CS253: Software Development with C++

Fall 2021

Random Numbers

Show Lecture.RandomNumbers as a slide show.

CS253 Random Numbers

Philosophy

“Computers can’t do anything truly random. Only a person can do that.”

Old Stuff

Traditional Method

Traditional random number generators work like this:

unsigned long n = 1;
for (int i=0; i<5; i++) {
    n = n * 16807 % 2147483647;
    cout << n << '\n';
}
16807
282475249
1622650073
984943658
1144108930

Overview

Generators

EngineDescription
default_random_engineDefault random engine
minstd_randMinimal Standard minstd_rand generator
minstd_rand0Minimal Standard minstd_rand0 generator
mt19937Mersenne Twister 19937 generator
mt19937_64Mersenne Twister 19937 generator (64 bit)
ranlux24_baseRanlux 24 base generator
ranlux48_baseRanlux 48 base generator
ranlux24Ranlux 24 generator
ranlux48Ranlux 48 generator
knuth_bKnuth-B generator
random_deviceTrue random number generator

Default Engine

Define a random-number generator, and use () to generate a number. This is not a function call, because gen is an object, not a function. It’s operator().







🤨 That sequence looks familiar …

#include <random>
#include <iostream>
using namespace std;

int main() {
    default_random_engine gen;
    for (int i=0; i<5; i++)
        cout << gen() << '\n';
}
16807
282475249
1622650073
984943658
1144108930

I won’t bother with the #includes in subsequent examples.

Mersenne Twister

mt19937_64 gen;
cout << "range is " << gen.min() << "…" << gen.max() << "\n\n";
for (int i=0; i<3; i++)
    cout << gen() << '\n';
range is 0…18446744073709551615

14514284786278117030
4620546740167642908
13109570281517897720

Ranges

Generators have varying ranges:

ranlux24 rl;
minstd_rand mr;
random_device rd;
mt19937_64 mt;

cout << "ranlux24:      " << rl.min() << "…" << rl.max() << '\n'
     << "minstd_rand:   " << mr.min() << "…" << mr.max() << '\n'
     << "random_device: " << rd.min() << "…" << rd.max() << '\n'
     << "mt19937_64:    " << mt.min() << "…" << mt.max() << '\n';
ranlux24:      0…16777215
minstd_rand:   1…2147483646
random_device: 0…4294967295
mt19937_64:    0…18446744073709551615

Hey, look! Zero is not a possible return value for minstd_rand.

Save/Restore

A generator can save & restore state to an I/O stream:

ranlux24 gen;
cout << gen() << ' ';
cout << gen() << endl;
ofstream("state") << gen;
system("wc -c state");
cout << gen() << ' ';
cout << gen() << '\n';
ifstream("state") >> gen;
cout << gen() << ' ';
cout << gen() << '\n';
15039276 16323925
209 state
14283486 7150092
14283486 7150092
endl! Isn’t that a sin? 😈 🔥

Needed to flush output before wc ran.

True randomness

random_device a, b, c;
cout << a() << '\n'
     << b() << '\n'
     << c() << '\n';
1138115686
340295239
2964067141

Cloudflare

The hosting service Cloudflare uses a unique source of randomness.

a picture of Cloudfare’s wall of lava lamps

Seeding

minstd_rand a, b, c(123);
cout << a() << ' ' << a() << '\n';
cout << b() << ' ' << b() << '\n';
cout << c() << ' ' << c() << '\n';
48271 182605794
48271 182605794
5937333 985676192

Seed with process ID

auto seed = getpid();
minstd_rand a(seed);
for (int i=0; i<5; i++)
    cout << a() << '\n';
397287351
428752411
1007725242
1253068385
839610933

Seed with time

// seconds since start of 1970
auto seed = time(nullptr);
minstd_rand a(seed);
for (int i=0; i<5; i++)
    cout << a() << '\n';
1448276731
583437663
1018883915
874977371
1471789992

Seed with more accurate time

Nanoseconds make more possibilities:

auto seed = chrono::high_resolution_clock::now()
            .time_since_epoch().count();
cout << "Seed: " << seed << '\n';
minstd_rand a(seed);
for (int i=0; i<5; i++)
    cout << a() << '\n';
Seed: 1719760853764503232
1511995495
1155312203
72522070
314496360
491892917

Better Seeding

Seed with random_device

random_device rd;
auto seed = rd();
minstd_rand0 a(seed);
for (int i=0; i<5; i++)
    cout << a() << '\n';
794713811
1554220784
1945118227
458482908
550909320

You can seed with random_device, if you know that it’s truly random.

Not good enough.

Caution

Resist the urge to create your own distribution using division or modulus. It’s harder than you think. Just use the standard distributions.

minstd_rand r;
int first_half = 0;
for (int i=0; i<100'000'000; i++)
    if (r() % 1'000'000'000 < 500'000'000)
        first_half++;
cout << first_half << '\n';
53435616
Why didn’t that work?

minstd_rand, on this computer, produces a number 1…2,147,483,646. If you take that mod a billion, the range 0…147,473,646 gets represented an extra time.

