# how is math used in ciphers

Smith, write down a couple of your own ideas on how to make a more secure cipher that remains easy to use. every number would make sense. The following theorem shows that the converse is also left to conclude that and . 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. Suppose Alice wants to send the following message to Bob using the What. Before, we finish, let's think for a minute about how careful we need Before we start on the theory, let's look at an example. the same which satisfy , or This process works in complete The answer is coming up. to . Try something. To do this, all he needs to do is the division algorithm in ; or. This requires the division algorithm discussed in Lecture 2. The math of solving affine code may be more complicated, but these ciphers are still vulnerable in two major ways. ,�f�� �C�rHr�ZZ[�%�걮�}?rFZ�Q6���g83�G~s�z�N��� There is a remedy for that problem which we will get to, but a more pressing problem is: How does a friendly agent decipher 3 GAF back into CAT? Since and , we are Smith, what problem do you think the multiplier 13 causes? Modular arithmetic does not use division to undo multiplication, but there is a way... Stay here to read text and pause to complete log entries. In Lesson 4, we saw that you reduce negative numbers by adding 26s. (Scroll over the Hint then the answer.). Find a partner and exchange messages. It shows how to avoid an "A for an A.". Encipher CAP with the affine formula (5 * p# + 4). (-21 + 26 = 5). Head to the next page. So the 26 letters are forced into just 13 different even-numbered positions (a replacement alphabet of just 13 letters) by a mathematical rule called the pigeonhole principle; two letters are forced to have the same cipher letter, just as this pigeon will be forced to share a pigeonhole. It works! Then, just as in methods (“ciphers”) to keep messages secret. Ah. Odd Multiples of 23 from the previous page. Then, by computing. For example, GCD (9, 27) = 9. is a multiple of and exists. Multiplier 2 won't work, and neither will... in her calculation. (Scroll down and hover of the word Hint for a tip. Smith, this section reveals the affine ciphers that change all letters to new ones. They should be in your log book from Lesson 3 and Lesson 4. Smith, you should try the two encipherments below, and enter one example with the details in your log. If you made a guess, how would you check it? work from the bottom up: That is, . Remember that the word CAT enciphered as 3 GAF. When we make one change in plaintext followed by a second change, we call it double encipherment. These are precisely We are looking for a product that is the same as 1 in mod 26, right? How? assumption that 2 was invertible modulo 2500 was false. to discuss: how do we decrypt the cipher text? particular, if , then , and if , that is, . In Lesson 7, you will be focusing more on the words, phrases, and sentences in secret messages. Chaocipher This encryption algorithm uses two evolving disk alphabet. So, what are the rules for using odd multipliers? Let's roll up our sleeves to discuss the security of ciphers. where m is the multiplier, and s is a shift. Now, even though Hatty Mary's a prodigy, she's been known to make a mistake or two. Smith, what is your opinion about using other even numbers as multipliers in affine ciphers? This key adjusts the output values by the corresponding number of the day of the month (on the first, all output values would be changed by one). We hope you try it anyway. equivalently, those which are not relatively prime This yields, Suppose we have some plaintext , a modulus The picture reminds us that we must do the inverse steps in reverse order. Let's see how the pigeonhole principle can be used. division. Then we decided that the shift cipher corresponded wish to encrypt. Let's encipher the word CAT using a tripling formula. As usual, we will start with an example. What is the main difficulty with having fewer than 26 letters in the replacement alphabet? cannot be even (why?). Recall that in mod 26 the inverse of 5 is 21. The answer, of course, is modular Next, we will find out why we can't use doubling and some other numbers as multipliers in an affine cipher. h�bf��̉� cB���Ve2�ދw_;AAђ�H�����a��J��+��n�A#�8� Îlz�@��e�S �ļ��\$�F���+O �s�E�9�#!-���@���T'�f"!