OpenFHE 使用样例

参考文献：

编译 OpenFHE
OpenFHE 源码解析：PKE 部分
[ABB+22] Al Badawi A, Bates J, Bergamaschi F, et al. Openfhe: Open-source fully homomorphic encryption library[C]//Proceedings of the 10th Workshop on Encrypted Computing & Applied Homomorphic Cryptography. 2022: 53-63.

文章目录

VSCode 的配置
- 编辑器
- 调试器
GCC 的配置
BFV
- 代码
- 结果
- - 使用 `BEHZ` 乘法
  - 使用 `HPSPOVERQLEVELED` 乘法
BGV
- 代码
- 结果
- - 使用 `FIXEDMANUAL` 模切换
  - 使用 `FLEXIBLEAUTO` 模切换
CKKS
- 代码
- 结果
- - 使用 `FIXEDMANUAL` 模切换
  - 使用 `FLEXIBLEAUTO` 模切换

VSCode 的配置

编辑器

c_cpp_properties.json 如下：

{"configurations": [{"name": "Linux","includePath": ["${workspaceFolder}/**","/usr/local/include/openfhe/pke","/usr/local/include/openfhe/core","/usr/local/include/openfhe","/usr/local/include/openfhe/binfhe"],"defines": [],"compilerPath": "/usr/bin/gcc","cStandard": "c17","cppStandard": "c++14","intelliSenseMode": "macos-gcc-x64"}],"version": 4
}

调试器

tasks.json 如下：

{"tasks": [{"type": "cppbuild","label": "C/C++: gcc 生成活动文件","command": "/usr/bin/gcc","args": ["-fdiagnostics-color=always","-g","${file}","-o","${fileDirname}/${fileBasenameNoExtension}","-I/usr/local/include/openfhe","-I/usr/local/include/openfhe/pke","-I/usr/local/include/openfhe/core","-I/usr/local/include/openfhe/binfhe","-I/usr/local/include/openfhe/cereal","-L/usr/local/lib","-lstdc++","-lm","-lssl","-lntl","-fopenmp","-fPIC","-lOPENFHEcore","-lOPENFHEpke","-lOPENFHEbinfhe",],"options": {"cwd": "${fileDirname}"},"problemMatcher": ["$gcc"],"group": {"kind": "build","isDefault": true},"detail": "调试器生成的任务。"}],"version": "2.0.0"
}

GCC 的配置

CMakeLists.txt 如下：

cmake_minimum_required(VERSION 3.10)project(MyCode CXX)set(CMAKE_CXX_COMPILER "g++")
set(CMAKE_CXX_STANDARD 17)
set(CMAKE_CXX_STANDARD_REQUIRED ON)#配置编译选项
add_compile_options(-march=native -O3 -lstdc++ -lm -lssl -lntl -pthread -fopenmp -fPIC-mavx2 -mno-avx256-split-unaligned-load -mno-avx256-split-unaligned-store-Wno-unused-command-line-argument -Wwrite-strings -v
)#配置头文件和静态库的路径
add_compile_options(-I/usr/local/include/openfhe-I/usr/local/include/openfhe/core-I/usr/local/include/openfhe/pke-I/usr/local/include/openfhe/binfhe-I/usr/local/include/openfhe/cereal-L/usr/local/lib
)#SET(EXECUTABLE_OUTPUT_PATH ${PROJECT_SOURCE_DIR}/bin)
#SET(LIBRARY_OUTPUT_PATH ${PROJECT_SOURCE_DIR}/lib/)set(CMAKE_EXE_LINKER_FLAGS "-Wl,--copy-dt-needed-entries") #递归获取动态库的依赖项add_executable(ckks ckks.cpp)
target_link_libraries(ckks OPENFHEcore OPENFHEpke OPENFHEbinfhe)add_executable(use_ckks use_ckks.cpp)
target_link_libraries(use_ckks OPENFHEcore OPENFHEpke OPENFHEbinfhe)add_executable(use_bgv use_bgv.cpp)
target_link_libraries(use_bgv OPENFHEcore OPENFHEpke OPENFHEbinfhe)add_executable(use_bfv use_bfv.cpp)
target_link_libraries(use_bfv OPENFHEcore OPENFHEpke OPENFHEbinfhe)

BFV

代码

BFV 方案使用 MSD 纠错码，不需要模切换。在 Ciphertext<DCRTPoly> 密文中，GetLevel() 总是等于 $0$ ，GetNoiseScaleDeg() 记录了乘法深度，GetElements().size() 是密文中环元素向量的长度。

