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[3.2] Create a simple benchmarking framework and use it to benchmark crypto primitives #17
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Original file line number | Diff line number | Diff line change |
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file(GLOB BENCHMARK "*.cpp") | ||
add_executable( benchmark ${BENCHMARK} ) | ||
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target_link_libraries( benchmark fc Boost::program_options ) | ||
target_include_directories( benchmark PUBLIC | ||
"${CMAKE_CURRENT_SOURCE_DIR}" | ||
) |
Original file line number | Diff line number | Diff line change |
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#include <iostream> | ||
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#include <fc/crypto/alt_bn128.hpp> | ||
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#include <benchmark.hpp> | ||
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namespace benchmark { | ||
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using bytes = std::vector<char>; | ||
using g1g2_pair = std::vector<std::string>; | ||
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void add_benchmarking() { | ||
std::vector<unsigned char> point1_raw = { | ||
12,175,215,144,98,95,151,228,179,85,31,170,172,159,40,255,250,252,68,28,235,65,172,180,69,164,153,29,187,239,220,201, // x | ||
18,102,219,76,79,148,120,28,39,149,42,172,41,248,120,249,255,69,42,51,160,239,13,219,239,183,77,174,217,158,130,10,}; // y | ||
std::vector<unsigned char> point2_raw = { | ||
12,144,10,32,104,85,103,36,222,232,48,152,108,217,40,145,230,48,8,54,0,7,134,164,7,10,139,110,95,205,124,121, // x | ||
22,254,176,251,18,168,78,220,142,100,102,113,58,176,83,186,212,62,154,138,235,135,34,46,237,117,54,36,198,40,79,73,}; // y | ||
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bytes point1, point2; | ||
point1.insert(point1.begin(), point1_raw.begin(), point1_raw.end()); | ||
point2.insert(point2.begin(), point2_raw.begin(), point2_raw.end()); | ||
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auto f = [&]() { | ||
auto res = fc::alt_bn128_add(point1, point2); | ||
if (std::holds_alternative<fc::alt_bn128_error>(res)) { | ||
std::cout << "alt_bn128_add failed: " | ||
<< (int)std::get<fc::alt_bn128_error>(res) << std::endl; | ||
} | ||
}; | ||
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benchmarking("alt_bn128_add", f); | ||
} | ||
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void mul_benchmarking() { | ||
std::vector<unsigned char> point_raw = { | ||
12,175,215,144,98,95,151,228,179,85,31,170,172,159,40,255,250,252,68,28,235,65,172,180,69,164,153,29,187,239,220,201, // x | ||
18,102,219,76,79,148,120,28,39,149,42,172,41,248,120,249,255,69,42,51,160,239,13,219,239,183,77,174,217,158,130,10,}; // y | ||
std::vector<unsigned char> scaler_raw = { | ||
25,62,182,170,104,140,135,90,37,150,0,77,2,77,146,71,54,101,113,69,177,216,157,4,229,213,33,215,169,99,150,91, }; // scaler size 256 bits | ||
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bytes point, scaler; | ||
point.insert(point.begin(), point_raw.begin(), point_raw.end()); | ||
scaler.insert(scaler.begin(), scaler_raw.begin(), scaler_raw.end()); | ||
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auto f = [&]() { | ||
auto res = fc::alt_bn128_mul(point, scaler); | ||
if (std::holds_alternative<fc::alt_bn128_error>(res)) { | ||
std::cout << "alt_bn128_mul failed: " | ||
<< (int)std::get<fc::alt_bn128_error>(res) << std::endl; | ||
} | ||
}; | ||
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benchmarking("alt_bn128_mul", f); | ||
} | ||
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void pair_benchmarking() { | ||
std::vector<unsigned char> g1_g2_pairs_raw = { | ||
/* pair 1 */ | ||
12,175,215,144,98,95,151,228,179,85,31,170,172,159,40,255,250,252,68,28,235,65,172,180,69,164,153,29,187,239,220,201,18,102,219,76,79,148,120,28,39,149,42,172,41,248,120,249,255,69,42,51,160,239,13,219,239,183,77,174,217,158,130,10, | ||
