1.5. PROBLEMS 17

are r rows and c columns, this means she can send a message of rc letters.

In terms of security and frustrating an attacker, which of the following is

the best choice for rc and why: 1331, 1369, 1800, or 10201?

Exercise 1.5.15. Research and write a brief description about one of

the following:

• The Black Chamber.

• The technological treason in the Falcon and the Snowman case.

• Cryptography during Prohibition and the role of Elizabeth Smith

Friedman.

• Echelon.

• The Kryptos sculpture at the NSA.

Exercise 1.5.16. A major theme of this book is the need to do compu-

tations quickly. The Babylonians worked base 60; this meant they needed to

know multiplication tables from 0 × 0 all the way to 59 × 59, far more than

we learn today (since we work base 10, we only go up to 9 × 9).

(a) Calculate how many multiplications Babylonians must memorize or

write down.

(b) The number in part (a) can almost be cut in half, as xy = yx. Using

this observation, how many multiplications must be memorized or written

down?

(c) As it is painful and expensive to lug clay tablets around, there was a

pressing need to trim these numbers as much as possible. The Babylonians

made the remarkable observation that

xy =

(x + y)2 − x2 − y2

2

.

Show this formula is true, and thus reduces multiplication to squaring, sub-

tracting, and division by 2.

Remark. The above formula shows that the Babylonians need only

learn the squares and can deduce the remaining products by elementary

operations. This is an early example of a “look-up table”, where some

calculations are done and stored, then used to deduce the rest. This exercise

shows that the standard way to do a problem is sometimes not the most

practical.