Show Lecture.RandomNumbers as a slide show.

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';

3350296188
164000301
2137125806

• 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';

547170176
550191243
351228404
1910380066
860880059

• 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';

1234141199
2033449149
1588817950
747779139
1141679893

• 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: 1727408365815059533
1222982743
274531323
1927390543
1634862172
702844656

• 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';

1512994236
540260325
594422759
369384669
2020392053

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_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';
}

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

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';
}

21.086 24.494 22.773 19.101 18.666 22.606 19.500 21.566 24.407 23.742 
21.834 23.499 22.952 24.999 22.558 21.046 18.931 20.872 24.515 23.794 
22.025 22.018 22.180 20.144 24.334 22.023 20.929 18.270 24.228 24.982 
23.634 24.294 22.410 22.056 21.563 18.095 23.609 18.928 19.899 24.864 
24.085 24.516 20.204 21.120 23.653 19.034 23.261 18.204 20.925 19.093 

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: 42.0184%

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: jdUxiU^mAaDVy^j|\D^jvz]^MZTsnY}F
Password: ZbNZAUQnRdPW_GTWgTYPow]GnnPWMMjF
Password: WpHZfyJhq{RQspHq\]GQVoSGInd_c|KX
Password: OmcOhb~iIlm]`~AC}EuduX\d[WsRQwhI
Password: HR}h]hUZaN_VRa[AxsXuW_{i}UkcmW|m
Password: \gsmcJ\TloBnLhg{OEijhJhQuv]|Mjvo
Password: z|TIBhlLvuYL}v`GxoFxMOOHKSqmT]F[
Password: a~JD^xjYP]|Lwctudo|QFrZpigSPvPFJ

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.