to deal with. So to test our odd multiples, we divide each product by 26 and look for that same fractional portion past the decimal point. (-21 + 26 = 5). the previous example? The problem is remedied by using a combination of two or more operations. Here are some warmup exercises to prepare you for the Lesson Quiz. In order for an agent to decipher the message, he or she must use the inverse process. In that case, we would have run into some After viewing the movie about the GCF at, write down how to use the prime factorizations of 52 and 130 to get GCF(52, 130). exponential cipher. Finish your inverses table and have your mentor check it. The math of solving affine code may be more complicated, but these ciphers are still vulnerable in two major ways. Dividing by 2 doesn't yield the correct plaintext either. {-1, 18, 18, 17} * 21 = {-21, 378, 378, 357}. For example, if we had only 3 possible multipliers and only 4 possible shifts, there would be 12 possible formulas. 'T' is the second most common letter and 'A' is the third most commonly used letter. We hope you try the Digging Deeper activity that follows. Instead, we find another number to multiply the cipher numbers, called an inverse, in order to undo the effect of tripling and recover the original plaintext letter. Now suppose that we have chosen and which are and so 5 and 21 are inverses of one another. Back in Lesson 4, you saw that the formula for the shift is a simple addition to the position numbers in mod 26. Let's hear about Hatty Mary's big oops! Then we decided that the shift cipher corresponded In regular arithmetic, you may have seen this example: Numbers called reciprocals undo multiplications by using multiplication, not division. Good grief! wish to encrypt. cipher. To do this, all he needs to do is the division algorithm in Agent Smith, in your log book, please make a note of why 3 has an inverse, and 2 does not. Then, in Lecture 2, we learned how to To see why this means that 2 The Caesar shift is easy to use but also easy to break. But there is no solution. Smith, can you see that B and O are both enciphered as C? But one of the 26 possible remainders. Ask your mentor for a link where you can read about Sarah Flanner who uses the pigeonhole principle to prove that there are two people in Dublin, Ireland, that have the same number of hairs on their heads! No dogs or hens here: let's encipher the word CAT. Which of the three ciphers took the longest to deal with? In Lesson 4, we saw that you reduce negative numbers by adding 26s. Justify your answer. The math of solving affine code may be more complicated, but these ciphers are still vulnerable in two major ways. And it's this information that can help you to crack codes. I analyzed the message in this way: "What I did was take C's position number, 2, and divided it by 2 to get B's position number, 1. Multiplier 2 won't work, and neither will... in her calculation. You may need to go back to Lesson 2 and look it up. relatively prime to . They were enciphered by the formula: Try enciphering and deciphering with other formulas—you choose which ones. Tell why this difference is meaningful. Decipher GZZY. Please enter the following problems in your log book, and have your mentor check your work. Let's encipher the word CAT using a tripling formula. true. Smith, you won't need to multiply the entire alphabet, but if you enter 13 times the position numbers up to 11 in your log and reduce them in mod 26, you will see the problem. multiplication cipher. and want to know whether or not a number is invertible modulo . Much In particular, it Jefferson wheel This one uses a cylinder with sev… "Mary must have doubled 14, which is letter O. TOTO, Dorothy's dog, in the Wizard of Oz stories. "Using doubling, Mary had caused a problem because both B and O encipher as the letter C in the ciphertext. Explain the way to decipher a secret message if it was made by tripling the position numbers. You can learn much from a funny little story involving Hatty Mary, the child agent-prodigy, and one of her rare mistakes. If you would like to double check your work, the CryptoClub has an Affine Department. integer. Bazeries This system combines two grids commonly called (Polybius) and a single key for encryption. Smith (that's you, by the way), you will look for an inverse for 2. If is prime and In other words, how do exponent that you used. is a positive integer, then . But alas, agents are human, and they can make errors that render a cipher useless. One of the messages above is a question. xڭXK��6��W(�TnbE�H= In modular arithmetic, instead of dividing, we multiply by the inverse. Since this is an optional section, this material will not be in the Lesson Quiz nor will it be in the Unit Test. For example. Alberti This uses a set of two mobile circular disks which can rotate easily. Agent Smith, it is said that a good cipher is easy to use and hard to break. As usual, we will start with an example. Write down any ideas you might have about fixing that weakness. 17th century French Mathematician Pierre de Fermat. This requires the division algorithm discussed in Lecture Next, we will find out why we can't use doubling and some other numbers as multipliers in an affine cipher. This is essentially the This time she won't be making any mistake. There is also no solution for. Try something. Since there are 26 letters in the English alphabet, let’s relate the letters a-z by numbers 0-25 as shown by the diagram below. Cryptology & Mathematics: Secret Codes & Number SecretsLesson 6: Affine Ciphers and Modular Inverses. Of The formula that combines these operations looks like. cannot be even (why?). That is, Theorem: Let be a positive integer and "The other cipher letter, M, has two possible origins also. the example, we get. Make a list of them in your log. Smith, what problem do you think the multiplier 13 causes?

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