对于 CoefPackedPlaintext 打包，明文模数可以是任意的整数，不仅仅是素数或者素数幂。

系数打包的 BFV 用例，

#include <vector>
#include <iostream>
#include "openfhe.h"
#include "cputimer.h"using namespace std;
using namespace lbcrypto;#define endl2 endl << endl
#define pn puts("")#define print_param(ct)                                         \cout << "Level = " << ct->GetLevel() << " | "               \<< "Depth = " << ct->GetNoiseScaleDeg() << " | "       \<< "SKDeg = " << ct->GetElements().size() - 1 << " | " \<< endl;#define print_pt(pt) \cout << "Result: " << pt << endl2;int main()
{GetFrequency();cout << endl;size_t ringDim = 16384;       // 必须是二的幂次size_t plaintextModulus = 17; // 系数打包可以是任意整数size_t multDepth = 10;size_t maxRelinSkDeg = 4;CCParams<CryptoContextBFVRNS> parameters;parameters.SetSecurityLevel(HEStd_NotSet); // 如果不设置安全级别，则需要手动给出环维度parameters.SetRingDim(ringDim);parameters.SetPlaintextModulus(plaintextModulus);parameters.SetMultiplicativeDepth(multDepth);parameters.SetMaxRelinSkDeg(maxRelinSkDeg);parameters.SetKeySwitchTechnique(HYBRID);// BEHZ，整数指令的 RNS 扩展，乘法速度很慢// HPSPOVERQLEVELED，浮点指令的 RNS 扩展，并且隐式模切换parameters.SetMultiplicationTechnique(HPSPOVERQLEVELED);CryptoContext<DCRTPoly> cc = GenCryptoContext(parameters);cc->Enable(PKE);cc->Enable(KEYSWITCH);cc->Enable(LEVELEDSHE);cout << "CyclotomicOrder = " << cc->GetCyclotomicOrder() << endl<< "RingDimension = " << cc->GetRingDimension() << endl<< "Level = " << cc->GetKeyGenLevel() << endl<< "Modulus = " << cc->GetModulus() << endl<< "ModulusBits = " << cc->GetModulus().GetLengthForBase(2) << endl<< "RootOfUnity = " << cc->GetRootOfUnity() << endl<< "PlaintextModulus = " << cc->GetEncodingParams()->GetPlaintextModulus() << endl<< "BatchSize = " << cc->GetEncodingParams()->GetBatchSize() << endl<< "PlaintextGenerator = " << cc->GetEncodingParams()->GetPlaintextGenerator() << endl<< endl2;/***************************************************************************************************/auto keys = cc->KeyGen();cc->EvalMultKeysGen(keys.secretKey);cc->EvalRotateKeyGen(keys.secretKey, {1, -2});// 生成元 g = 5 mod N, ord(g) = N/2map<usint, EvalKey<DCRTPoly>> evk_map = *cc->EvalAutomorphismKeyGen(keys.secretKey, {1u, 5u, 25u});vector<int64_t> x1 = {1, 2, 3, 4, 5, 6, 7, 8};vector<int64_t> x2 = {1, 0, 0, 0, 0, 0, 0, -1};Plaintext pt1 = cc->MakeCoefPackedPlaintext(x1);Plaintext pt2 = cc->MakeCoefPackedPlaintext(x2);print_pt(pt1);print_pt(pt2);/***************************************************************************************************/auto ct1 = cc->Encrypt(keys.publicKey, pt1);auto ct2 = cc->Encrypt(keys.publicKey, pt2);Plaintext pt;print_param(ct1);pn;cc->Decrypt(keys.secretKey, ct1, &pt);print_pt(pt);cc->Decrypt(keys.secretKey, ct2, &pt);print_pt(pt);Ciphertext<DCRTPoly> ct;cout.precision(8);/***************************************************************************************************/Timer("+", ct = ct1 + ct2;); // 比 cc->EvalAdd 要快print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Add #1", ct = cc->EvalAdd(ct1, ct2);); // 首先会 type 检查print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Scale", ct = cc->EvalMult(ct, cc->MakeCoefPackedPlaintext({0, 2}));); // 必须把 scale 先构造为 Plaintextprint_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Add #2", ct = cc->EvalAdd(ct, ct2);); // 相同 Level 不同 ScalingFactorInt 的加法，正确print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("*", ct = ct1 * ct2;); // 和 cc->EvalMult 一样，但是不检查 typeprint_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Mult", ct = cc->EvalMult(ct1, ct2);); // 总是自动密钥切换，和 cc->EvalMultAndRelinearize 完全一样print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Add #3", ct = cc->EvalAdd(ct, ct2);); // 不同 Level 不同 ScalingFactorInt 的加法，正确print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Mult And Relinearize", ct = cc->EvalMultAndRelinearize(ct1, ct2);); // 和 cc->EvalMult 基本一样print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("Mult No Relin #1", ct = cc->EvalMultNoRelin(ct1, ct2);); // 这个才是不做密钥切换的，SKDeg 增长print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Mult No Relin #2", ct = cc->EvalMultNoRelin(ct, ct2);); // 不同 ScalingFactor 不同 Level 的乘法，正确print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Relinear", cc->RelinearizeInPlace(ct);); // 占据几乎全部的时间print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("Rotate(1)", ct = cc->EvalRotate(ct1, 1);); // 正数是左旋print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Rotate(-2)", ct = cc->EvalRotate(ct1, -2);); // 负数是右旋print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("Automorphism(1)", ct = cc->EvalAutomorphism(ct1, 1, evk_map);); // 不旋转print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(5)", ct = cc->EvalAutomorphism(ct1, 5, evk_map);); // 数量 N 的明文槽，由 (-5) 生成print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(25)", ct = cc->EvalAutomorphism(ct1, 25, evk_map);); // 5^2 mod Nprint_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/return 0;
}

结果

使用 `BEHZ` 乘法

CPU Frequency = 2918913687CyclotomicOrder = 2048
RingDimension = 1024
Level = 0
Modulus = 1532495540865662914854395520533418584588770550682052609
ModulusBits = 180
RootOfUnity = 0
PlaintextModulus = 17
BatchSize = 1024
PlaintextGenerator = 0Result: ( 1 2 3 4 5 6 7 8 ... )Result: ( 1 0 0 0 0 0 0 -1 ... )Level = 0 | Depth = 1 | SKDeg = 1 | Result: ( 1 2 3 4 5 6 7 8 ... )Result: ( 1 0 0 0 0 0 0 -1 ... )Running Time of <+>:         99032 cycles (  0.033928 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 2 2 3 4 5 6 7 7 ... )Running Time of <Add #1>:          2736272 cycles (  0.937428 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 2 2 3 4 5 6 7 7 ... )Running Time of <Scale>:           2735077 cycles (  0.937019 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 0 4 4 6 8 -7 -5 -3 -3 ... )Running Time of <Add #2>:          1133253 cycles (  0.388245 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 1 4 4 6 8 -7 -5 -4 -3 ... )Running Time of <*>:     253469802 cycles ( 86.837032 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 7 -2 -3 -4 -5 -6 -7 -8 ... )Running Time of <Mult>:  187670910 cycles ( 64.294779 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 7 -2 -3 -4 -5 -6 -7 -8 ... )Running Time of <Add #3>:           435927 cycles (  0.149346 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 2 2 3 4 5 6 7 6 -2 -3 -4 -5 -6 -7 -8 ... )Running Time of <Mult And Relinearize>:  126839959 cycles ( 43.454508 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 7 -2 -3 -4 -5 -6 -7 -8 ... )Running Time of <Mult No Relin #1>:      130843571 cycles ( 44.826119 ms)
Level = 0 | Depth = 2 | SKDeg = 2 | 
Result: ( 1 2 3 4 5 6 7 7 -2 -3 -4 -5 -6 -7 -8 ... )Running Time of <Mult No Relin #2>:      172007707 cycles ( 58.928672 ms)
Level = 0 | Depth = 3 | SKDeg = 3 | 
Result: ( 1 2 3 4 5 6 7 6 -4 -6 -8 7 5 3 2 2 3 4 5 6 7 8 ... )Running Time of <Relinear>:      208823169 cycles ( 71.541399 ms)
Level = 0 | Depth = 3 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 6 -4 -6 -8 7 5 3 2 2 3 4 5 6 7 8 ... )Running Time of <Rotate(1)>:     115388270 cycles ( 39.531237 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 0 7 0 0 0 0 8 ... )Running Time of <Rotate(-2)>:     70254530 cycles ( 24.068725 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -8 ... )Running Time of <Automorphism(1)>:        83246053 cycles ( 28.519532 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 8 ... )Running Time of <Automorphism(5)>:        74563676 cycles ( 25.545009 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 0 7 0 0 0 0 8 ... )Running Time of <Automorphism(25)>:       67644689 cycles ( 23.174611 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 ... )

