9.3 Modular Arithmetic 565 Modular Arithmetic and Groups Modular arithmetic systems under the operation of addition are commutative groups, as illustrated in Example 6. brings us to the first nonworking day. The two N’s shown at the beginning of the letters below are actually the end days of the previous cycle. NNWWWWWWNN 4 days today earlier than today ↑ ↑ Therefore, 124 days ago Jackson was not working. 7 Now try Exercise 59 Did You Know? The Enigma A cipher, or secret code, actually has group properties: A welldefined operation turns plain text into a cipher, and an inverse operation allows it to be deciphered. During World War II, the Germans used an encrypting device based on a modulo 26 system (for the 26 letters of the alphabet) known as the Enigma. It has four rotors, each with the 26 letters. The rotors were wired to one another so that if an A were typed into the machine, the first rotor would contact a different letter on the second rotor, which contacted a different letter on the third rotor, and so on. The French secret service obtained the wiring instructions, and Polish and British cryptoanalysts were able to crack the ciphers using group theory. This breakthrough helped to keep the Allies abreast of the deployment of the German Navy. The 2014 movie The Imitation Game is a true story about breaking the Enigma. Example 6 A Commutative Group Construct a mod 5 addition table and show that the mathematical system is a commutative group. Assume that the associative property holds for the given operation. Solution The set of elements in modulo 5 arithmetic is {0, 1, 2, 3, 4}; the binary operation is .+ + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 For this system to be a commutative group, it must satisfy the five properties of a commutative group. 1. Closure : Every entry in the table is a member of the set {0, 1, 2, 3, 4}, so the system is closed under addition. 2. Identity element : An easy way to determine whether there is an identity element is to look for a row in the table that is identical to the elements at the top of the table. Note that the row next to 0 is identical to the top of the table, which indicates that 0 might be the identity element. Now look at the column under the 0 at the top of the table. If this column is identical to the far-left column, then 0 is the identity element. Since the column under 0 is the same as the far-left column, 0 is the additive identity in modulo 5 arithmetic. + = + = + = + = + = + = Element Identity Element 0 0 0 1 0 1 2 0 2 3 0 3 4 0 4
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