46,24,234,176,148,82,198,136,44,30,157,18,217,25,49,241,63,197,2,151,14,150,136,210,114,4,74,124,145,225,131,139,13,60,0,50,236,103,39,15,150,226,246,189,209,113,0,46,151,39,41,6,87,119,228,213,45,225,29,234,161,110,43,87,3,237,74,227,93,23,171,129,49,118,228,173,154,111,168,220,161,46,140,103,155,56,168,207,254,90,228,76,188,232,25,34,44,102,159,35,193,216,177,31,131,247,226,19,170,222,26,227,86,37,81,149,202,144,188,126,120,168,9,195,32,169,147,161, | ||
/* pair,2,*/ | ||
12,144,10,32,104,85,103,36,222,232,48,152,108,217,40,145,230,48,8,54,0,7,134,164,7,10,139,110,95,205,124,121,22,254,176,251,18,168,78,220,142,100,102,113,58,176,83,186,212,62,154,138,235,135,34,46,237,117,54,36,198,40,79,73, | ||
0,186,148,242,227,252,208,242,165,128,1,160,59,2,6,195,30,54,69,118,8,28,223,92,254,98,110,219,90,92,7,113,26,8,119,98,122,61,119,91,216,107,29,179,122,218,203,6,84,109,119,140,44,12,57,157,203,90,14,108,218,54,94,44,42,77,118,169,232,192,189,60,129,213,156,157,215,160,91,214,138,130,119,145,167,89,237,149,225,85,151,33,89,181,224,235,2,91,213,225,37,226,214,14,255,84,254,117,142,211,28,164,78,254,64,11,126,217,25,124,203,220,52,181,39,32,78,132, | ||
/* pair 3 */ | ||
48,58,116,183,56,3,62,111,104,127,0,7,202,204,146,2,236,192,214,213,231,100,12,17,104,47,76,49,149,39,35,179,2,162,218,104,52,130,61,134,223,111,123,225,186,122,171,2,144,236,110,29,26,142,113,183,238,44,130,30,76,212,52,15, | ||
19,206,16,79,220,158,153,194,12,48,10,79,173,10,97,199,113,25,255,87,220,3,102,235,164,170,50,240,177,237,223,205,11,75,211,28,143,229,192,35,171,167,172,238,138,235,82,134,111,165,144,29,118,150,179,21,158,9,202,2,242,109,33,148,14,188,33,145,213,186,0,126,178,10,131,168,121,66,121,193,106,209,176,34,176,41,145,227,3,55,245,150,82,218,232,155,31,153,213,183,157,2,159,247,25,69,49,215,219,36,46,5,192,205,201,182,72,189,84,62,61,47,136,81,51,65,231,161, | ||
/* pair 4 */ | ||
27,31,150,92,17,82,135,210,46,161,0,24,16,199,200,165,29,165,157,168,222,9,83,17,44,27,64,226,208,112,223,128,20,253,182,130,45,130,249,9,168,206,153,86,197,214,59,248,241,191,93,113,70,113,247,244,43,214,240,246,24,38,33,50, | ||
23,76,241,195,191,99,69,237,123,173,212,42,74,85,138,108,39,80,105,135,226,11,84,237,39,73,180,224,42,230,246,76,26,61,86,57,253,213,223,2,93,42,38,186,38,38,206,38,209,138,153,181,57,89,16,187,234,2,110,23,12,14,252,240,32,231,237,147,16,62,220,6,160,64,154,167,57,238,243,112,255,35,80,36,100,173,21,58,96,244,77,245,75,202,24,63,35,178,46,215,219,69,120,221,70,119,111,195,145,45,109,84,206,76,254,205,98,167,30,72,117,5,128,129,156,128,196,192, | ||
/* pair 5 */ | ||
33,230,106,254,43,77,114,126,9,205,25,63,9,170,62,237,149,102,198,184,60,31,185,213,103,120,71,249,28,106,238,107,1,133,60,180,58,152,18,177,142,248,46,186,90,34,94,200,235,158,242,255,209,32,143,133,102,195,19,107,153,42,207,228, | ||
30,43,165,137,133,231,242,156,65,126,94,159,26,27,78,164,165,159,225,150,220,158,247,118,118,170,195,122,12,125,8,34,39,57,159,231,167,233,3,47,243,142,174,224,162,38,147,65,74,191,98,144,126,125,81,177,75,140,161,27,34,97,46,249,8,65,220,15,139,144,78,12,72,239,54,163,39,6,30,147,154,208,89,111,170,126,6,214,32,181,195,140,215,53,218,239,44,43,229,0,169,242,16,227,163,185,224,21,142,203,135,66,60,53,91,80,48,112,8,13,226,206,58,249,163,52,196,53, | ||
/* pair 6 */ | ||
14,157,100,136,255,206,130,101,56,187,235,39,67,209,67,124,40,174,153,135,155,26,166,170,118,193,244,21,70,71,120,150,3,189,254,120,183,98,23,175,175,28,195,233,165,224,35,95,247,132,40,64,26,208,70,206,247,92,134,105,118,181,21,242, | ||
16,159,200,208,93,99,234,139,8,222,239,67,3,187,15,125,5,144,190,196,146,171,15,218,92,36,37,130,2,201,241,182,36,31,104,164,95,29,73,231,87,50,174,142,209,72,31,75,48,13,70,237,118,193,39,85,97,88,228,50,77,163,209,112,7,202,214,212,235,15,224,247,229,179,101,15,72,130,125,8,62,227,46,20,109,53,26,168,10,237,222,85,169,133,181,196,25,36,112,182,41,208,39,132,100,181,199,189,44,99,238,140,6,187,19,197,99,93,243,63,159,132,211,0,230,118,153,54, | ||
