page 3 5. Cryptology is the science of making and breaking codes. This lab explores how the idea of functions and their inverses can be used to encode and decode messages. To encode and decode a message, first replace each letter of the alphabet with a positive integer using the following scheme, thus rewriting the original message as numbers instead of words: F-6 K-11 P-16 A-1 L-12 Q-17 U-21 B-2 G-7 M-13 R-18 V-22 H-8 C-3 N-14 S-19 W-23 2-26 Blank-27 1-9 D-4 J-10 O-15 T-20 X-24 Y-25 E-5 ENCODING: A one-to-one function can be used to encode a numerical message. that the function f(x)=3x+4 will be the encoding function. This function simply describes the For example, suppose you want to send the message MATH to a friend, and you have decided change the letters to corresponding numbers as shown above: 13 1 20 8. Then use these as the procedure used to create the encoded message-in this case multiply by 3 and add 4. First input values in f(x). (13) 3(13)+4=43 (1)=7 1(20)=64 1(8)=28 So the encoded message that you send to your friend is: 43 7 64 28 NOTE: A graphing calculator can be used to evaluate a function as above: a. Press [Y=1 on the calculator. Type 3x + 4. b. Choose [2nd][WINDOW] (TBLSET). Set the independent variable to Indpnt: Ask. c. Then choose [2nd][GRAPH] (TABLE). Input each x value (13 1 20 8). The calculator will return the function values (43 7 64 28) which is the encoded message. DECODING an encoded message: Now it is up to your friend to decode the message. Decoding is the process that "undoes" the encoding process. If f(x) encodes the message, what will decode it? The inverse of f(x) or f'(x)! So f'(x) will be the decoding function. In the example, the inverse of f(x) = 3x + 4 can be shown to be f'(x)=3 Take the encoded message (43 7 64 28) and use these values as input values in f'(x). Again, the calculator can be used to decode - simply enter y = x-4 and use the TABLE feature as described above. 3 f1(43) = 13 f¹(7)=1 f'(64)=20 f1(28)=8 That's the original numerical message! The last step is to convert back to letters using the tabl given previously. Now your friend knows the message that you sent: MATH. Math 1314 Critical Thinking Assessment page 3 5.2. You intercepted the following message from Boris and Natasha. 25 19 19 30 41 17 15 26 27 41 15 28 18 41 18 29 41 34 22 19 41 27 15 34 22 You do not know the encoding or decoding function but from previous work with Boris and Natasha you know their encoding and decoding words are always linear and have 1 as the coefficient of x You also know this message consists of 6 words. Using the encoding system described above, a. How would you decode the message? Explain your strategy in complete sentences using proper grammar and correct spelling. b. Identify the decoding function. c. What is the message?