使用 `HPSPOVERQLEVELED` 乘法

CPU Frequency = 2918959071CyclotomicOrder = 2048
RingDimension = 1024
Level = 0
Modulus = 1532495540865662914854395520533418584588770550682052609
ModulusBits = 180
RootOfUnity = 0
PlaintextModulus = 17
BatchSize = 1024
PlaintextGenerator = 0Result: ( 1 2 3 4 5 6 7 8 ... )Result: ( 1 0 0 0 0 0 0 -1 ... )Level = 0 | Depth = 1 | SKDeg = 1 | Result: ( 1 2 3 4 5 6 7 8 ... )Result: ( 1 0 0 0 0 0 0 -1 ... )Running Time of <+>:        115828 cycles (  0.039681 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 2 2 3 4 5 6 7 7 ... )Running Time of <Add #1>:          4293174 cycles (  1.470789 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 2 2 3 4 5 6 7 7 ... )Running Time of <Scale>:            894332 cycles (  0.306387 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 0 4 4 6 8 -7 -5 -3 -3 ... )Running Time of <Add #2>:           987738 cycles (  0.338387 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 1 4 4 6 8 -7 -5 -4 -3 ... )Running Time of <*>:     236847696 cycles ( 81.141150 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 7 -2 -3 -4 -5 -6 -7 -8 ... )Running Time of <Mult>:  162997593 cycles ( 55.841000 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 7 -2 -3 -4 -5 -6 -7 -8 ... )Running Time of <Add #3>:           311286 cycles (  0.106643 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 2 2 3 4 5 6 7 6 -2 -3 -4 -5 -6 -7 -8 ... )Running Time of <Mult And Relinearize>:  152945198 cycles ( 52.397171 ms)
Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 7 -2 -3 -4 -5 -6 -7 -8 ... )Running Time of <Mult No Relin #1>:      108039418 cycles ( 37.012995 ms)
Level = 0 | Depth = 2 | SKDeg = 2 | 
Result: ( 1 2 3 4 5 6 7 7 -2 -3 -4 -5 -6 -7 -8 ... )Running Time of <Mult No Relin #2>:      183090958 cycles ( 62.724743 ms)
Level = 0 | Depth = 3 | SKDeg = 3 | 
Result: ( 1 2 3 4 5 6 7 6 -4 -6 -8 7 5 3 2 2 3 4 5 6 7 8 ... )Running Time of <Relinear>:      116302449 cycles ( 39.843809 ms)
Level = 0 | Depth = 3 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 6 -4 -6 -8 7 5 3 2 2 3 4 5 6 7 8 ... )Running Time of <Rotate(1)>:      88811197 cycles ( 30.425640 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 0 7 0 0 0 0 8 ... )Running Time of <Rotate(-2)>:     87308745 cycles ( 29.910918 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -8 ... )Running Time of <Automorphism(1)>:        87723653 cycles ( 30.053060 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 8 ... )Running Time of <Automorphism(5)>:        97097626 cycles ( 33.264470 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 0 7 0 0 0 0 8 ... )Running Time of <Automorphism(25)>:       80256150 cycles ( 27.494784 ms)
Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 ... )

BGV

代码

BGV 方案使用 LSD 纠错码，需要模切换。在 Ciphertext<DCRTPoly> 密文中，GetScalingFactorInt() 记录了扭曲因子，GetLevel() 记录了乘法深度，GetNoiseScaleDeg() 记录了扭曲因子的指数，GetElements().size() 是密文中环元素向量的长度。

