1.5. PROBLEMS 15
Chapter 3 we’ll see similar issues on a larger scale when we explore Enigma
Another important takeaway is the need for speed and eﬃciency. In a
battle situation, one does not have thirty minutes to leisurely communicate
with headquarters. Decisions need to be made in real time. It’s precisely
for such reasons that the Navajo code talkers played such an important role,
as they allowed U.S. units the ability to quickly communicate under fire.
Of course, this code was designed to communicate very specific information.
In modern applications we have a very rich set of information we want to
encode and protect, and it becomes critically important to have eﬃcient
ways to both encrypt and decrypt.
Another theme, which will play a central role throughout much of this
book, is replacing message text with numbers. We saw a simple recipe in the
work of Polybius; we’ll see more involved methods later. The monumental
advances in the subject allow us to use advanced mathematical methods and
results in cryptography.
We end with one last comment. Though there are many threads which
we’ll pursue later, an absolutely essential point comes from the Soviet efforts
to read our ciphers. Even though the cipher machines in Moscow used
double encryption, the Soviets were able to circumvent electronic filters by
“cleaning” the power lines. This story serves as a powerful warning: in
cryptography you have to defend against all possible attacks, and not just
the expected ones. We’ll see several schemes that appear safe and secure,
only to see how a little more mathematics and a different method of attack
are able to quickly break them.
Exercise 1.5.1. Use the Polybius checkerboard to encode:
(a) Men coming from the south.
(b) King has called a cease fire.
Exercise 1.5.2. Use the Polybius checkerboard to encode:
(a) Fire when ready.
(b) Luke, I am your father.
Exercise 1.5.3. Use the Polybius checkerboard to decode:
(a) 13, 54, 13, 32, 11, 14, 15, 44.
(b) 33, 45, 35, 32, 54, 33, 41, 51, 44.
(c) 23, 15, 32, 15, 34, 35, 21, 45, 43, 35, 54.
Exercise 1.5.4. Use the Polybius checkerboard to decode:
(a) 43, 44, 15, 11, 31, 23, 34, 32, 15.
(b) 35, 34, 31, 54, 12, 24, 45, 43.
Exercise 1.5.5. Use the Polybius checkerboard to decode
23, 22, 22, 22, 33, 25, 43.