Generating Random Data in Python (Guide)

The random Module

Probably the most widely known tool for generating random data in Python is its random module, which uses the Mersenne Twister PRNG algorithm as its core generator.

Earlier, you touched briefly on random.seed(), and now is a good time to see how it works. First, let’s build some random data without seeding. The random.random() function returns a random float in the interval [0.0, 1.0). The result will always be less than the right-hand endpoint (1.0). This is also known as a semi-open range:

>>> # Don't call `random.seed()` yet
>>> import random
>>> random.random()
0.35553263284394376
>>> random.random()
0.6101992345575074

If you run this code yourself, I’ll bet my life savings that the numbers returned on your machine will be different. The default when you don’t seed the generator is to use your current system time or a “randomness source” from your OS if one is available.

With random.seed(), you can make results reproducible, and the chain of calls after random.seed() will produce the same trail of data:

>>> random.seed(444)
>>> random.random()
0.3088946587429545
>>> random.random()
0.01323751590501987

>>> random.seed(444)  # Re-seed
>>> random.random()
0.3088946587429545
>>> random.random()
0.01323751590501987

Notice the repetition of “random” numbers. The sequence of random numbers becomes deterministic, or completely determined by the seed value, 444.

Let’s take a look at some more basic functionality of random. Above, you generated a random float. You can generate a random integer between two endpoints in Python with the random.randint() function. This spans the full [x, y] interval and may include both endpoints:

>>> random.randint(0, 10)
7
>>> random.randint(500, 50000)
18601

With random.randrange(), you can exclude the right-hand side of the interval, meaning the generated number always lies within [x, y) and will always be smaller than the right endpoint:

>>> random.randrange(1, 10)
5

If you need to generate random floats that lie within a specific [x, y] interval, you can use random.uniform(), which plucks from the continuous uniform distribution:

>>> random.uniform(20, 30)
27.42639687016509
>>> random.uniform(30, 40)
36.33865802745107

To pick a random element from a non-empty sequence (like a list or a tuple), you can use random.choice(). There is also random.choices() for choosing multiple elements from a sequence with replacement (duplicates are possible):

>>> items = ['one', 'two', 'three', 'four', 'five']
>>> random.choice(items)
'four'

>>> random.choices(items, k=2)
['three', 'three']
>>> random.choices(items, k=3)
['three', 'five', 'four']

To mimic sampling without replacement, use random.sample():

>>> random.sample(items, 4)
['one', 'five', 'four', 'three']

You can randomize a sequence in-place using random.shuffle(). This will modify the sequence object and randomize the order of elements:

>>> random.shuffle(items)
>>> items
['four', 'three', 'two', 'one', 'five']

If you’d rather not mutate the original list, you’ll need to make a copy first and then shuffle the copy. You can create copies of Python lists with the copy module, or just x[:] or x.copy(), where x is the list.

Before moving on to generating random data with NumPy, let’s look at one more slightly involved application: generating a sequence of unique random strings of uniform length.

It can help to think about the design of the function first. You need to choose from a “pool” of characters such as letters, numbers, and/or punctuation, combine these into a single string, and then check that this string has not already been generated. A Python set works well for this type of membership testing:

import string

def unique_strings(k: int, ntokens: int,
               pool: str=string.ascii_letters) -> set:
    """Generate a set of unique string tokens.

    k: Length of each token
    ntokens: Number of tokens
    pool: Iterable of characters to choose from

    For a highly optimized version:
    https://stackoverflow.com/a/48421303/7954504
    """

    seen = set()

    # An optimization for tightly-bound loops:
    # Bind these methods outside of a loop
    join = ''.join
    add = seen.add

    while len(seen) < ntokens:
        token = join(random.choices(pool, k=k))
        add(token)
    return seen

''.join() joins the letters from random.choices() into a single Python str of length k. This token is added to the set, which can’t contain duplicates, and the while loop executes until the set has the number of elements that you specify.

Let’s try this function out:

>>> unique_strings(k=4, ntokens=5)
{'AsMk', 'Cvmi', 'GIxv', 'HGsZ', 'eurU'}

>>> unique_strings(5, 4, string.printable)
{"'O*1!", '9Ien%', 'W=m7<', 'mUD|z'}

For a fine-tuned version of this function, this Stack Overflow answer uses generator functions, name binding, and some other advanced tricks to make a faster, cryptographically secure version of unique_strings() above.