对于 PackedPlaintext 打包，明文模数必须是满足 NTT 的大素数，槽旋转对应的自同构指标由 5 mod{2*N} 生成。

槽打包的 BGV 用例，

#include <vector>
#include <iostream>
#include "openfhe.h"
#include "cputimer.h"using namespace std;
using namespace lbcrypto;#define endl2 endl << endl
#define pn puts("")#define print_param(ct)                                                            \cout << "ScalingFactorInt = " << ct->GetScalingFactorInt().ToString() << " | " \<< "Level = " << ct->GetLevel() << " | "                                  \<< "Depth = " << ct->GetNoiseScaleDeg() << " | "                          \<< "SKDeg = " << ct->GetElements().size() - 1 << " | "                    \<< endl;#define print_pt(pt) \cout << "Result: " << pt << endl2;int main()
{GetFrequency();cout << endl;size_t ringDim = 16384;          // 必须是二的幂次size_t plaintextModulus = 65537; // 槽打包必须是大素数size_t multDepth = 10;size_t maxRelinSkDeg = 4;CCParams<CryptoContextBGVRNS> parameters;parameters.SetSecurityLevel(HEStd_NotSet); // 如果不设置安全级别，则需要手动给出环维度parameters.SetRingDim(ringDim);parameters.SetPlaintextModulus(plaintextModulus);parameters.SetMultiplicativeDepth(multDepth);parameters.SetMaxRelinSkDeg(maxRelinSkDeg);parameters.SetKeySwitchTechnique(HYBRID);// FIXEDMANUAL 手动模切换，ScalingFactorInt 总是 1// FLEXIBLEAUTO 自动模切换，ScalingFactorInt 会在 [0, plaintextModulus] 内变换parameters.SetScalingTechnique(FLEXIBLEAUTO);CryptoContext<DCRTPoly> cc = GenCryptoContext(parameters);cc->Enable(PKE);cc->Enable(KEYSWITCH);cc->Enable(LEVELEDSHE);cout << "CyclotomicOrder = " << cc->GetCyclotomicOrder() << endl<< "RingDimension = " << cc->GetRingDimension() << endl<< "Level = " << cc->GetKeyGenLevel() << endl<< "Modulus = " << cc->GetModulus() << endl<< "ModulusBits = " << cc->GetModulus().GetLengthForBase(2) << endl<< "RootOfUnity = " << cc->GetRootOfUnity() << endl<< "PlaintextModulus = " << cc->GetEncodingParams()->GetPlaintextModulus() << endl<< "BatchSize = " << cc->GetEncodingParams()->GetBatchSize() << endl<< "PlaintextGenerator = " << cc->GetEncodingParams()->GetPlaintextGenerator() << endl<< endl2;/***************************************************************************************************/auto keys = cc->KeyGen();cc->EvalMultKeysGen(keys.secretKey);cc->EvalRotateKeyGen(keys.secretKey, {1, -2});// 生成元 g = 5 mod N, ord(g) = N/2map<usint, EvalKey<DCRTPoly>> evk_map = *cc->EvalAutomorphismKeyGen(keys.secretKey, {1u, 5u, 25u});vector<int64_t> x1 = {1, 2, 3, 4, 5, 6, 7, 8};vector<int64_t> x2 = {4, 3, 2, 1, 1, 2, 3, 4};Plaintext pt1 = cc->MakePackedPlaintext(x1);Plaintext pt2 = cc->MakePackedPlaintext(x2);print_pt(pt1);print_pt(pt2);/***************************************************************************************************/auto ct1 = cc->Encrypt(keys.publicKey, pt1);auto ct2 = cc->Encrypt(keys.publicKey, pt2);Plaintext pt;print_param(ct1);pn;cc->Decrypt(keys.secretKey, ct1, &pt);print_pt(pt);cc->Decrypt(keys.secretKey, ct2, &pt);print_pt(pt);Ciphertext<DCRTPoly> ct;cout.precision(8);/***************************************************************************************************/Timer("+", ct = ct1 + ct2;); // 比 cc->EvalAdd 要快print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Add #1", ct = cc->EvalAdd(ct1, ct2);); // 首先会 type 检查print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Scale", ct = cc->EvalMult(ct, cc->MakePackedPlaintext(vector<int64_t>(cc->GetEncodingParams()->GetBatchSize(), 4)));); // 必须把 scale 先构造为 Plaintextprint_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Add #2", ct = cc->EvalAdd(ct, ct2);); // 相同 Level 不同 ScalingFactorInt 的加法，正确print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("*", ct = ct1 * ct2;); // 和 cc->EvalMult 一样，但是不检查 typeprint_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Mult", ct = cc->EvalMult(ct1, ct2);); // 总是自动密钥切换，和 cc->EvalMultAndRelinearize 完全一样print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Composed Mult #1", ct = cc->ComposedEvalMult(ct1, ct2);); // 密钥切换 + 模切换print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Composed Mult #2", ct = cc->ComposedEvalMult(ct, ct2);); // FLEXIBLEAUTO  在乘法之前做模切换，EXT 第一次乘法之前也做模切换print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Add #3", ct = cc->EvalAdd(ct, ct2);); // 不同 Level 不同 ScalingFactorInt 的加法，正确print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Mult And Relinearize", ct = cc->EvalMultAndRelinearize(ct1, ct2);); // 和 cc->EvalMult 基本一样print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("Mult No Relin #1", ct = cc->EvalMultNoRelin(ct1, ct2);); // 这个才是不做密钥切换的，SKDeg 增长print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);// cc->ModReduceInPlace(ct);Timer("Mult No Relin #2", ct = cc->EvalMultNoRelin(ct, ct2);); // 不同 ScalingFactor 不同 Level 的乘法，正确print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Relinear", cc->RelinearizeInPlace(ct);); // 占据几乎全部的时间print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("ModReduce", cc->ModReduceInPlace(ct);); // 降低一层的缩放print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("Rotate(1)", ct = cc->EvalRotate(ct1, 1);); // 正数是左旋print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Rotate(-2)", ct = cc->EvalRotate(ct1, -2);); // 负数是右旋print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("Automorphism(1)", ct = cc->EvalAutomorphism(ct1, 1, evk_map);); // 不旋转print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(5)", ct = cc->EvalAutomorphism(ct1, 5, evk_map);); // 数量 N 的明文槽，由 (-5) 生成print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(25)", ct = cc->EvalAutomorphism(ct1, 25, evk_map);); // 5^2 mod Nprint_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/return 0;
}

结果

使用 `FIXEDMANUAL` 模切换

CPU Frequency = 2919164586CyclotomicOrder = 2048
RingDimension = 1024
Level = 0
Modulus = 101915945247153194422621674945726031044678923861288323567523751344699549609876260625831851757818456743873649587405440263849024730822259572880063879169
ModulusBits = 495
RootOfUnity = 0
PlaintextModulus = 65537
BatchSize = 1024
PlaintextGenerator = 0Result: ( 1 2 3 4 5 6 7 8 ... )Result: ( 4 3 2 1 1 2 3 4 ... )ScalingFactorInt = 1 | Level = 0 | Depth = 1 | SKDeg = 1 | Result: ( 1 2 3 4 5 6 7 8 ... )Result: ( 4 3 2 1 1 2 3 4 ... )Running Time of <+>:        928928 cycles (  0.318217 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 5 5 5 5 6 8 10 12 ... )Running Time of <Add #1>:           918166 cycles (  0.314530 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 5 5 5 5 6 8 10 12 ... )Running Time of <Scale>:           1406846 cycles (  0.481934 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 20 20 20 20 24 32 40 48 ... )Running Time of <Add #2>:           721196 cycles (  0.247056 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 24 23 22 21 25 34 43 52 ... )Running Time of <*>:      98448862 cycles ( 33.725012 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 4 6 6 4 5 12 21 32 ... )Running Time of <Mult>:   74771761 cycles ( 25.614096 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 4 6 6 4 5 12 21 32 ... )Running Time of <Composed Mult #1>:      103001350 cycles ( 35.284530 ms)
ScalingFactorInt = 1 | Level = 1 | Depth = 1 | SKDeg = 1 | 
Result: ( 4 6 6 4 5 12 21 32 ... )Running Time of <Composed Mult #2>:       97931103 cycles ( 33.547647 ms)
ScalingFactorInt = 1 | Level = 2 | Depth = 1 | SKDeg = 1 | 
Result: ( 16 18 12 4 5 24 63 128 ... )Running Time of <Add #3>:           660204 cycles (  0.226162 ms)
ScalingFactorInt = 1 | Level = 2 | Depth = 1 | SKDeg = 1 | 
Result: ( 20 21 14 5 6 26 66 132 ... )Running Time of <Mult And Relinearize>:  121328656 cycles ( 41.562801 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 4 6 6 4 5 12 21 32 ... )Running Time of <Mult No Relin #1>:        1237702 cycles (  0.423992 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 2 | SKDeg = 2 | 
Result: ( 4 6 6 4 5 12 21 32 ... )Running Time of <Mult No Relin #2>:        1619596 cycles (  0.554815 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 3 | SKDeg = 3 | 
Result: ( 16 18 12 4 5 24 63 128 ... )Running Time of <Relinear>:      207136845 cycles ( 70.957577 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 3 | SKDeg = 1 | 
Result: ( 16 18 12 4 5 24 63 128 ... )Running Time of <ModReduce>:        991996 cycles (  0.339822 ms)
ScalingFactorInt = 1 | Level = 1 | Depth = 2 | SKDeg = 1 | 
Result: ( 16 18 12 4 5 24 63 128 ... )Running Time of <Rotate(1)>:      82565072 cycles ( 28.283802 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ... )Running Time of <Rotate(-2)>:    118772481 cycles ( 40.687148 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 0 0 1 2 3 4 5 6 7 8 ... )Running Time of <Automorphism(1)>:        98264131 cycles ( 33.661730 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 8 ... )Running Time of <Automorphism(5)>:       104197049 cycles ( 35.694133 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ... )Running Time of <Automorphism(25)>:       96996770 cycles ( 33.227578 ms)
ScalingFactorInt = 1 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 ... )

