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CS253 Random Numbers

Inclusion

To use C++ random numbers, you need to:

    
#include <random>

To use old C random numbers (don’t ), you need to:

    
#include <cstdlib>

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

Patron Saint of Randomness

• 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';

778082440
2409031190
4114114827

• 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';

2128048799
310805931
592337359
1119380131
762261334

• 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';

1645474067
1848520215
1973765415
370864663
584466281

• 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: 1732266086563710003
1427866592
1060611967
790114577
311175647
1259029219

• 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';

440497032
1057485615
586068733
1697190389
1801068469

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 hack your own distribution—it’s hard. 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 2 6 5 6 1 1 2 2 4 4 6 3 2 3 5 2 6 1 1 1 4 2 4 2 6 4 2 3 6 1 2 2 5 2 6 3 2 3 4 
2 6 6 4 4 4 5 6 4 2 4 1 4 6 6 6 4 6 1 6 2 5 3 6 4 4 4 5 1 3 4 6 5 5 4 6 1 5 5 3 
1 2 6 2 3 4 3 2 1 2 2 6 6 4 2 4 2 6 1 1 3 3 3 1 1 6 1 6 3 3 6 4 5 5 1 1 5 1 3 3 
6 1 5 6 4 6 1 3 3 4 3 2 5 4 3 1 4 4 5 1 3 2 1 2 5 5 5 1 2 1 4 6 5 5 3 2 5 3 2 6 
2 5 1 5 1 2 6 4 4 1 4 2 2 2 1 6 3 5 1 1 5 2 2 4 2 6 6 1 6 2 1 6 2 5 2 4 1 3 1 3 
6 4 4 1 6 5 5 3 5 5 1 4 5 2 5 6 5 1 5 4 3 5 3 2 6 3 3 5 2 3 5 3 3 3 4 2 2 6 4 4 
6 4 2 1 5 4 2 3 6 4 4 2 3 3 3 1 6 3 6 6 2 2 5 1 4 4 5 5 5 3 4 4 6 3 3 5 1 4 2 3 
4 4 5 3 4 3 1 2 4 5 6 5 4 3 5 2 2 3 2 5 4 4 3 1 6 2 3 1 2 3 3 3 6 1 5 6 2 3 2 4 
4 4 6 3 1 5 4 5 5 6 2 2 3 1 5 5 2 5 4 4 3 3 2 6 6 3 5 5 5 6 5 1 2 3 5 3 6 6 1 2 
2 5 6 4 2 6 5 4 2 2 1 2 1 1 3 6 6 1 2 2 2 2 5 6 4 5 1 6 2 5 5 6 6 2 6 6 1 1 2 1 

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';
}

20.605 18.641 19.202 22.767 22.879 21.182 21.226 22.260 21.077 20.630 
22.521 19.338 19.710 24.092 24.074 20.406 24.183 21.663 18.471 23.200 
20.861 24.698 24.623 21.250 19.342 24.500 22.821 24.829 19.563 22.032 
21.938 19.347 20.633 18.718 19.659 23.042 19.070 22.691 22.089 18.599 
18.455 19.530 19.521 24.520 23.007 21.104 18.309 21.649 24.817 24.835 

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: 41.9189%

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: UIsOjz]pFubIkDsMVshuPpVFocXzp\BF
Password: UYeduC\XVpIRhZ}LLMyNNK}}^{cCFubO
Password: `JVXxsZpAFGDwcqfp[aDWUMiYv{y}wAX
Password: Wvqvm}y\~VVhJM\Vxthu[iMOEsa{lwZu
Password: }EPF]GzFVolfyNLzSweSCK^pt}B{R|~A
Password: }VvlND_INhd~_Xhq[vf[{zbtkDLbkzWc
Password: axB{hCwljfwAIYEQajfhHD]OAiaXSew]
Password: FBwQ]B_]YNoLnwzOFw~yZ`j`nVZYBarw

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.