PRNGs for Arrays: numpy.random

One thing you might have noticed is that a majority of the functions from random return a scalar value (a single int, float, or other object). If you wanted to generate a sequence of random numbers, one way to achieve that would be with a Python list comprehension:

>>> [random.random() for _ in range(5)]
[0.021655420657909374,
 0.4031628347066195,
 0.6609991871223335,
 0.5854998250783767,
 0.42886606317322706]

But there is another option that is specifically designed for this. You can think of NumPy’s own numpy.random package as being like the standard library’s random, but for NumPy arrays. (It also comes loaded with the ability to draw from a lot more statistical distributions.)

Take note that numpy.random uses its own PRNG that is separate from plain old random. You won’t produce deterministically random NumPy arrays with a call to Python’s own random.seed():

>>> import numpy as np
>>> np.random.seed(444)
>>> np.set_printoptions(precision=2)  # Output decimal fmt.

Without further ado, here are a few examples to whet your appetite:

>>> # Return samples from the standard normal distribution
>>> np.random.randn(5)
array([ 0.36,  0.38,  1.38,  1.18, -0.94])

>>> np.random.randn(3, 4)
array([[-1.14, -0.54, -0.55,  0.21],
       [ 0.21,  1.27, -0.81, -3.3 ],
       [-0.81, -0.36, -0.88,  0.15]])

>>> # `p` is the probability of choosing each element
>>> np.random.choice([0, 1], p=[0.6, 0.4], size=(5, 4))
array([[0, 0, 1, 0],
       [0, 1, 1, 1],
       [1, 1, 1, 0],
       [0, 0, 0, 1],
       [0, 1, 0, 1]])

In the syntax for randn(d0, d1, ..., dn), the parameters d0, d1, ..., dn are optional and indicate the shape of the final object. Here, np.random.randn(3, 4) creates a 2d array with 3 rows and 4 columns. The data will be i.i.d., meaning that each data point is drawn independent of the others.

Another common operation is to create a sequence of random Boolean values, True or False. One way to do this would be with np.random.choice([True, False]). However, it’s actually about 4x faster to choose from (0, 1) and then view-cast these integers to their corresponding Boolean values:

>>> # NumPy's `randint` is [inclusive, exclusive), unlike `random.randint()`
>>> np.random.randint(0, 2, size=25, dtype=np.uint8).view(bool)
array([ True, False,  True,  True, False,  True, False, False, False,
       False, False,  True,  True, False, False, False,  True, False,
        True, False,  True,  True,  True, False,  True])

What about generating correlated data? Let’s say you want to simulate two correlated time series. One way of going about this is with NumPy’s multivariate_normal() function, which takes a covariance matrix into account. In other words, to draw from a single normally distributed random variable, you need to specify its mean and variance (or standard deviation).

To sample from the multivariate normal distribution, you specify the means and covariance matrix, and you end up with multiple, correlated series of data that are each approximately normally distributed.

However, rather than covariance, correlation is a measure that is more familiar and intuitive to most. It’s the covariance normalized by the product of standard deviations, and so you can also define covariance in terms of correlation and standard deviation:

So, could you draw random samples from a multivariate normal distribution by specifying a correlation matrix and standard deviations? Yes, but you’ll need to get the above into matrix form first. Here, S is a vector of the standard deviations, P is their correlation matrix, and C is the resulting (square) covariance matrix:

This can be expressed in NumPy as follows:

def corr2cov(p: np.ndarray, s: np.ndarray) -> np.ndarray:
    """Covariance matrix from correlation & standard deviations"""
    d = np.diag(s)
    return d @ p @ d

Now, you can generate two time series that are correlated but still random:

>>> # Start with a correlation matrix and standard deviations.
>>> # -0.40 is the correlation between A and B, and the correlation
>>> # of a variable with itself is 1.0.
>>> corr = np.array([[1., -0.40],
...                  [-0.40, 1.]])