使用 `FLEXIBLEAUTO` 模切换

CPU Frequency = 2919001454CyclotomicOrder = 2048
RingDimension = 1024
Level = 0
Modulus = 11090766327926935557496534795450603891404351373393750077990741913791395272295156970046992897184754308334673051240449197703818432513
ModulusBits = 433
RootOfUnity = 0
PlaintextModulus = 65537
BatchSize = 1024
PlaintextGenerator = 0Result: ( 1 2 3 4 5 6 7 8 ... )Result: ( 4 3 2 1 1 2 3 4 ... )ScalingFactorInt = 20731 | Level = 0 | Depth = 1 | SKDeg = 1 | Result: ( 1 2 3 4 5 6 7 8 ... )Result: ( 4 3 2 1 1 2 3 4 ... )Running Time of <+>:        876028 cycles (  0.300112 ms)
ScalingFactorInt = 20731 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 5 5 5 5 6 8 10 12 ... )Running Time of <Add #1>:           882874 cycles (  0.302458 ms)
ScalingFactorInt = 20731 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 5 5 5 5 6 8 10 12 ... )Running Time of <Scale>:           1067962 cycles (  0.365866 ms)
ScalingFactorInt = 48252 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 20 20 20 20 24 32 40 48 ... )Running Time of <Add #2>:           903446 cycles (  0.309505 ms)
ScalingFactorInt = 48252 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 24 23 22 21 25 34 43 52 ... )Running Time of <*>:      79417215 cycles ( 27.206980 ms)
ScalingFactorInt = 48252 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 4 6 6 4 5 12 21 32 ... )Running Time of <Mult>:   96562770 cycles ( 33.080754 ms)
ScalingFactorInt = 48252 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 4 6 6 4 5 12 21 32 ... )Running Time of <Composed Mult #1>:       74693744 cycles ( 25.588800 ms)
ScalingFactorInt = 48252 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 4 6 6 4 5 12 21 32 ... )Running Time of <Composed Mult #2>:       98139791 cycles ( 33.621015 ms)
ScalingFactorInt = 48252 | Level = 1 | Depth = 2 | SKDeg = 1 | 
Result: ( 16 18 12 4 5 24 63 128 ... )Running Time of <Add #3>:           923506 cycles (  0.316377 ms)
ScalingFactorInt = 48252 | Level = 1 | Depth = 2 | SKDeg = 1 | 
Result: ( 20 21 14 5 6 26 66 132 ... )Running Time of <Mult And Relinearize>:   94704513 cycles ( 32.444147 ms)
ScalingFactorInt = 48252 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ( 4 6 6 4 5 12 21 32 ... )Running Time of <Mult No Relin #1>:        1258456 cycles (  0.431126 ms)
ScalingFactorInt = 48252 | Level = 0 | Depth = 2 | SKDeg = 2 | 
Result: ( 4 6 6 4 5 12 21 32 ... )Running Time of <Mult No Relin #2>:        2055410 cycles (  0.704148 ms)
ScalingFactorInt = 48252 | Level = 1 | Depth = 2 | SKDeg = 3 | 
Result: ( 16 18 12 4 5 24 63 128 ... )Running Time of <Relinear>:      179685843 cycles ( 61.557298 ms)
ScalingFactorInt = 48252 | Level = 1 | Depth = 2 | SKDeg = 1 | 
Result: ( 16 18 12 4 5 24 63 128 ... )Running Time of <ModReduce>:          6682 cycles (  0.002289 ms)
ScalingFactorInt = 48252 | Level = 1 | Depth = 2 | SKDeg = 1 | 
Result: ( 16 18 12 4 5 24 63 128 ... )Running Time of <Rotate(1)>:     109469187 cycles ( 37.502272 ms)
ScalingFactorInt = 20731 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ... )Running Time of <Rotate(-2)>:     74563389 cycles ( 25.544142 ms)
ScalingFactorInt = 20731 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 0 0 1 2 3 4 5 6 7 8 ... )Running Time of <Automorphism(1)>:        84518442 cycles ( 28.954573 ms)
ScalingFactorInt = 20731 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 1 2 3 4 5 6 7 8 ... )Running Time of <Automorphism(5)>:       104449053 cycles ( 35.782460 ms)
ScalingFactorInt = 20731 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ... )Running Time of <Automorphism(25)>:       75254525 cycles ( 25.780914 ms)
ScalingFactorInt = 20731 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ( 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 ... )

CKKS

代码

CKKS 方案不使用纠错码，但是把浮点数缩放为整数，需要重缩放。在 Ciphertext<DCRTPoly> 密文中，GetScalingFactor() 记录了缩放因子，GetLevel() 记录了乘法深度，GetNoiseScaleDeg() 记录了缩放因子的指数，GetElements().size() 是密文中环元素向量的长度。

对于 CKKSPackedPlaintext 打包，仅支持 Real() 部分的加解密。SetScalingModSize() 用于设置缩放因子的比特数，SetBatchSize() 用于设置明文槽的个数。对于 BatchSize <= RingDim/2，打包成维度 2*BatchSize 的子环，槽旋转对应的自同构指标由 5 mod{4*BatchSize} 生成。

