Python is a fantastic tool for handling large amounts of data in an efficient way. This makes it a key tool for gleaning insights from a wide variety of sources.
Unsurprisingly, Python has a handy built-in repository of arithmetic and logical functions. It also supports more complex mathematics through modular libraries like NumPy and SciPy.
While most of the built-in mathematical functions that Python supports are well-known to any high-school graduate, the MOD function stands out. Unlike trigonometric or exponential functions, MOD is something altogether more fundamental.
The Python MOD function is easy to call. Simply use “%” in an arithmetic format to return the modular element of the equation in question. You can do this with any expression that evaluates to a numeric type.
But in order to delve deeper into what that means, it’s important to comprehensively cover what modular functions are, how they work, and what you would want to use them for in Python.
What is a Modular Function?
Modular functions are a basic operation of arithmetic – albeit one so unusual that it wasn’t described until the 19th century. Unlike adding, subtracting, multiplying, and dividing, modular functions focus on what’s left out of an equation.
Modular functions point to the value left out of a divided sum after division. The easiest way to understand the way modular functions is using the face of a clock. There are 24 hours in a day, but only 12 on a clock face. People tell the time of day by calculating the congruent number modulo 12 for the current hour of the day.
Immediately after noon, when the clock strikes 12:01, it physically wraps around its perimeter to restart counting time from the start. This means that in the mathematical system the clock face asserts, 13 is congruent with 1. By the same system, 23 is congruent with 11. In all cases, the remainder of the division equation is the answer of the modular function.
In computer science, modular functions are incredibly important for a wide variety of reasons. They help programmers control bitwise operations involving cyclic data structures. They allow for mathematical descriptions of logical behavior like the exclusive disjunction.
The powerful thing about using Python MOD equations in computer programming is that high-level programming languages like Python allow you to process many all types of numeric operands – not just integers.
When to Use Modular Functions
Whenever you want to add or subtract a number and another number but have the result be within a certain range, modular functions are your key operator.
Here is an example of a simple program that performs modular functions in Python:
a = input('Do you want to continue? ')
if a.lower() != 'y':
a = float(input('Enter a number : '))
b = float(input('Enter another number : '))
print(a, ' % ', b, ' = ', a % b)
print(b, ' % ', a, ' = ', b % a)
The program asks the user if he or she wants to continue. Pressing “y” causes the program to ask for two numbers and to print the remainder of dividing them using the modulo operation.
One of the fascinating things about the Python modulo operation is the fact that it can work with integers as well as floating-point numbers equally well. If you enter the numbers “1.2” and “2.1” into the program described above, it will give you the result, “0.9000000000000001” making maximum use of its available bit-depth.
The only time when Python will generate an exception is when you attempt to divide by zero. If the divider operand – the second number in the modulo operation – is zero, the code won’t run. The laws of mathematics simply don’t provide a defined answer for that operation. Here’s an example:
a = 12
b = 0
print(a, ' % ', b, ' = ', a % b)
except ZeroDivisionError as z:
print('Cannot divide by zero! Sorry! ')
Python Mod Applications
Some immediate applications for modular functions obviously center around loop counters. If you want to create a program that operates inside a loop but does not operate on every iteration of the loop, then you could design it to only operate on a specific modulo of that loop. That would look like this:
if k % n == 0: begin_program()
Where “k” is the loop counter and “n” is the iteration that you wish the engage the begin_program command on. This gives you a lot of power over timing certain program operations.
Another option is to use modular functions to turn a list into a cyclic snippet of code. Doing so would look like this:
def following(L, k):
N = len(L)
return L[k % N]
Now the list in question will behave cyclically, returning to its beginning state at the end of every cycle. Python’s MOD functions allow you to quickly and efficiently manage cyclic behaviors. You could use this, for instance, to increase an index while wrapping back to the beginning after reaching the end of an array.
Use Modular Exponentiation for Large Numbers
Modular functions can become expensive in terms of operation time when it comes to very large numbers. The time it takes to calculate the modulo of a very large number gets longer as the number gets larger. This led computer researchers and mathematicians to look for more efficient ways to perform these functions.
In Python, the secret to performing modular functions on very large numbers is called modular exponentiation. This operation features logarithmic running time and is functionally similar to binary exponentiation.
In Python, the built-in function pow() performs modular exponentiation. Instead of solving the basic modular equation A % B, it solves A^B % M, which is better-suited to numbers of exponential size.
This makes modular exponentiation a key element of cryptography coding. A large part of cybersecurity encryption standards focus on computing the modular exponent of very large numbers. Many developers choose Python for this work because Python has an efficient built-in set of computing functions, including MOD and pow().
A simplified version of RSA encryption could look like the Python Software Foundation’s example:
from mod import Mod
# Two RSA keys
public_key = Mod(3, 61423)
private_key = Mod(40619, 61423)
# The secret message
top_secret_message = 666
# RSA encryption
encrypted = top_secret_message**public_key
# RSA decryption
decrypted = encrypted**private_key
# The secret message has been encrypted and decrypted correctly
assert decrypted == top_secret_message
By importing a more powerful library for modular functions, this code is able to set and assert an encryption and decryption routine in only a handful of lines.
However, the mod library is not the most powerful mathematics library for doing math with Python. The best tool for this purpose is NumPy – a dedicated library of mathematics operations that includes complex modular functions for arrays among many other useful tools.
Using Python NumPy for Complex Operations
Python’s built-in arithmetic library is more than enough for most mathematical operations involving the modular function. However, there are cases when it breaks down or produces results too slowly to be of practical use.
For instance, with a more advanced version of the RSA encryption described above, it is easy to arrive at astronomical numbers. If an engine were processing a very large array of functions similar to these, it would need substantial computing power arranged in a tightly organized array to make any sense of them in a convenient time frame.
NumPy can simplify very large calculations by asserting a homogenous multidimensional array for same-type elements to fit along any of its axes. The ability to calculate entire axes at a time vastly reduces the number of resources that go into each individual calculation, providing a stable and powerful tool for Big Data analyses and cybersecurity research.
Otherwise, you would be stuck using Python lists for the purpose, which would hugely increase both the time needed to complete each calculation and the amount of storage needed to hold all of the information during calculation.
Since Python allows modular operations to work on elements in axes, you can compute the remainder complimentary to a floor_divide(x1, x2) function for an entire set of cells with a single command.
NumPy offers a variety of parameters for dealing with modular arithmetic:
- array_like(x1, x2) establishes the dividend and divisor arrays. x1 refers to the dividend and x2 to the divisor.
- ndarray describes the location where the result will be stored. This is the output of the resulting modular array function. It must have a shape congruent with the relative shapes of the dividend and divisor arrays. Otherwise, the program will return a freshly allocated array.
- **kwargs is a keyword-only argument that applies the operation to all pairs described. Read more about these in SciPy’s ufunc documentation.
In terms of returns, the ndarray return of the quotient of floor_divide(x1, x2) will return a scalar variable if both x1 and x2 are scalar variables.
Python offers many ways to manipulate large arrays of data in mathematically complex ways. Refer to the official NumPy and SciPy documentation and reference materials for more information and examples of how this mathematical engine works.