>>> # Standard deviations/means of A and B, respectively
>>> stdev = np.array([6., 1.])
>>> mean = np.array([2., 0.5])
>>> cov = corr2cov(corr, stdev)

>>> # `size` is the length of time series for 2d data
>>> # (500 months, days, and so on).
>>> data = np.random.multivariate_normal(mean=mean, cov=cov, size=500)
>>> data[:10]
array([[ 0.58,  1.87],
       [-7.31,  0.74],
       [-6.24,  0.33],
       [-0.77,  1.19],
       [ 1.71,  0.7 ],
       [-3.33,  1.57],
       [-1.13,  1.23],
       [-6.58,  1.81],
       [-0.82, -0.34],
       [-2.32,  1.1 ]])
>>> data.shape
(500, 2)

You can think of data as 500 pairs of inversely correlated data points. Here’s a sanity check that you can back into the original inputs, which approximate corr, stdev, and mean from above:

>>> np.corrcoef(data, rowvar=False)
array([[ 1.  , -0.39],
       [-0.39,  1.  ]])

>>> data.std(axis=0)
array([5.96, 1.01])

>>> data.mean(axis=0)
array([2.13, 0.49])

Before we move on to CSPRNGs, it might be helpful to summarize some random functions and their numpy.random counterparts:

Now that you’ve covered two fundamental options for PRNGs, let’s move onto a few more secure adaptations.

os.urandom(): About as Random as It Gets

Python’s os.urandom() function is used by both secrets and uuid (both of which you’ll see here in a moment). Without getting into too much detail, os.urandom() generates operating-system-dependent random bytes that can safely be called cryptographically secure:

• On Unix operating systems, it reads random bytes from the special file /dev/urandom, which in turn “allow access to environmental noise collected from device drivers and other sources.” (Thank you, Wikipedia.) This is garbled information that is particular to your hardware and system state at an instance in time but at the same time sufficiently random.

• On Windows, the C++ function CryptGenRandom() is used. This function is still technically pseudorandom, but it works by generating a seed value from variables such as the process ID, memory status, and so on.

With os.urandom(), there is no concept of manually seeding. While still technically pseudorandom, this function better aligns with how we think of randomness. The only argument is the number of bytes to return:

>>> os.urandom(3)
b'\xa2\xe8\x02'

>>> x = os.urandom(6)
>>> x
b'\xce\x11\xe7"!\x84'

>>> type(x), len(x)
(bytes, 6)

Before we go any further, this might be a good time to delve into a mini-lesson on character encoding. Many people, including myself, have some type of allergic reaction when they see bytes objects and a long line of \x characters. However, it’s useful to know how sequences such as x above eventually get turned into strings or numbers.

os.urandom() returns a sequence of single bytes:

>>> x
b'\xce\x11\xe7"!\x84'

But how does this eventually get turned into a Python str or sequence of numbers?

First, recall one of the fundamental concepts of computing, which is that a byte is made up of 8 bits. You can think of a bit as a single digit that is either 0 or 1. A byte effectively chooses between 0 and 1 eight times, so both 01101100 and 11110000 could represent bytes. Try this, which makes use of Python f-strings introduced in Python 3.6, in your interpreter:

>>> binary = [f'{i:0>8b}' for i in range(256)]
>>> binary[:16]
['00000000',
 '00000001',
 '00000010',
 '00000011',
 '00000100',
 '00000101',
 '00000110',
 '00000111',
 '00001000',
 '00001001',
 '00001010',
 '00001011',
 '00001100',
 '00001101',
 '00001110',
 '00001111']

This is equivalent to [bin(i) for i in range(256)], with some special formatting. bin() converts an integer to its binary representation as a string.

Where does that leave us? Using range(256) above is not a random choice. (No pun intended.) Given that we are allowed 8 bits, each with 2 choices, there are 2 ** 8 == 256 possible bytes “combinations.”

This means that each byte maps to an integer between 0 and 255. In other words, we would need more than 8 bits to express the integer 256. You can verify this by checking that len(f'{256:0>8b}') is now 9, not 8.