槽打包的 CKKS 用例，

#include <vector>
#include <iostream>
#include "openfhe.h"
#include "cputimer.h"using namespace std;
using namespace lbcrypto;#define endl2 endl << endl
#define pn puts("")#define print_param(ct)                                           \cout << "ScalingFactor = " << ct->GetScalingFactor() << " | " \<< "Level = " << ct->GetLevel() << " | "                 \<< "Depth = " << ct->GetNoiseScaleDeg() << " | "         \<< "SKDeg = " << ct->GetElements().size() - 1 << " | "   \<< endl;#define print_pt(pt)    \cout << "Result: "; \PrintComplex(pt);   \cout << "Estimated precision in bits: " << pt->GetLogPrecision() << endl2;void PrintComplex(Plaintext pt)
{vector<complex<double>> value = pt->GetCKKSPackedValue();size_t len = pt->GetLength();cout << "(";while (--len > 0)if (value[len] != complex<double>(0, 0))break;for (int i = 0; i <= len; i++){cout << value[i] << ", ";}cout << " ... ); "<< "Estimated precision: " << pt->GetLogPrecision() << " bits" << endl;
}int main()
{GetFrequency();cout << endl;size_t ringDim = 16384;   // 必须是二的幂次size_t batchSize = 8;     // 必须是 N/2 的因子，二的幂次size_t scaleModSize = 50; // 仅支持至多 60-bit 尾数size_t multDepth = 10;size_t maxRelinSkDeg = 4;CCParams<CryptoContextCKKSRNS> parameters;parameters.SetSecurityLevel(HEStd_NotSet); // 如果不设置安全级别，则需要手动给出环维度parameters.SetRingDim(ringDim);parameters.SetBatchSize(batchSize);parameters.SetScalingModSize(scaleModSize);parameters.SetMultiplicativeDepth(multDepth);parameters.SetMaxRelinSkDeg(maxRelinSkDeg);parameters.SetKeySwitchTechnique(HYBRID);// FIXEDMANUAL 手动模切换，ScalingFactor 会变// FLEXIBLEAUTO 自动模切换，ScalingFactor 总是 2^scaleModSizeparameters.SetScalingTechnique(FIXEDMANUAL);CryptoContext<DCRTPoly> cc = GenCryptoContext(parameters);cc->Enable(PKE);cc->Enable(KEYSWITCH);cc->Enable(LEVELEDSHE);cc->Enable(ADVANCEDSHE);cout << "CyclotomicOrder = " << cc->GetCyclotomicOrder() << endl<< "RingDimension = " << cc->GetRingDimension() << endl<< "Level = " << cc->GetKeyGenLevel() << endl<< "Modulus = " << cc->GetModulus() << endl<< "ModulusBits = " << cc->GetModulus().GetLengthForBase(2) << endl<< "RootOfUnity = " << cc->GetRootOfUnity() << endl<< "PlaintextModulus = " << cc->GetEncodingParams()->GetPlaintextModulus() << endl<< "BatchSize = " << cc->GetEncodingParams()->GetBatchSize() << endl<< "PlaintextGenerator = " << cc->GetEncodingParams()->GetPlaintextGenerator() << endl<< endl2;/***************************************************************************************************/auto keys = cc->KeyGen();cc->EvalMultKeysGen(keys.secretKey);cc->EvalRotateKeyGen(keys.secretKey, {1, -2});// 生成元 g = 5 mod{4*batchSize}, ord(g) = batchSizemap<usint, EvalKey<DCRTPoly>> evk_map = *cc->EvalAutomorphismKeyGen(keys.secretKey, {1u, 5u, 25u, 29u, 17u, 21u, 9u, 13u});vector<complex<double>> x1 = {{0.25, 0.5}, {0.5, 1}, {0.75, 1.5}, {1, 2}, {2, 4}, {3, 6}, {4, 8}, {5, 10}};vector<double> x2 = {5, 4, 3, 2, 1, 0.75, 0.5, 0.25};Plaintext pt1 = cc->MakeCKKSPackedPlaintext(x1);Plaintext pt2 = cc->MakeCKKSPackedPlaintext(x2);print_pt(pt1); // 明文编码了 Re 以及 Imprint_pt(pt2);/***************************************************************************************************/auto ct1 = cc->Encrypt(keys.publicKey, pt1);auto ct2 = cc->Encrypt(keys.publicKey, pt2);Plaintext pt;print_param(ct1);pn;cc->Decrypt(keys.secretKey, ct1, &pt); // 但是仅加密 Re 部分，丢弃 Im 部分，复制在 Z_{4*batchSize}^* = <-1, 5> 中的 <5> 和 <-5> 两行print_pt(pt);cc->Decrypt(keys.secretKey, ct2, &pt);print_pt(pt);Ciphertext<DCRTPoly> ct;cout.precision(8);/***************************************************************************************************/Timer("+", ct = ct1 + ct2;); // 比 cc->EvalAdd 要快print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Add #1", ct = cc->EvalAdd(ct1, ct2);); // 首先会 type 检查print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Scale", ct = cc->EvalMult(ct1, 4);); // 速度接近 operator+print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);cc->RescaleInPlace(ct); // 相同 Level 不同 ScalingFactor 的加法，错误Timer("Add #2", ct = cc->EvalAdd(ct, ct2);); // 相同 ScalingFactor 不同 Level 的加法，正确print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("*", ct = ct1 * ct2;); // 和 cc->EvalMult 一样，但是不检查 typeprint_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Mult", ct = cc->EvalMult(ct1, ct2);); // 总是自动密钥切换，和 cc->EvalMultAndRelinearize 完全一样print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Composed Mult #1", ct = cc->ComposedEvalMult(ct1, ct2);); // 密钥切换 + 模切换print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Composed Mult #2", ct = cc->ComposedEvalMult(ct, ct2);); // FLEXIBLEAUTO  在乘法之前做模切换，EXT 第一次乘法之前也做模切换print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Mult And Relinearize", ct = cc->EvalMultAndRelinearize(ct1, ct2);); // 和 cc->EvalMult 基本一样print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("Mult No Relin #1", ct = cc->EvalMultNoRelin(ct1, ct2);); // 这个才是不做密钥切换的，SKDeg 增长print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);// cc->RescaleInPlace(ct);Timer("Mult No Relin #2", ct = cc->EvalMultNoRelin(ct, ct2);); // 不同 ScalingFactor 不同 Level 的乘法，正确print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Relinear", cc->RelinearizeInPlace(ct);); // 占据几乎全部的时间print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("ModReduce", cc->ModReduceInPlace(ct);); // 降低一层的缩放，和 cc->RescaleInPlace 完全一样print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Rescale", cc->RescaleInPlace(ct);); // 降低一层的缩放，和 cc->ModReduceInPlace 完全一样print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("Rotate(1)", ct = cc->EvalRotate(ct1, 1);); // 正数是左旋print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Rotate(-2)", ct = cc->EvalRotate(ct1, -2);); // 负数是右旋print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/Timer("Automorphism(1)", ct = cc->EvalAutomorphism(ct1, 1, evk_map);); // 不旋转print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(5)", ct = cc->EvalAutomorphism(ct1, 5, evk_map);); // 次数 4*batchSize 的子环上，数量 batchSize 的明文槽，由 (-5) 生成print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(25)", ct = cc->EvalAutomorphism(ct1, 25, evk_map);); // 5^2 mod 32print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(29)", ct = cc->EvalAutomorphism(ct1, 29, evk_map);); // 5^3 mod 32print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(17)", ct = cc->EvalAutomorphism(ct1, 17, evk_map););print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(21)", ct = cc->EvalAutomorphism(ct1, 21, evk_map););print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(9)", ct = cc->EvalAutomorphism(ct1, 9, evk_map););print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);Timer("Automorphism(13)", ct = cc->EvalAutomorphism(ct1, 13, evk_map););print_param(ct);cc->Decrypt(keys.secretKey, ct, &pt);print_pt(pt);/***************************************************************************************************/return 0;
}