Distributions

uniform_int_distribution

auto seed = random_device()();  //❓❓❓
mt19937 gen(seed);
uniform_int_distribution<int> dist(1,6);
for (int y=0; y<10; y++) {
    for (int x=0; x<40; x++)
        cout << dist(gen) << ' ';
    cout << '\n';
}
6 5 4 2 1 4 6 6 5 6 5 5 4 5 4 1 4 2 4 6 3 3 6 1 1 1 4 2 5 6 1 4 5 2 4 3 1 2 4 5 
4 2 2 3 1 2 6 1 6 1 5 2 1 1 5 3 3 2 5 4 2 6 3 4 1 3 3 2 6 4 2 1 2 5 3 4 2 4 6 3 
6 3 6 3 2 5 3 2 4 5 4 2 1 2 6 3 6 4 6 6 3 5 4 5 2 4 3 3 1 5 2 3 4 3 6 6 2 6 1 2 
2 6 2 4 3 4 4 1 1 6 5 6 2 6 3 4 1 6 4 1 1 6 3 3 1 1 1 5 4 4 6 2 3 2 5 6 4 6 2 6 
6 6 1 4 5 1 2 4 1 6 4 5 5 3 2 5 2 4 2 5 3 3 4 3 1 5 4 1 2 6 5 5 5 4 3 3 2 4 4 5 
6 5 4 6 5 4 4 5 4 5 2 2 6 6 1 6 2 1 2 2 5 6 4 4 4 2 4 5 1 5 5 4 4 3 4 6 1 3 3 4 
3 3 6 3 5 6 2 4 6 4 3 2 2 5 3 4 2 3 6 2 3 3 6 1 6 3 4 5 2 5 5 4 2 3 5 5 6 4 4 5 
5 5 3 1 3 1 4 2 3 4 6 3 2 6 4 6 5 3 4 1 6 1 6 2 2 6 3 5 5 1 3 3 3 2 4 2 2 1 1 5 
4 2 1 6 1 5 6 4 2 6 6 1 5 6 4 1 3 5 2 4 2 3 3 5 1 4 2 4 3 1 2 1 5 2 4 4 4 3 3 5 
4 3 2 5 5 3 2 4 3 1 2 2 1 4 4 5 4 1 6 2 4 3 2 4 4 5 5 3 3 1 4 6 2 1 4 5 6 1 1 4 

uniform_real_distribution

auto seed = random_device()();
ranlux48 gen(seed);
uniform_real_distribution<> dist(18.0, 25.0);
for (int y=0; y<5; y++) {
    for (int x=0; x<10; x++)
        cout << fixed << setprecision(3) << dist(gen) << ' ';
    cout << '\n';
}
21.292 18.907 22.207 23.716 20.900 18.694 18.371 24.751 19.930 22.258 
21.844 19.363 23.243 19.619 21.130 20.055 21.148 18.138 22.479 24.941 
22.244 21.285 22.918 21.111 23.069 23.236 23.846 24.506 19.088 23.030 
23.744 23.468 21.771 20.338 20.497 21.138 24.185 22.342 20.351 20.965 
24.963 24.429 22.905 20.303 18.190 22.935 18.891 21.849 23.319 20.462 
OMG—what’s that <> doing there?

uniform_real_distribution’s template argument defaults to double, because … real.

Boolean Values

Yield true 42% of time:

random_device rd;
knuth_b gen(rd());
bernoulli_distribution dist(0.42);
constexpr int nrolls = 1'000'000;

int count=0;
for (int i=0; i<nrolls; i++)
    if (dist(gen))
        count++;

cout << "true: " << count*100.0/nrolls << "%\n";
true: 42.1154%

Histogram

random_device rd;
mt19937_64 gen(rd());
normal_distribution<> dist(21.5, 1.5);
map<int,int> tally;
for (int i=0; i<10000; i++)
    tally[dist(gen)]++;
for (auto p : tally)
    cout << p.first << ": " << string(p.second/100,'#') << '\n';
15: 
16: 
17: 
18: ###
19: ###########
20: #####################
21: ###########################
22: ####################
23: ###########
24: ###
25: 
26: 
27: 

Passwords

random_device rd;
auto seed = rd();
ranlux24 gen(seed);
uniform_int_distribution<char> dist('A','~');
for (int y=0; y<8; y++) {
    string pw;
    for (int x=0; x<32; x++)
        pw += dist(gen);
    cout << "Password: " << pw << '\n';
}
Password: AcICjeSLSuMDeEKDQqDbaCO]roNnV^Sh
Password: L`NiDzKptyPLR]~nR{RnYZ`fbZfMUeJa
Password: Xz|\sKzvHu_UXpQYmXQLDVMA~^Vwm~xW
Password: eT^}X\F{PvUPz~m\{JWBDOSpAZLrKH|t
Password: g`rMr]C||KXATzir{gAloFce~\ZFzuCK
Password: Ddn]eciSNKwG`CswMC^gvuieAasPJeNX
Password: Aon_[~`JCXaAqShzh][D~vgx|Lcf~qgh
Password: UMl\BwCQfa~|[{OnOEjYAUzRHu`^qCZe

Even though we’re using uniform_int_distribution, which has int right there in its name, it’s uniform_int_distribution<char>, so we get characters. Think of them as 8-bit integers that display differently.