Okay, now let’s get back to the bytes data type that you saw above, by constructing a sequence of the bytes that correspond to integers 0 through 255:

>>> bites = bytes(range(256))

If you call list(bites), you’ll get back to a Python list that runs from 0 to 255. But if you just print bites, you get an ugly looking sequence littered with backslashes:

>>> bites
b'\x00\x01\x02\x03\x04\x05\x06\x07\x08\t\n\x0b\x0c\r\x0e\x0f\x10\x11\x12\x13\x14\x15'
 '\x16\x17\x18\x19\x1a\x1b\x1c\x1d\x1e\x1f !"#$%&\'()*+,-./0123456789:;<=>?@ABCDEFGHIJK'
 'LMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~\x7f\x80\x81\x82\x83\x84\x85\x86'
 '\x87\x88\x89\x8a\x8b\x8c\x8d\x8e\x8f\x90\x91\x92\x93\x94\x95\x96\x97\x98\x99\x9a\x9b'

 # ...

These backslashes are escape sequences, and \xhh represents the character with hex value hh. Some of the elements of bites are displayed literally (printable characters such as letters, numbers, and punctuation). Most are expressed with escapes. \x08 represents a keyboard’s backspace, while \x13 is a carriage return (part of a new line, on Windows systems).

If you need a refresher on hexadecimal, Charles Petzold’s Code: The Hidden Language is a great place for that. Hex is a base-16 numbering system that, instead of using 0 through 9, uses 0 through 9 and a through f as its basic digits.

Finally, let’s get back to where you started, with the sequence of random bytes x. Hopefully this makes a little more sense now. Calling .hex() on a bytes object gives a str of hexadecimal numbers, with each corresponding to a decimal number from 0 through 255:

>>> x
b'\xce\x11\xe7"!\x84'

>>> list(x)
[206, 17, 231, 34, 33, 132]

>>> x.hex()
'ce11e7222184'

>>> len(x.hex())
12

One last question: how is b.hex() 12 characters long above, even though x is only 6 bytes? This is because two hexadecimal digits correspond precisely to a single byte. The str version of bytes will always be twice as long as far as our eyes are concerned.

Even if the byte (such as \x01) does not need a full 8 bits to be represented, b.hex() will always use two hex digits per byte, so the number 1 will be represented as 01 rather than just 1. Mathematically, though, both of these are the same size.

With that under your belt, let’s touch on a recently introduced module, secrets, which makes generating secure tokens much more user-friendly.

Python’s Best Kept secrets

Introduced in Python 3.6 by one of the more colorful PEPs out there, the secrets module is intended to be the de facto Python module for generating cryptographically secure random bytes and strings.

You can check out the source code for the module, which is short and sweet at about 25 lines of code. secrets is basically a wrapper around os.urandom(). It exports just a handful of functions for generating random numbers, bytes, and strings. Most of these examples should be fairly self-explanatory:

>>> n = 16

>>> # Generate secure tokens
>>> secrets.token_bytes(n)
b'A\x8cz\xe1o\xf9!;\x8b\xf2\x80pJ\x8b\xd4\xd3'
>>> secrets.token_hex(n)
'9cb190491e01230ec4239cae643f286f'  
>>> secrets.token_urlsafe(n)
'MJoi7CknFu3YN41m88SEgQ'

>>> # Secure version of `random.choice()`
>>> secrets.choice('rain')
'a'

Now, how about a concrete example? You’ve probably used URL shortener services like tinyurl.com or bit.ly that turn an unwieldy URL into something like https://bit.ly/2IcCp9u. Most shorteners don’t do any complicated hashing from input to output; they just generate a random string, make sure that string has not already been generated previously, and then tie that back to the input URL.

Let’s say that after taking a look at the Root Zone Database, you’ve registered the site short.ly. Here’s a function to get you started with your service:

# shortly.py

from secrets import token_urlsafe

DATABASE = {}

def shorten(url: str, nbytes: int=5) -> str:
    ext = token_urlsafe(nbytes=nbytes)
    if ext in DATABASE:
        return shorten(url, nbytes=nbytes)
    else:
        DATABASE.update({ext: url})
        return f'short.ly/{ext}

Is this a full-fledged real illustration? No. I would wager that bit.ly does things in a slightly more advanced way than storing its gold mine in a global Python dictionary that is not persistent between sessions.

However, it’s roughly accurate conceptually:

>>> urls = (
...     'https://realpython.com/',
...     'https://docs.python.org/3/howto/regex.html'
... )

>>> for u in urls:
...     print(shorten(u))
short.ly/p_Z4fLI
short.ly/fuxSyNY

>>> DATABASE
{'p_Z4fLI': 'https://realpython.com/',
 'fuxSyNY': 'https://docs.python.org/3/howto/regex.html'}

The bottom line here is that, while secrets is really just a wrapper around existing Python functions, it can be your go-to when security is your foremost concern.