Transposition Ciphers

Transcription

1 Transposition Ciphers A transposition cipher is one in which plaintext symbols are rearranged (that is, permuted or transposed) to produce ciphertext. The method of rearrangement, shared by Alice and Bob, may be typographical or mathematical in nature. 1

2 Simple Columnar Transposition The key information is the number k of columns. Encipherment: Plaintext is written in lines k letters wide and then transcribed column by column left to right to produce ciphertext. 2

3 Decipherment: If n is the length of the ciphertext, it is written column by column left to right down in a k (n DIV k) rectangular array with a tail of length n MOD k as shown. Transcribing row by row produces plaintext. k M E S S A G E... A N D M O R E... A N D S T I L... L A S T n MOD k n DIV k 3

4 Example: Suppose the key (number of columns) is 8. Encipher the plaintext NOTHING IN THE WORLD IS MORE DANGEROUS THAN SINCERE IGNORANCE AND CONSCIENTIOUS STUPIDITY Martin Luther King, Jr. 4

5 Solution: N O T H I N G I N T H E W O R L D I S M O R E D A N G E R O U S T H A N S I N C E R E I G N O R A N C E A N D C O N S C I E N T I O U S S T U P I D I T Y Ciphertext: NNDAT EAOII OTINH RNNOD THSGA ECSUI HEMEN IECST IWORS GAISY NOROI NNETG REUNO DNUIL DSCRC TP 5

6 Example: If the number of columns is 7 and the ciphertext is GPSDO AILTI VRVAA WETEC NITHM EDLHE TALEA ONNE find the plaintext. 6

7 Solution: There are 39 letters in the ciphertext, so there are 39 DIV 7 = 5 full rows and one partial row with 39 MOD 7 = 4 letters. G I V E M E A P L A C E T O S T A N D A N D I W I L L M O V E T H E E A R T H Plaintext: GIVE ME A PLACE TO STAND AND I WILL MOVE THE EARTH Archimedes 7

8 Keyword Columnar Transposition The order of transcription of the columns is determined by the alphabetical order of letters in the keyword. If repeated letters are allowed in the keyword, then the column under the leftmost occurence of the letter is transcribed first, followed successively by those to the right. Decipherment is as for the simple columnar transposition except that columns are written into the template in the order specified by the alphabetical order of the keyword s letters. 8

9 Example: With keyword QUICK, encipher the plaintext DO NOT SEEK THE TRUTH, ONLY CEASE TO CHERISH YOUR OPINIONS Zen Proverb 9

10 Solution Work: Q U I C K D O N O T S E E K T H E T R U T H O N L Y C E A S E T O C H E R I S H Y O U R O P I N I O N S Ciphertext: OKRNA CSRIN ETOEO IUNTT ULSHH OODSH TYEEY PNOEE HCTRO IS 10

11 Example: Decipher NSCRG LEXCT OEFNE HNRTL HOAHT OEICY NOIOT TEEGK SGWAO IHIAA NRWEN OTKRT DDPE if the keyword was k = COMPARE. 11

12 Solution: There are 64 letters in the ciphertext and 7 letters in the keyword, so it fits in a template with 64 DIV 7 = 9 full rows and a short row with 64 MOD 7 = 1 letter. 12

13 C O M P A R E T E C H N O L O G Y I S T H E K N A C K O F S O A R R A N G I N G T H E W O R L D T H A T W E D O N O T E X P E R I E N C E I T 13

14 C O M P A R E T E C H N O L O G Y I S T H E K N A C K O F S O A R R A N G I N G T H E W O R L D T H A T W E D O N O T E X P E R I E N C E I T Plaintext: TECHNOLOGY IS THE KNACK OF SO ARRANGING THE WORLD THAT WE DO NOT EXPERIENCE IT Max Frisch (1911 ), Rollo May, The Cry for Myth, Norton, NY, p

15 Cryptanalysis For a simple columnar transposition, cryptanalysis is relatively easy: try deciphering with various numbers of columns until intelligible plaintext shows up. 15

16 Example: Cryptanalyze this! SISEE RMIHI GHNST SEANA VAGOI MDNGN IRIMM OEDTG N Solution k = 2? S A I G S O E I E M.. gives SAIGSOEIEM..., which is not English. 16

17 k = 3? S T G I S N S E I E A R E N I... gives STGISNSEIEARENI..., which is not English. 17

18 k = 4? S H A R I N G I S S O M E T I M E S M O R E D E M A N D I N G T H A N G I V I N G gives SHARINGISSOMETIMESMOREDEMANDINGTHANGIVING Mary Bateson 18

19 Cryptanalysis of a Keyword Columnar Transposition Basic idea: determine the number of columns and then anagram the columns to find an intelligibile message. Example: Suppose that SRUGE PEDIA SDWXC SATLS NRNMT EAEEE AOMSU ITENI KHBBI IDEER ENEW was produced by a keyword columnar transposition and that EXPERIENCE was part of the plaintext. Find the plaintext. 19

20 Solution: k=2? The plaintext EXPERIENCE would encipher either * E E X X P P E E R as R I or I E E N N C C E E * In either case, the sequences EPREC and XEINE should appear. They do not, so this is not the keyword length. 20

21 k=3? The plaintext EXPERIENCE would encipher either as or * E X P E R I E N C E * or * * E X P E R I E N C E E X P E R I E N C E In any case, the sequences EEEE, XRN, and PIC should appear. They do not, so this is not the keyword length. 21

22 Continuing this way and hoping that the keyword has fewer than 10 letters, we find that assuming k = 7 implies * * * * * * * E X P E R I E N C E * * * * The sequences EN, XC, and PE should appear in the ciphertext. They do! SRUGE PEDIA SDWXC SATLS NRNMT EAEEE AOMSU ITENI KHBBI IDEER ENEW 22

23 7 Columns fit in this template; determine the long and short columns

24 The columns are either M I S A T I E S A R T E K E U S U L A H R I D G S E B E T W E N E B N E X P R E I E N C E N A I W S D M O D if MSU ITEN and EER ENEW are short, or M I S A T I D S A R T E K E U S U L A H E I D G S E B R T W E N E B E E X P R E I N N C E N A I E S D M O W if MSU ITEN and IKHBBII are short, or 24

25 M I S A T K E S A R T E H E U S U L A B R I D G S E B E T W E N E I N E X P R E I E N C E N A D W I S D M O if KHBBI ID and EER ENEW are short. 25

26 Anagramming the third set of columns gives M I S T A K E S A R E T H E U S U A L B R I D G E S B E T W E E N I N E X P E R I E N C E A N D W I S D O M 26