Mathematical Perspectives: Secret Codes: Dr. Morton. Ch 2: Substitution Ciphers

Transcription

1 Mathematical Perspectives: Secret Codes: Dr. Morton Ch 2: Substitution Ciphers 1. Definition: In a simple substitution ciphers, users agree upon a rearrangement, or permutation (meaning, each possible plaintext letter in the original alphabet is paired with one and only one possible ciphertext letter, and vice versa), of the alphabet letters, yielding a collection of correspondences to be used for converting plaintext letters into ciphertext letters. One could include numbers, punctuation marks, and other characters too, if one wanted. These are easy to break, but have a rich history. 1

2 2.1 Keyword Substitution Ciphers 1. Definition: A random cipher alphabet is one in which the permutation is chosen randomly. 2. Example: Use the simple substitution cipher below to encrypt hello. Plaintext a b c d e f g h i j k l m Z A M B C Q W T E G H I J Plaintext n o p q r s t u v w x y z D F R N O X V K P S L Y U Simple Keyword Substitution Ciphers 3. Definition: In a simple keyword substitution cipher, Alice and Bob agree on a keyword. Delete spaces and duplicate letters and then use the remaining letters of the alphabet in order. 4. Example: Use the simple keyword substitution with keyword DARTH VADER to encrypt the message star wars. The keyword to be used is: Then the cipher alphabet is: Plaintext a b c d e f g h i j k l m Plaintext n o p q r s t u v w x y z Keyword Columnar Substitution Ciphers 5. Definition: In a keyword columnar substitution cipher, we start the same as before, using a keyword with spaces and duplicate letters deleted. These are the headings of an array. Then use the columns of the array to give the cipher text alphabet. 2

3 6. Example: Use the columnar substation cipher with keyword DARTH VADER to again encrypt the message star wars: Plaintext a b c d e f g h i j k l m Plaintext n o p q r s t u v w x y z 7. Example: Use the keyword columnar substation cipher with keyword BAG to encrypt the message star shopping: Plaintext a b c d e f g h i j k l m Plaintext n o p q r s t u v w x y z 3

4 2.2 A Maplet for Substitution Ciphers 1. Almost every topic in this bookis completely covered in onesection, with one or more Maplets for the topic introduced in the following section. Maplets are like applets, but use (and require) the engine of the software package Maple, a computer algebra system first developed at the University of Waterloo in Ontario, Canada. All Maplets demonstrated in this book will work in both PC and Mac environments with versions of Maple going back through at least 13, and are available for download at After downloading a Maplet onto a computer on which Maple is available, doubleclicking the filename or icon of the Maplet should cause a Maplet Launcher to open the Maplet. 2. Example: Use the Maplets to redo the following examples from the previous section: (a) Use the simple keyword substitution with keyword DARTH VADER to encrypt the message star wars. (b) Use the columnar substation cipher with keyword DARTH VADER to again encrypt the message star wars. (c) Use the columnar substation cipher with keyword DARTH VADER to encrypt the message: Episode IV A New Hope It is a period of civil war. Rebel spaceships, striking from a hidden base, have won their first victory against the evil Galactic Empire. During the battle, Rebel spies managed to steal secret plans to the Empire s ultimate weapon, the DEATH STAR, an armored space station with enough power to destroy an entire planet. Pursued by the Empire s sinister agents, Princess Leia races home aboard her starship, custodian of the stolen plans that can save her people and restore freedom to the galaxy... 4

5 2.3 Cryptanalysis of Substitution Ciphers 1. It seems like substitution ciphers would be impossible to break. How many possible keys are there? 2. Here are the frequencies of letters in the English language: a b c d e f g h i 8.17% 1.49% 2.78 % 4.25% 12.7 % 2.23% 2.02% 6.09 % 6.97 % j k l m n o p q r 0.15% 0.77% 4.03% 2.41 % 6.75 % 7.51 % 1.93 % 0.10 % 5.99 % s t u v w x y z 6.33 % 9.06 % 2.76 % 0.98 % 2.36 % 0.15 % 1.97 % 0.07 % 3. The most common bigrams (as measured by Cornell University: th he in er an re nd at on nt 1.52% 1.28% 0.94% 0.94% 0.82% 0.68% 0.63% 0.59% 0.57% 0.56% ha es st en ed to it ou ea hi 0.56% 0.56% 0.55% 0.55% 0.53%.0.52% 0.50% 0.50% 0.47% 0.46% is or ti as te et ng of al de 0.46% 0.43% 0.34% 0.33% 0.27% 0.19% 0.18% 0.16% 0.09% 0.09% se le sa si ar ve ra ld ur 0.08% 0.08% 0.06% 0.05% 0.04% 0.04% 0.04% 0.02% 0.02% 4. Most common trigraphs are the, and, tha, ent, ion, tio, for, nde, has, nce. 5. Most common repeated letters are ll, ee, ss, tt, oo, mm, ff. 5

