Implementing-cryptography-using-python

Chapter 4 ■ Cryptographic Math and Frequency Analysis 117
Solving Systems of Linear Equations
In this section, you will learn how to solve systems of linear equations using
Python. Linear equations can increase the level of difficulty in breaking cryptographic
schemes by using large matrices to encode and decode messages. The
first matrix is typically referred to as an encoding matrix. While we could start
writing programs from scratch, linear equations have been made much simpler
using the NumPy library. NumPy also provides methods that will allow
you to multiply matrices without having to worry about how the rows and
columns are being multiplied. NumPy also provides linear algebra libraries to
find inverses and determinates quickly. The inverse of a matrix is known as the
decoding matrix. For our example, we examine the following:
1a + 1b = 35
2a + 4b = 94
These two equations will look like the following. Notice that the values from
the first equation fill in the top row, while the values from the second fill the
second row:
11 a
A X B
24 b
35
94
When we write this in matrices form, it should be in the form of AX = B.
To solve this equation (and solve for X), you multiply both sides of the equation
for the inverse of A (shown as A −1 ). A −1 multiplied by A will produce the identity
matrix: A −1 AX = A −1 B. This will allow us to isolate X such that X = A −1 B.
Completed, it should resemble the following:
11
24
a
b
35
94
Here is the Python code that will produce the matrices shown previously:
import numpy as np
# solve the following linear equation
print ('1a + 1b = 35')
print ('2a + 4b = 94')
A = np.matrix([[1,1],[2,4]])
B = np.matrix([[35],[94]])