结果

使用 `FIXEDMANUAL` 模切换

CPU Frequency = 2919133983CyclotomicOrder = 2048
RingDimension = 1024
Level = 0
Modulus = 3773962425391299449455856105110342697983390191203387110210545988123360807414630715088298851847265236596671914551723351177669086733035396939530237480795634800286060664833
ModulusBits = 561
RootOfUnity = 0
PlaintextModulus = 50
BatchSize = 8
PlaintextGenerator = 0Result: ((0.25,0.5), (0.5,1), (0.75,1.5), (1,2), (2,4), (3,6), (4,8), (5,10),  ... ); Estimated precision: 50 bits
Estimated precision in bits: 50Result: ((5,0), (4,0), (3,0), (2,0), (1,0), (0.75,0), (0.5,0), (0.25,0),  ... ); Estimated precision: 50 bits
Estimated precision in bits: 50ScalingFactor = 1.1259e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | Result: ((0.25,0), (0.5,0), (0.75,0), (1,0), (2,0), (3,0), (4,0), (5,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Result: ((5,0), (4,0), (3,0), (2,0), (1,0), (0.75,0), (0.5,0), (0.25,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <+>:        232698 cycles (  0.079715 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((5.25,0), (4.5,0), (3.75,0), (3,0), (3,0), (3.75,0), (4.5,0), (5.25,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Add #1>:           293886 cycles (  0.100676 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((5.25,0), (4.5,0), (3.75,0), (3,0), (3,0), (3.75,0), (4.5,0), (5.25,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Scale>:            218742 cycles (  0.074934 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ((1,0), (2,0), (3,0), (4,0), (8,0), (12,0), (16,0), (20,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Add #2>:           138844 cycles (  0.047563 ms)
ScalingFactor = 1.1258999e+15 | Level = 1 | Depth = 1 | SKDeg = 1 | 
Result: ((6,0), (6,0), (6,0), (6,0), (9,0), (12.75,0), (16.5,0), (20.25,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <*>:      88682107 cycles ( 30.379595 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ((1.25,0), (2,0), (2.25,0), (2,0), (2,0), (2.25,0), (2,0), (1.25,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Mult>:   83510222 cycles ( 28.607876 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ((1.25,0), (2,0), (2.25,0), (2,0), (2,0), (2.25,0), (2,0), (1.25,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Composed Mult #1>:      125045991 cycles ( 42.836674 ms)
ScalingFactor = 1.1258999e+15 | Level = 1 | Depth = 1 | SKDeg = 1 | 
Result: ((1.25,0), (2,0), (2.25,0), (2,0), (2,0), (2.25,0), (2,0), (1.25,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Composed Mult #2>:      126722970 cycles ( 43.411152 ms)
ScalingFactor = 1.1258999e+15 | Level = 2 | Depth = 1 | SKDeg = 1 | 
Result: ((6.25,0), (8,0), (6.75,0), (4,0), (2,0), (1.6875,0), (1,0), (0.3125,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Mult And Relinearize>:  100906735 cycles ( 34.567353 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ((1.25,0), (2,0), (2.25,0), (2,0), (2,0), (2.25,0), (2,0), (1.25,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Mult No Relin #1>:         477430 cycles (  0.163552 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 2 | 
Result: ((1.25,0), (2,0), (2.25,0), (2,0), (2,0), (2.25,0), (2,0), (1.25,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Mult No Relin #2>:        1077588 cycles (  0.369146 ms)
ScalingFactor = 1.4272477e+45 | Level = 0 | Depth = 3 | SKDeg = 3 | 
Result: ((6.25,0), (8,0), (6.75,0), (4,0), (2,0), (1.6875,0), (1,0), (0.3125,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Relinear>:      162462043 cycles ( 55.654192 ms)
ScalingFactor = 1.4272477e+45 | Level = 0 | Depth = 3 | SKDeg = 1 | 
Result: ((6.25,0), (8,0), (6.75,0), (4,0), (2,0), (1.6875,0), (1,0), (0.3125,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <ModReduce>:        390542 cycles (  0.133787 ms)
ScalingFactor = 1.2676506e+30 | Level = 1 | Depth = 2 | SKDeg = 1 | 
Result: ((6.25,0), (8,0), (6.75,0), (4,0), (2,0), (1.6875,0), (1,0), (0.3125,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Rescale>:          294642 cycles (  0.100935 ms)
ScalingFactor = 1.1258999e+15 | Level = 2 | Depth = 1 | SKDeg = 1 | 
Result: ((6.25,0), (8,0), (6.75,0), (4,0), (2,0), (1.6875,0), (1,0), (0.3125,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Rotate(1)>:      85623396 cycles ( 29.331780 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((0.5,0), (0.75,0), (1,0), (2,0), (3,0), (4,0), (5,0), (0.25,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Rotate(-2)>:     85083932 cycles ( 29.146977 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((4,0), (5,0), (0.25,0), (0.5,0), (0.75,0), (1,0), (2,0), (3,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(1)>:        84371770 cycles ( 28.903014 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((0.25,0), (0.5,0), (0.75,0), (1,0), (2,0), (3,0), (4,0), (5,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(5)>:        32532042 cycles ( 11.144415 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((0.5,0), (0.75,0), (1,0), (2,0), (3,0), (4,0), (5,0), (0.25,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(25)>:      102897566 cycles ( 35.249347 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((0.75,0), (1,0), (2,0), (3,0), (4,0), (5,0), (0.25,0), (0.5,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(29)>:      154811323 cycles ( 53.033305 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((1,0), (2,0), (3,0), (4,0), (5,0), (0.25,0), (0.5,0), (0.75,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(17)>:       79522168 cycles ( 27.241699 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((2,0), (3,0), (4,0), (5,0), (0.25,0), (0.5,0), (0.75,0), (1,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Automorphism(21)>:       72500099 cycles ( 24.836167 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((3,0), (4,0), (5,0), (0.25,0), (0.5,0), (0.75,0), (1,0), (2,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(9)>:        82870142 cycles ( 28.388605 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((4,0), (5,0), (0.25,0), (0.5,0), (0.75,0), (1,0), (2,0), (3,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(13)>:       84619741 cycles ( 28.987961 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((5,0), (0.25,0), (0.5,0), (0.75,0), (1,0), (2,0), (3,0), (4,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41