/* pair 7 */ | ||
31,60,154,8,188,169,224,107,69,183,223,15,60,32,61,51,81,226,85,38,72,88,241,202,216,204,253,98,0,250,116,167,21,255,0,81,80,3,108,196,240,102,114,33,33,122,116,155,188,19,232,217,42,217,54,195,91,187,88,188,249,218,240,146, | ||
12,28,171,60,8,166,151,19,125,237,128,127,207,183,19,1,188,189,105,181,170,172,8,180,30,175,4,108,191,252,197,161,37,122,14,151,117,34,172,227,236,154,84,99,189,197,91,46,43,148,174,218,83,61,196,48,131,45,242,61,92,242,56,251,39,125,111,11,24,37,76,62,151,59,183,3,156,187,249,232,229,56,230,171,55,2,246,70,74,102,0,156,185,237,126,90,0,30,248,164,199,146,66,237,158,134,139,168,128,189,126,55,35,198,48,18,112,40,63,70,254,246,88,111,254,67,213,223, | ||
/* pair 8 */ | ||
27,106,52,232,120,148,201,234,161,123,171,160,223,151,6,105,206,47,94,147,47,22,27,21,248,20,223,51,116,192,29,221,25,149,109,33,131,198,32,101,157,94,62,164,205,151,23,108,214,43,231,7,95,228,183,1,242,205,19,62,124,79,48,203, | ||
37,186,150,204,63,241,145,219,169,250,247,76,247,18,135,175,140,58,209,133,113,144,185,191,101,197,252,253,50,126,188,177,45,103,105,194,21,229,145,210,89,135,244,248,238,64,102,155,129,232,252,32,208,82,51,131,113,100,25,174,253,19,211,18,6,77,66,232,173,42,146,120,106,72,92,144,51,233,117,184,113,60,33,118,209,83,119,93,170,253,126,41,58,13,85,111,30,233,117,238,183,18,219,38,216,163,244,98,219,254,156,189,57,246,220,178,17,190,88,84,13,41,8,236,181,155,85,198, | ||
/* pair 9 */ | ||
25,62,182,170,104,140,135,90,37,150,0,77,2,77,146,71,54,101,113,69,177,216,157,4,229,213,33,215,169,99,150,91,5,185,157,10,152,12,220,171,39,188,1,48,87,129,192,101,36,179,99,212,123,207,13,76,89,78,8,154,75,239,117,189, | ||
19,87,226,242,101,183,176,42,195,135,40,225,195,207,13,62,60,37,28,251,13,165,96,198,173,250,251,107,25,141,77,68,45,115,128,233,135,104,231,12,192,206,60,249,137,191,114,141,3,235,34,189,208,170,23,59,231,12,197,116,5,188,52,20,11,153,186,15,96,240,54,123,212,242,168,141,151,187,17,38,185,238,196,242,26,194,193,18,34,107,95,238,108,154,140,224,13,76,71,149,81,56,85,83,56,59,203,183,170,10,167,200,128,135,105,19,93,203,131,146,116,209,62,27,81,80,234,172, | ||
/* pair 10 */ | ||
30,204,212,77,212,159,135,173,195,194,165,112,62,248,180,204,66,73,253,99,65,111,39,171,19,211,171,203,35,66,146,20,33,241,46,6,167,133,80,76,238,165,59,232,120,58,211,157,50,212,86,191,95,6,134,164,36,227,79,58,119,98,108,171, | ||
48,49,12,141,166,151,158,56,136,255,197,138,114,195,39,59,71,236,82,57,149,249,170,55,187,95,193,171,14,124,45,87,7,157,47,178,153,237,194,157,142,194,100,14,40,51,61,201,244,93,61,196,154,59,14,135,209,72,102,186,14,228,228,152,40,246,109,82,93,249,92,105,191,121,77,108,20,87,3,87,167,173,171,255,189,34,155,239,218,95,181,153,222,20,120,195,27,28,47,82,113,3,218,129,54,210,185,165,206,99,126,61,217,19,237,5,12,90,148,246,128,231,63,53,37,223,204,195 | ||
}; | ||
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// benchmarking 1 pair of points | ||
bytes g1_g2_1_pair; | ||
g1_g2_1_pair.insert(g1_g2_1_pair.begin(), g1_g2_pairs_raw.begin(), g1_g2_pairs_raw.begin() + 384); | ||
auto f_1_pair = [&]() { | ||
auto res = fc::alt_bn128_pair(g1_g2_1_pair, [](){}); | ||
if (std::holds_alternative<fc::alt_bn128_error>(res)) { | ||
std::cout << "alt_bn128_pair 1 pair failed: " | ||
<< (int)std::get<fc::alt_bn128_error>(res) << std::endl; | ||
} | ||
}; | ||
benchmarking("alt_bn128_pair (1 pair)", f_1_pair); | ||
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// benchmarking 10 pair of points | ||
bytes g1_g2_10_pairs; | ||
g1_g2_10_pairs.insert(g1_g2_10_pairs.begin(), g1_g2_pairs_raw.begin(), g1_g2_pairs_raw.end()); | ||
auto f_10_pairs = [&]() { | ||
auto res = fc::alt_bn128_pair(g1_g2_10_pairs, [](){}); | ||
if (std::holds_alternative<fc::alt_bn128_error>(res)) { | ||
std::cout << "alt_bn128_pair 10 pairs failed: " | ||
<< (int)std::get<fc::alt_bn128_error>(res) << std::endl; | ||
} | ||
}; | ||
