------------------------------+
/
| Encryption parameters :
| scheme: CKKS
| poly_modulus_degree: 8192
| coeff_modulus size: 200 (40 + 40 + 40 + 40 + 40) bits
\
Number of slots: 4096
Input vector:
[ 0.0000000, 0.0002442, 0.0004884, ..., 0.9995116, 0.9997558, 1.0000000 ]
Line 173 --> Encode and encrypt.
Line 181 --> Rotate 2 steps left.
+ Decrypt and decode ...... Correct.
[ 0.0004884, 0.0007326, 0.0009768, ..., 1.0000000, -0.0000000, 0.0002442 ]
+---------------------------------------------------------+
| The following examples should be executed while reading |
| comments in associated files in native/examples/. |
+---------------------------------------------------------+
| Examples | Source Files |
+----------------------------+----------------------------+
| 1. BFV Basics | 1_bfv_basics.cpp |
| 2. Encoders | 2_encoders.cpp |
| 3. Levels | 3_levels.cpp |
| 4. CKKS Basics | 4_ckks_basics.cpp |
| 5. Rotation | 5_rotation.cpp |
| 6. Serialization | 6_serialization.cpp |
| 7. Performance Test | 7_performance.cpp |
+----------------------------+----------------------------+
[ 82 MB] Total allocation from the memory pool
结语
本文我们分析了SEAL对于rotation的实现机制。总体来说还是比前几篇难度低一些，有一些集大成者的感觉。这也是从底层向上层分析的收获吧。