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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';

4108057877
1388469865
789537133

• 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';

1997992470
1603932600
202609309
513426301
1639689191

• 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';

515725883
973662269
1971772304
798167697
348791060

• 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: 1719786022039148641
1434901385
1334688644
98640877
522528268
766590613

• 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';

409305393
805618810
150945335
760058238
1066073710

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';
}

3 3 3 3 1 3 5 6 2 6 4 2 4 5 2 6 4 6 1 2 2 1 6 3 2 4 1 2 4 1 4 4 4 2 4 4 4 4 4 4 
5 3 1 1 5 2 6 5 1 6 5 1 1 2 4 5 5 1 4 4 6 1 4 3 2 6 3 2 3 5 4 6 2 6 4 6 4 2 1 1 
4 3 1 2 2 4 5 1 3 3 1 5 4 6 2 6 2 3 4 4 6 3 2 4 6 5 1 5 2 4 1 6 1 5 1 1 3 2 5 4 
2 4 4 4 1 1 3 4 3 1 6 2 5 3 4 2 1 4 5 5 1 3 6 4 6 6 6 6 1 3 4 6 1 5 5 4 1 6 2 3 
4 3 4 6 3 2 4 1 1 5 3 5 3 6 3 6 5 5 2 2 5 2 5 1 1 4 3 1 5 3 6 1 3 2 3 3 6 1 3 6 
6 2 1 2 4 5 4 5 4 2 2 2 3 2 1 4 5 2 5 6 2 2 3 1 4 5 3 5 6 5 6 5 6 4 5 1 1 6 6 1 
6 5 5 4 4 6 6 3 6 2 2 2 6 6 4 6 6 4 5 4 3 5 3 4 5 2 6 1 5 3 1 4 4 6 3 2 5 2 1 3 
2 1 2 6 1 4 6 3 2 6 4 5 2 3 4 4 3 6 3 6 4 1 6 4 3 4 2 6 2 2 4 6 2 3 6 6 5 1 5 5 
6 3 2 4 4 6 3 3 1 6 2 5 2 6 6 5 1 1 6 3 2 4 6 2 2 1 3 3 1 1 1 5 5 3 3 2 6 1 4 3 
2 5 2 3 1 2 3 3 2 3 1 2 5 2 2 6 4 1 1 1 2 3 3 1 1 5 4 1 1 6 2 3 3 5 1 2 4 5 6 6 

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';
}

20.995 23.664 23.183 21.804 23.120 23.749 19.439 23.606 19.662 23.740 
20.671 20.377 23.536 19.224 21.416 23.123 19.665 19.788 24.114 21.083 
21.938 19.110 23.763 19.043 19.074 18.792 19.932 23.213 21.755 19.530 
21.505 18.655 20.068 18.704 21.666 23.556 19.105 24.319 23.628 23.337 
22.051 22.750 22.148 23.439 23.910 20.151 21.655 21.640 19.674 18.782 

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: 42.0101%

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: Ykh`K|N`~PfN_GX~[SXk|R~~q~[zcAc^
Password: J[NtOpqQDCkP\TxehFZuvelsBf_fTKoH
Password: UAv~Fb[WfiJyV~GIYx^M|mVYixBUkwZD
Password: BlEiNptpAgkIvlKbDyLdFxLOGaRPMFLG
Password: uz[_Z}tEFn]UZZwQ{yXZLyW_IpDLfpDl
Password: zJ_^QwXN}QD`NuVrFIplKKflN{A|\HpP
Password: hdUOR}sK`nVaxE{Icx|LtqLcmtdyRJYZ
Password: rdSWT~MMyJsQnKz|YdY_tvpMFxeJ}Mcs

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.