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benchmarking("alt_bn128_pair (10 pairs)", f_10_pairs); | ||
} | ||
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void alt_bn_128_benchmarking() { | ||
add_benchmarking(); | ||
mul_benchmarking(); | ||
pair_benchmarking(); | ||
} | ||
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} // benchmark |
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#include <chrono> | ||
#include <iostream> | ||
#include <iomanip> | ||
#include <locale> | ||
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#include <benchmark.hpp> | ||
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namespace benchmark { | ||
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// update this map when a new feature is supported | ||
// key is the name and value is the function doing benchmarking | ||
std::map<std::string, std::function<void()>> features { | ||
{ "alt_bn_128", alt_bn_128_benchmarking }, | ||
{ "modexp", modexp_benchmarking }, | ||
{ "key", key_benchmarking }, | ||
{ "hash", hash_benchmarking }, | ||
}; | ||
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// values to control cout format | ||
constexpr auto name_width = 28; | ||
constexpr auto runs_width = 5; | ||
constexpr auto time_width = 13; | ||
constexpr auto ns_width = 2; | ||
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uint32_t num_runs = 1; | ||
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std::map<std::string, std::function<void()>> get_features() { | ||
return features; | ||
} | ||
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void set_num_runs(uint32_t runs) { | ||
num_runs = runs; | ||
} | ||
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void printt_header() { | ||
std::cout << std::left << std::setw(name_width) << "function" | ||
<< std::setw(runs_width) << "runs" | ||
<< std::setw(time_width + ns_width) << std::right << "average" | ||
<< std::setw(time_width + ns_width) << "minimum" | ||
<< std::setw(time_width + ns_width) << "maximum" | ||
<< std::endl << std::endl; | ||
} | ||
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void print_results(std::string name, uint32_t runs, uint64_t total, uint64_t min, uint64_t max) { | ||
std::cout.imbue(std::locale("")); | ||
std::cout | ||
<< std::setw(name_width) << std::left << name | ||
<< std::setw(runs_width) << runs | ||
// std::fixed for not printing 1234 in 1.234e3. | ||
// setprecision(0) for not printing fractions | ||
<< std::right << std::fixed << std::setprecision(0) | ||
<< std::setw(time_width) << total/runs << std::setw(ns_width) << " ns" | ||
<< std::setw(time_width) << min << std::setw(ns_width) << " ns" | ||
<< std::setw(time_width) << max << std::setw(ns_width) << " ns" | ||
<< std::endl; | ||
} | ||
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void benchmarking(std::string name, const std::function<void()>& func) { | ||
uint64_t total {0}, min {std::numeric_limits<uint64_t>::max()}, max {0}; | ||
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for (auto i = 0U; i < num_runs; ++i) { | ||
auto start_time = std::chrono::high_resolution_clock::now(); | ||
func(); | ||
auto end_time = std::chrono::high_resolution_clock::now(); | ||
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uint64_t duration = std::chrono::duration_cast<std::chrono::nanoseconds>(end_time - start_time).count(); | ||
total += duration; | ||
min = std::min(min, duration); | ||
max = std::max(max, duration); | ||
} | ||
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print_results(name, num_runs, total, min, max); | ||
} | ||
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} // benchmark |
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#pragma once | ||
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#include <functional> | ||
#include <map> | ||
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namespace benchmark { | ||
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void set_num_runs(uint32_t runs); | ||
