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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';
}

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

Engine	Description
default_random_engine	Default random engine
minstd_rand	Minimal Standard minstd_rand generator
minstd_rand0	Minimal Standard minstd_rand0 generator
mt19937	Mersenne Twister 19937 generator
mt19937_64	Mersenne Twister 19937 generator (64 bit)
ranlux24_base	Ranlux 24 base generator
ranlux48_base	Ranlux 48 base generator
ranlux24	Ranlux 24 generator
ranlux48	Ranlux 48 generator
knuth_b	Knuth-B generator
random_device	True 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';
}

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

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';

3350188630
420138396
2500820400

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';

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';

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.

Y2038

int biggest = 0x7fffffff;
time_t epoch = 0,
       now = time(nullptr),
       end = biggest,
       endp1 = biggest + 1;
cout << "epoch:" << setw(12) << epoch << ' ' << ctime(&epoch);
cout << "now:  " << setw(12) << now   << ' ' << ctime(&now);
cout << "end:  " << setw(12) << end   << ' ' << ctime(&end);
cout << "end+1:" << setw(12) << endp1 << ' ' << ctime(&endp1);

epoch:           0 Wed Dec 31 17:00:00 1969
now:    1714203066 Sat Apr 27 01:31:06 2024
end:    2147483647 Mon Jan 18 20:14:07 2038
end+1: -2147483648 Fri Dec 13 13:45:52 1901

I hope that nobody’s still using 32-bit signed time representations by then!

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: 1714203066014798178
261633426
2103262086
2127257734
780012962
174905851

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';

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.
This is harder than you think.
Your home-grown code will by off by one, or have some bias because the range of the generator isn’t a perfect multiple of what you want.
Just use the standard distributions.

Distributions

Uniform:
- uniform_int_distribution
- uniform_real_distribution
Bernoulli (yes/no) trials:
Piecewise distributions:

Related to Normal distribution:
Rate-based 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';
}

5 1 1 6 5 1 5 2 3 2 3 6 2 3 5 6 6 1 2 1 4 2 3 5 3 4 1 5 4 3 2 4 5 3 6 4 1 5 6 4 
5 5 1 5 5 3 4 6 6 5 1 4 6 5 5 6 4 6 4 5 3 2 6 5 4 1 4 2 3 1 2 1 5 6 4 3 6 6 6 2 
4 2 2 3 3 4 1 5 4 3 1 1 1 4 3 2 6 6 2 1 1 3 5 6 4 3 6 1 1 4 4 5 1 2 5 4 6 2 3 3 
6 2 1 5 2 4 1 4 2 2 2 2 5 1 1 3 5 2 2 4 2 2 4 1 1 6 5 2 2 1 3 1 1 3 2 1 4 3 4 3 
4 5 2 2 6 4 3 3 4 3 4 2 6 5 5 2 2 4 2 1 5 1 5 4 1 2 4 3 3 3 1 4 6 4 3 2 4 6 2 1 
3 3 5 6 5 3 5 1 4 6 2 3 1 4 2 2 5 2 2 4 6 2 1 5 5 1 1 6 6 6 2 3 2 1 5 1 4 4 2 2 
3 6 3 1 6 6 4 3 2 5 1 5 2 2 3 2 2 6 2 4 2 2 6 2 5 2 4 3 4 1 2 4 2 2 6 5 1 5 3 4 
4 1 3 5 6 5 1 5 3 5 5 4 4 6 2 2 3 1 4 3 6 1 1 4 4 2 3 6 2 4 2 5 1 1 4 3 3 4 3 3 
1 3 6 1 2 3 6 5 3 6 6 4 6 2 1 5 5 4 4 2 2 3 3 3 2 2 3 6 3 4 2 5 5 3 2 5 1 4 1 3 
1 1 4 4 6 6 2 4 4 2 1 6 5 3 4 2 1 2 4 1 5 1 1 2 2 2 4 1 2 4 4 2 4 3 5 6 6 2 4 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';
}

20.512 24.728 18.533 22.482 19.890 23.017 24.929 19.635 22.365 21.944 
21.156 23.784 22.177 22.816 20.378 22.220 21.752 20.557 21.832 24.071 
20.919 24.384 21.033 18.749 23.679 24.796 23.272 18.805 24.677 20.476 
18.892 23.275 22.617 23.986 22.754 20.475 22.919 21.020 18.437 19.882 
23.074 23.164 24.131 21.560 22.570 21.808 24.463 23.725 23.813 18.175

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: 42.0689%

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('!','~');
for (int y=0; y<8; y++) {
    string pw;
    for (int x=0; x<32; x++)
        pw += dist(gen);
    cout << "Password: " << pw << '\n';
}

Password: &K)s/{qaQ_bJSDfMt|UWzns+7R@jA&e(
Password: Y]ZHKl@q0fk4@B~tx)HGfdeG7QlMk(,b
Password: rbuiYt-LX\}z>*(/maR}`8_s\bK5>jKd
Password: F%1kFP8iw<l+5gYf7k9p/ux"}MA>9|p9
Password: [gH]Lku&AR-_.>q~An?{igx{ITFQl)50
Password: ]y6jAKOB>n9G]F&6HwBU"$>BtrTIU'\r
Password: NTxV&Lf-?[%<c"t\|rL4p-ERJSvpmMm-
Password: >>3IB0CQA9b,o-^*oo62|5gTg9~`[]=[

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.

CS253: Software Development with C++

Fall 2020

Random Numbers