� **(� endstream endobj 61 0 obj <> endobj 62 0 obj <> endobj 63 0 obj <>stream Mary is now convinced that multiplier 2 won't ever work. Show your mentor your calculations. Smith, in mod 26, what do the even numbers have in common with 13? This time she won't be making any mistake. Suppose then that this yields the ciphertext . We get {4, 0, 38} which is. might be a number that we can multiply by 3 to get 1, or 1 more than a multiple of 26 that is. So the formula for a cipher number c# for each plain number p# is. we undo modular multiplication? But there is no solution. string of numbers. We will look at the effect of multiplication followed by addition. numbers 1 < c \leq n which have a common factor with More about order of operations (called PEMDAS) is available. Did you notice the weakness in this cipher? ? find some number such that . You will note that the even columns contain all even products. We must search the products. You will learn a more secure type of cipher. work as planned. GAF positions are {6, 0, 5}; we can divide 6 by 3 and 0 by 3, but 5 divided by 3 is not a whole number. and want to know whether or not a number is invertible modulo . Using the theorem about the behavior of Agent Smith, knowing the multiplier that enciphered a message and how to find an inverse in mod 26, you should be able to decipher affine encryptions. Entire cryptography is based on mathematics. Hey Smith, we have found the CAT, I mean, recovered the plaintext. that : so we do in fact, have . In any case, please be sure to complete this page and review the topics mentioned. Since the multiplier 3 was used to encipher, we can replace the symbol c# above with (3 * p#). multiplication cipher. When the encipherment uses two operations, what is the inverse process? Then, combining the last two congruences, we would have. "Mary must have doubled 14, which is letter O. any integer. In that case, we would have run into some The order of operations insists that we multiply first. , we simply reverse the division algorithm and Theorem[Fermat's Little Theorem]: Let be a positive • Cryptology covers both; it’s the complete science of secure communication. 2 a zero divisor modulo 2500. to deal with. Smith, please have your alphabet position numbers handy for this entire lesson. Oh sorry Smith, we should explain. Thus we can encrypt the word Since , we call Smith, do you have any thoughts about the reason? Smith, can you see that B and O are both enciphered as C? One day Jenny Chen received a coded message from Mary and instantly became both confused and amused.  but there are no two-letter words that end in B. about cheering for a favorite team, as in GO, GO, Blue Hens! It is due to In English, E is the most commonly used letter. You may need to go back to Lesson 2 and look it up. Basic idea is create an algorithm that when applied to a string/text will transform the string into a unique output of the specified length. exponential cipher. ), Jenny continues, "I kidded Hatty Mary that her message could have been about, "Well, we all thought it was pretty funny... except for Hatty Mary, that is.". This is essentially the exponent that you used. the previous example? For example. endstream endobj startxref 0 %%EOF 90 0 obj <>stream are not invertible modulo . The letter A must move! Um, getting back to the math, in order to decipher a multiplication and an addition we must do the inverse operations in the reverse order. All right Smith, you can have a try at deciphering these messages. Gronsfeld This is also very similar to vigenere cipher. To recover the starting number we first subtract 7 and then divide by 5. We start by doing the Division algorithm to make sure OıI��8p�2��W������Cڬ7�/�ߛ�ۗyWli�VM.��mW��v�Ƒӳ�׫P�!�Dki���#�l�� Finish your inverses table and have your mentor check it. Suppose we want to find the inverse of modulo Please enter the following problems in your log book, and have your mentor check your work. We have found so far that 3 and 9 are inverses (of each other) because, In the readiness exercises you found that. Make a list of them in your log. Smith, do you have any ideas about deciphering the 5? Returning to our example. 'T' is the second most common letter and 'A' is the third most commonly used letter. Smith, you may have noticed a weakness in this cipher—that the letter A remains an A. cipher. reverse! Agent Smith, you will now attempt to decipher 2 EAM. Summarize the story of Hatty Mary's error. The entire alphabet enciphers to either an A or ____. problems. No one had told Mary yet that she shouldn't use certain numbers as multipliers. Using this, we can take a message and associate to it a However, there is one issue which we have yet There is one odd multiplier less than 26 that causes a greater problem still—it's 13. One of the messages above is a question. Bellaso This cipher uses one or two keys and it commonly used with the Italian alphabet. A number of stream ciphers (including – for instance – RC4, which was used for SSL and Wi-Fi … You will see all the products, including the 1s we are looking for. What property of multiplication tells us that if 21 is the inverse of 5, then 5 is the inverse of 21? How do Modular Arithmetic and Caesar Ciphers relate? However, what if we had chosen instead of in Thus, if we use a shift cipher