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CS253 Random Numbers

Philosophy

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

• Stop trying to prove your superiority.
  • If you believe that you have something special that distinguishes you from machines, you’re talking religion, not CS.
• My dog is pretty random.
• You’re somewhat predictable.
  • An online rock-paper-scissors program beats people 60% of the time over more than a million games, because people are lousy at being random.

Old Stuff

• There are several C random number generators, of varying degrees of standardization:
  • rand(), srand()
  • random(), srandom()
  • drand48(), erand48(), lrand48(), nrand48(), mrand48(), jrand48(), srand48()
• They still work ok, but avoid them for new C++ code.
• They mix up generation and distribution something terrible.
• Also, each family has a separate seeding function.
• Also also, there’s no way to save/restore state!

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

• It’s fast, simple, and good enough for many tasks. However …
  • What happens if n is zero?
  • What number always follows 16807?
  • How many possible states does this RNG (Random Number Generator) have?

Overview

• In C++, random numbers have:
  • Generators:
    Generate uniformly-distributed random integers, typically zero or one to a big number.
  • Distributions:
    Take uniformly-distributed random integers, and transform them into other distributions with different ranges.
• Examples:
  • Picking a card (uniform, but discrete)
  • Rolling 3d6 (bell-shaped, but discrete)
  • Human height (bell-shaped, continuous)

Generators

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().

#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

• Here’s a different, 64-bit generator.
• Use .min() and .max() to find out the range of a given generator.

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

True randomness

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

1138115686
340295239
2964067141

• random_device is, ideally, truly random, and not pseudo-random.
• Intel computers have an RDRAND instruction.
• It might depend on random things like human typing intervals, network packets arrival times, or radioactive decay.
• If true randomness isn’t available, it resorts to pseudo-random numbers.
• It could pause waiting for randomness to become available.
• Use it sparingly.

Cloudflare

The hosting service Cloudflare uses a unique source of randomness.

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

• Great—we can “seed” the random number generator with a value.
• This way, we can reproduce our pseudo-random sequences.
• Consider random testing: we want to be able to reproduce the sequence if we find an error.
• How to choose the random seed?
  • It should probably be … random.

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

• You can seed with your process id.
• OK for casual use, but the seed is easily guessed.
• Process IDs are usually 15- or 16-bit quantities, so there are generally only 32768 or 65536 of them. Somebody could easily try them all.

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

• You can seed with a time-related value.
• Two runs may occur within the same second, and so produce identical random sequences.
• OK for casual use, but the seed is easily guessed.
• There are only 86,400 seconds in a day. Somebody could easily try them all.

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

• There are 86,400,000,000,000 nanoseconds in a day.

Better Seeding

• Many generators have more than 32 or 64 bits of state.
• Therefore, you can seed them with more than 32 or 64 bits.
• If you’re doing something very important, and somebody guessing your seed, and hence predicting your sequence, would be catastrophic:
  • on-line poker 🂺 🂻 🂽 🂾 🂱
  • encryption of military communications ⚔️ 🔫 💣 🥆 ☢️
  • encrypted email re: extra-marital affairs 💔
• That’s beyond the scope of this discussion.

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.

• Great, so we know how to generate a number 1…2,147,483,646
  or perhaps 0…18,446,744,073,709,551,615
• How often do we want to do that?
• Sometimes, we want integers with different ranges.
• Or, perhaps we want floating-point numbers.
• Maybe spread out linearly, or a bell-shaped curve, Poisson, etc.
• This is a job for a distribution.

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

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 

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.