std::map<std::string, std::function<void()>> get_features(); | ||
void printt_header(); | ||
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void alt_bn_128_benchmarking(); | ||
void modexp_benchmarking(); | ||
void key_benchmarking(); | ||
void hash_benchmarking(); | ||
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void benchmarking(std::string name, const std::function<void()>& func); | ||
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} // benchmark |
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#include <fc/crypto/hex.hpp> | ||
#include <fc/crypto/sha3.hpp> | ||
#include <fc/utility.hpp> | ||
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#include <benchmark.hpp> | ||
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using namespace fc; | ||
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namespace benchmark { | ||
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void hash_benchmarking() { | ||
std::string small_message = "abcdefghijklmnopqrstuvwxyz0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ01"; | ||
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// build a large message | ||
constexpr auto large_msg_size = 4096; | ||
std::string large_message; | ||
large_message.reserve(large_msg_size); | ||
uint32_t num_concats = large_msg_size/small_message.length(); | ||
for (uint32_t i = 0; i < num_concats; ++i) { | ||
large_message += small_message; | ||
} | ||
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auto hash_small_msg = [&]() { | ||
fc::sha3::hash(small_message, true); | ||
}; | ||
benchmarking("sha3-256 (" + std::to_string(small_message.length()) + " bytes)", hash_small_msg); | ||
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auto hash_large_msg = [&]() { | ||
fc::sha3::hash(large_message, true); | ||
}; | ||
benchmarking("sha3-256 (" + std::to_string(large_message.length()) + " bytes)", hash_large_msg); | ||
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auto keccak_small_msg = [&]() { | ||
fc::sha3::hash(small_message, false); | ||
}; | ||
benchmarking("keccak256 (" + std::to_string(small_message.length()) + " bytes)", keccak_small_msg); | ||
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auto keccak_large_msg = [&]() { | ||
fc::sha3::hash(large_message, false); | ||
}; | ||
benchmarking("keccak256 (" + std::to_string(large_message.length()) + " bytes)", keccak_large_msg); | ||
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} | ||
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} // benchmark |
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#include <fc/crypto/public_key.hpp> | ||
#include <fc/crypto/private_key.hpp> | ||
#include <fc/crypto/signature.hpp> | ||
#include <fc/utility.hpp> | ||
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#include <benchmark.hpp> | ||
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using namespace fc::crypto; | ||
using namespace fc; | ||
using namespace std::literals; | ||
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namespace benchmark { | ||
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void k1_benchmarking() { | ||
auto payload = "Test Cases"; | ||
auto digest = sha256::hash(payload, const_strlen(payload)); | ||
auto key = private_key::generate<ecc::private_key_shim>(); | ||
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auto sign_f = [&]() { | ||
key.sign(digest); | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I would recommend disabling the canonical requirement for the signing. Otherwise the value could swing widely based on the private key you happened to randomly generate. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Good point! I will change to use fixed private keys so we can compare the values consistently across different runs. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. if you just pass false here for the second parameter, it'll be constant time for all private keys anyways There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Thanks! I am benchmarking both canonical and non-canonical cases. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. imo there is very little utility in the canonical case -- I would never look at it personally. It's dependent on both the private key and payload, both of which are fixed here, thus making the time spent always be some constant multiplied with the non-canonical time. Up to you though. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I will remove the canonical case. |
