1.5. PROBLEMS 15 Chapter 3 we’ll see similar issues on a larger scale when we explore Enigma and Ultra. 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. 1.5. Problems 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.

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