6 2.4 A Maplet for Cryptanalysis of Substitution Ciphers 1. Example: Let s cryptanalyze the following cipher text, which is from a random cipher alphabet, using the substitution cipher cryptanalysis maplet: QMOAAOVTDCJXOQMAANGATPVTDCSTLAOQMACPFCAGATJX OQMALYKOJNXOLCVTQMAVOMKNNCXJCQXTATVTAJXOBXOQKNB ATLXXBALQXLVAXTAJXOQMALKOPNXOLXTMVCLKOPQMOXTAVTQ MANKTLXJBXOLXOYMAOAQMACMKLXYCNVAXTAOVTDQXOSNAQM ABKNNXTAOVTDQXJVTLQMABXTAOVTDQXROVTDQMABKNNKTLVT QMALKOPTACCVTLQMABVTQMANKTLXJBXOLXOYMAOAQMACMKL XYCNVA 2. Now taking the most common single, bigrams, and trigraphs gives: (a) single letter gives: 1 graph A occurs 39 times 1 graph X occurs 29 times 1 graph T occurs 29 times 1 graph O occurs 27 times 1 graph M occurs 22 times 1 graph Q occurs 22 times 1 graph L occurs 19 times 1 graph V occurs 18 times 1 graph N occurs 15 times 1 graph C occurs 14 times 1 graph K occurs 13 times 1 graph B occurs 9 times 1 graph J occurs 9 times 1 graph D occurs 6 times (b) digraph gives: 2 graph MA occurs 16 times 2 graph QM occurs 16 times 2 graph VT occurs 13 times 2 graph XO occurs 12 times 2 graph LX occurs 8 times 2 graph TA occurs 8 times 2 graph XT occurs 7 times 2 graph TL occurs 7 times 2 graph AO occurs 7 times 2 graph TD occurs 6 times 2 graph QX occurs 5 times 2 graph OQ occurs 5 times 2 graph OV occurs 5 times 2 graph AB occurs 4 times 2 graph DQ occurs 4 times 2 graph BX occurs 4 times 2 graph XJ occurs 4 times 2 graph KN occurs 4 times 2 graph TQ occurs 4 times 2 graph OL occurs 4 times 6

7 2 graph KO occurs 4 times 2 graph AL occurs 4 times 2 graph AC occurs 4 times 2 graph AT occurs 4 times 2 graph JX occurs 4 times (c) trigraph gives: 3 graph QMA occurs 14 times 3 graph XTA occurs 6 times 3 graph VTD occurs 6 times 3 graph OVT occurs 5 times 3 graph MAB occurs 4 times 3 graph TDQ occurs 4 times 3 graph TQM occurs 4 times 3 graph VTQ occurs 4 times 3 graph XOL occurs 4 times 3 graph OQM occurs 4 times 3 graph XOQ occurs 4 times 3 graph JXO occurs 4 times 3 graph AOV occurs 4 times 3 graph DQX occurs 3 times 3 graph TAO occurs 3 times 3 graph AQM occurs 3 times 3 graph KTL occurs 3 times 3 graph NKT occurs 3 times 3 graph OLX occurs 3 times 3 graph KOP occurs 3 times 3 graph LKO occurs 3 times 3 graph TLX occurs 3 times 3 graph BXO occurs 3 times 3 graph KNN occurs 3 times 3 graph MAL occurs 3 times 3 graph MAC occurs 3 times (d) double Double letter sequence NN occurs 3 times Double letter sequence AA occurs 2 times (e) Example: Break the random substitution cipher M RHBCQGBPBH GYKGU NXCR QXG GYBKLDU AZ HXQJBQHBQO BGR XDDXQCQGR MQN LMWBQO GUCL RCC GUC EBOUG, AKG YMGUCY ACHMKRC BGR XDDXQCQGR CJCQGKMEEZ NBC MQN M QCT OCQ- CYMGBXQ OYXTR KD GUMG BR PMLBEBMY TBGU BG (f) Example: Break the random substitution cipher URTLT EAPP HT PAU- UPT LWHK MJQ QAKMCCYAJUSTJUK TVTLXERTLT MJQ ET MLT MPP MCU UY TOCTBU UYY SWBR HWU URTJ AG YJT KBRTST YG RMCCAJTKK GMAPK RWSMJ JMUWLT UWLJK UY MJYURTL AG URT GALKU BMPBWPMUAYJ AK ELYJZ ET SMFT M KTBYJQ HTUUTL ET GAJQ BYSGYLU KYSTERTLT 7

8 2.5 Playfair Ciphers 1. Definition: Playfair ciphers use one or more keywords. Spaces and duplicate letters in the keyword(s) are removed, and the resulting letters are then used to form an array of letters, similar to the array used in keyword columnar substitution ciphers, except that for Playfair ciphers this array must always have exactly five letters per row. Also, I and J are considered to be the same letter in Playfair arrays, so J is not included. 2. Example: Encypt Strive not to be a success, but rather to be of value using keyword Albert Einstein. First, fill in the 5x5 key matrix with i and j being treated as one letter: Now prepare the plaintext: Remove the spaces and divide the text into groups of two letters. If any letter is doubled in a group, insert an x and regroup. Add an extra x at the end to complete the last group if needed. Now to encrypt each digraph: (a) If the two letters are in the same row, replace each letter with the letter immediately to its right, with the matrix wrapping around from the last column to the first. (b) If the two letters are in the same column, replace each letter with the letter immediately below it, with the matrix wrapping around from the last row to the first. (c) If the two letters are not in the same row or column, replace each letter by the letter that is in its row and is in the column of the other letter. 3. To decrypt, instead of going right and down, go left and up. The third rule remains the same. 4. Example: Decipher CQBZARTGRATDPISR using the same keyword. 8

9 5. Example: Use a Playfair cipher with keyword spies to encrypt the plaintext message cloak and dagger and then to decrypt the cipher text message MSRIHROMI- TIY. 2.6 A Maplet for Playfair Ciphers 9