Implementing-cryptography-using-python

226 Chapter 8 ■ Cryptographic Applications and PKI
that uses the two secret numbers, 11 and 17, to unlock the mathematical
padlock:
C 1/e mod (p – 1) (q – 1) mod N = M
165 1/13 mod (11 – 1) (17 – 1) = 88
With your algorithm being public, one could argue that someone can take the
secret key and find the factors used to produce it. Remember, multiplying is a
one-way operation, meaning it is difficult to go backward. Greater difficulty is
introduced as the numbers become larger. It wouldn’t take very long to find that
11 and 17 were the secret numbers when examining 187 since the only numbers
that factor into 187 are 1, 11, 17, and 187.
If you come up with two larger numbers that are 300 digits long, then your
padlock is secure. Factoring large numbers is hugely time-consuming. While
the math can get tough, Cocks invented a super secure encryption system and,
more importantly, he solved the key distribution problem. In the end, the Cocks
equation was the most important breakthrough in the history of secret codes.
The GCHQ was unable to implement the algorithm and classified it top secret,
keeping it closely guarded. In fact, it would remain top secret for another 24
years, even though a similar algorithm was published by Rivest, Shamir, and
Adleman in 1977.
Ron Rivest, Adi Shamir, and Leonard Adleman, researchers at the Massachusetts
Institute of Technology (MIT), developed a similar equation to Cocks’s
algorithm four years after Cocks. Their breakthrough was also attributed to
Rivest’s knowledge of number theory. This time, the discovery was in the public
arena. Knowing nothing about the British discovery, the team named their cipher
Rivest, Shamir, Adleman, better known as RSA. RSA has become one of the most
important ciphers ever introduced. Almost every time you transmit credit card
details online, the data is encrypted using RSA, making RSA the cornerstone
of a multi-billion-dollar e-commerce revolution. So, while the Americans were
not the first to invent public-key cryptography, they were the first to bring it
out into the public domain where anyone can use it. You can now communicate
as privately as a nation-state. You can send messages that cannot be cracked
even by the combined efforts of all the world’s secret services. For the first time
ever, you have access to unbreakable code. Cryptography may be the science
of secrecy, but through your continued practice, it is no longer a secret science.
Exploring the Basics of RSA
This first example is a simple implementation of the RSA algorithm that can
encrypt and decrypt a message. Note that the keys here are far too small to be
of practical use since they are still relatively easy to factor:
import random