使用 `FLEXIBLEAUTO` 模切换

CPU Frequency = 2918792554CyclotomicOrder = 2048
RingDimension = 1024
Level = 0
Modulus = 3773962422988620268549387626283230612775297831378964384076336766063448441293238012679355400777370037636733749256861996141103106549700437881327817936926763295941309458433
ModulusBits = 560
RootOfUnity = 0
PlaintextModulus = 50
BatchSize = 8
PlaintextGenerator = 0Result: ((0.25,0.5), (0.5,1), (0.75,1.5), (1,2), (2,4), (3,6), (4,8), (5,10),  ... ); Estimated precision: 50 bits
Estimated precision in bits: 50Result: ((5,0), (4,0), (3,0), (2,0), (1,0), (0.75,0), (0.5,0), (0.25,0),  ... ); Estimated precision: 50 bits
Estimated precision in bits: 50ScalingFactor = 1.1259e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | Result: ((0.25,0), (0.5,0), (0.75,0), (1,0), (2,0), (3,0), (4,0), (5,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Result: ((5,0), (4,0), (3,0), (2,0), (1,0), (0.75,0), (0.5,0), (0.25,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <+>:        217702 cycles (  0.074586 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((5.25,0), (4.5,0), (3.75,0), (3,0), (3,0), (3.75,0), (4.5,0), (5.25,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Add #1>:           195658 cycles (  0.067034 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((5.25,0), (4.5,0), (3.75,0), (3,0), (3,0), (3.75,0), (4.5,0), (5.25,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Scale>:            336894 cycles (  0.115422 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ((1,0), (2,0), (3,0), (4,0), (8,0), (12,0), (16,0), (20,0),  ... ); Estimated precision: 39 bits
Estimated precision in bits: 39Running Time of <Add #2>:           474080 cycles (  0.162423 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ((6,0), (6,0), (6,0), (6,0), (9,0), (12.75,0), (16.5,0), (20.25,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <*>:     118372907 cycles ( 40.555437 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ((1.25,0), (2,0), (2.25,0), (2,0), (2,0), (2.25,0), (2,0), (1.25,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Mult>:  102517626 cycles ( 35.123300 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ((1.25,0), (2,0), (2.25,0), (2,0), (2,0), (2.25,0), (2,0), (1.25,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Composed Mult #1>:       68991449 cycles ( 23.636983 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ((1.25,0), (2,0), (2.25,0), (2,0), (2,0), (2.25,0), (2,0), (1.25,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Composed Mult #2>:       83294391 cycles ( 28.537277 ms)
ScalingFactor = 1.2676506e+30 | Level = 1 | Depth = 2 | SKDeg = 1 | 
Result: ((6.25,0), (8,0), (6.75,0), (4,0), (2,0), (1.6875,0), (1,0), (0.3125,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Mult And Relinearize>:   83116483 cycles ( 28.476324 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 1 | 
Result: ((1.25,0), (2,0), (2.25,0), (2,0), (2,0), (2.25,0), (2,0), (1.25,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Mult No Relin #1>:         661054 cycles (  0.226482 ms)
ScalingFactor = 1.2676506e+30 | Level = 0 | Depth = 2 | SKDeg = 2 | 
Result: ((1.25,0), (2,0), (2.25,0), (2,0), (2,0), (2.25,0), (2,0), (1.25,0),  ... ); Estimated precision: 40 bits
Estimated precision in bits: 40Running Time of <Mult No Relin #2>:        1388252 cycles (  0.475625 ms)
ScalingFactor = 1.2676506e+30 | Level = 1 | Depth = 2 | SKDeg = 3 | 
Result: ((6.25,0), (8,0), (6.75,0), (4,0), (2,0), (1.6875,0), (1,0), (0.3125,0),  ... ); Estimated precision: 38 bits
Estimated precision in bits: 38Running Time of <Relinear>:      161033258 cycles ( 55.171190 ms)
ScalingFactor = 1.2676506e+30 | Level = 1 | Depth = 2 | SKDeg = 1 | 
Result: ((6.25,0), (8,0), (6.75,0), (4,0), (2,0), (1.6875,0), (1,0), (0.3125,0),  ... ); Estimated precision: 38 bits
Estimated precision in bits: 38Running Time of <ModReduce>:          7684 cycles (  0.002633 ms)
ScalingFactor = 1.2676506e+30 | Level = 1 | Depth = 2 | SKDeg = 1 | 
Result: ((6.25,0), (8,0), (6.75,0), (4,0), (2,0), (1.6875,0), (1,0), (0.3125,0),  ... ); Estimated precision: 38 bits
Estimated precision in bits: 38Running Time of <Rescale>:            1646 cycles (  0.000564 ms)
ScalingFactor = 1.2676506e+30 | Level = 1 | Depth = 2 | SKDeg = 1 | 
Result: ((6.25,0), (8,0), (6.75,0), (4,0), (2,0), (1.6875,0), (1,0), (0.3125,0),  ... ); Estimated precision: 38 bits
Estimated precision in bits: 38Running Time of <Rotate(1)>:      26580736 cycles (  9.106758 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((0.5,0), (0.75,0), (1,0), (2,0), (3,0), (4,0), (5,0), (0.25,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Rotate(-2)>:     17406282 cycles (  5.963521 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((4,0), (5,0), (0.25,0), (0.5,0), (0.75,0), (1,0), (2,0), (3,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(1)>:        41046903 cycles ( 14.062974 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((0.25,0), (0.5,0), (0.75,0), (1,0), (2,0), (3,0), (4,0), (5,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(5)>:        37049927 cycles ( 12.693580 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((0.5,0), (0.75,0), (1,0), (2,0), (3,0), (4,0), (5,0), (0.25,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(25)>:       98023806 cycles ( 33.583684 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((0.75,0), (1,0), (2,0), (3,0), (4,0), (5,0), (0.25,0), (0.5,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Automorphism(29)>:       92120842 cycles ( 31.561284 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((1,0), (2,0), (3,0), (4,0), (5,0), (0.25,0), (0.5,0), (0.75,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Automorphism(17)>:       87246888 cycles ( 29.891432 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((2,0), (3,0), (4,0), (5,0), (0.25,0), (0.5,0), (0.75,0), (1,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41Running Time of <Automorphism(21)>:      136802988 cycles ( 46.869719 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((3,0), (4,0), (5,0), (0.25,0), (0.5,0), (0.75,0), (1,0), (2,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(9)>:       103675496 cycles ( 35.519995 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((4,0), (5,0), (0.25,0), (0.5,0), (0.75,0), (1,0), (2,0), (3,0),  ... ); Estimated precision: 42 bits
Estimated precision in bits: 42Running Time of <Automorphism(13)>:       65311725 cycles ( 22.376282 ms)
ScalingFactor = 1.1258999e+15 | Level = 0 | Depth = 1 | SKDeg = 1 | 
Result: ((5,0), (0.25,0), (0.5,0), (0.75,0), (1,0), (2,0), (3,0), (4,0),  ... ); Estimated precision: 41 bits
Estimated precision in bits: 41