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}; | ||
benchmarking("k1_sign", sign_f); | ||
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auto sig = key.sign(digest); | ||
auto recover_f = [&]() { | ||
public_key(sig, digest);; | ||
}; | ||
benchmarking("k1_recover", recover_f); | ||
} | ||
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void r1_benchmarking() { | ||
auto payload = "Test Cases"; | ||
auto digest = sha256::hash(payload, const_strlen(payload)); | ||
auto key = private_key::generate<r1::private_key_shim>(); | ||
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auto sign_f = [&]() { | ||
key.sign(digest); | ||
}; | ||
benchmarking("r1_sign", sign_f); | ||
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auto sig = key.sign(digest); | ||
auto recover_f = [&]() { | ||
public_key(sig, digest);; | ||
}; | ||
benchmarking("r1_recover", recover_f); | ||
} | ||
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static fc::crypto::webauthn::signature make_webauthn_sig(const fc::crypto::r1::private_key& priv_key, | ||
std::vector<uint8_t>& auth_data, | ||
const std::string& json) { | ||
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//webauthn signature is sha256(auth_data || client_data_hash) | ||
fc::sha256 client_data_hash = fc::sha256::hash(json); | ||
fc::sha256::encoder e; | ||
e.write((char*)auth_data.data(), auth_data.size()); | ||
e.write(client_data_hash.data(), client_data_hash.data_size()); | ||
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r1::compact_signature sig = priv_key.sign_compact(e.result()); | ||
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char buff[8192]; | ||
datastream<char*> ds(buff, sizeof(buff)); | ||
fc::raw::pack(ds, sig); | ||
fc::raw::pack(ds, auth_data); | ||
fc::raw::pack(ds, json); | ||
ds.seekp(0); | ||
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fc::crypto::webauthn::signature ret; | ||
fc::raw::unpack(ds, ret); | ||
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return ret; | ||
} | ||
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void wa_benchmarking() { | ||
static const r1::private_key priv = fc::crypto::r1::private_key::generate(); | ||
static const fc::sha256 d = fc::sha256::hash("sup"s); | ||
static const fc::sha256 origin_hash = fc::sha256::hash("fctesting.invalid"s); | ||
std::string json = "{\"origin\":\"https://fctesting.invalid\",\"type\":\"webauthn.get\", \"challenge\":\"" + fc::base64url_encode(d.data(), d.data_size()) + "\"}"; | ||
std::vector<uint8_t> auth_data(37); | ||
memcpy(auth_data.data(), origin_hash.data(), sizeof(origin_hash)); | ||
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auto sign = [&]() { | ||
make_webauthn_sig(priv, auth_data, json); | ||
}; | ||
benchmarking("webauthn_sign", sign); | ||
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auto sig = make_webauthn_sig(priv, auth_data, json); | ||
auto recover= [&]() { | ||
sig.recover(d, true); | ||
}; | ||
benchmarking("webauthn_recover", recover); | ||
} | ||
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void key_benchmarking() { | ||
k1_benchmarking(); | ||
r1_benchmarking(); | ||
wa_benchmarking(); | ||
} | ||
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} // benchmark |
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Is this supposed to be
printt_header
orprint_header
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Thanks for your sharp eyes! Sorry for this copy/paste error.