Grade 6 Math Circles. History of Cryptography - Solutions

Transcription

1 Faculty of Mathematics Waterloo, Ontario N2L 3G1 Problem Set Grade 6 Math Circles November 19/20, 2013 History of Cryptography - Solutions 1. A Caesar Cipher with a shift of 5 is used. ciphertext V W X Y Z A B C D E F G H ciphertext I J K L M N O P Q R S T U The ciphertext letter K corresponds to the plaintext letter P. Continuing this: K P O V G D O O G Z W J J B T D I O J D O P U T A L I T T L E B O O G Y I N T O I T How do you get a kleenex to dance? Put a little boogy into it! 2. The Atbash cipher is used. A B C D E F G H I J K L M Z Y X W V U T S R Q P O N The ciphertext letter L corresponds to the plaintext letter O. Continuing this: 1

2 L K G R N F H K I R N V O P T I M U S P R I M E What kind of numbers transform? Optimus Prime! 3. A mixed alphabet cipher is used with keyword BIRD and keyletter P. ciphertext K L M N O P Q S T U V W X ciphertext Y Z B I R D A C E F G H J The ciphertext letter K corresponds to the plaintext letter A. Continuing this: K B Z W H Q Z Y A P O L Y G O N What do you call a dead parrot? A polygon! 4. A mixed alphabet cipher is used with keyword SLEEP and keyletter S. ciphertext F G H I J K M N O Q R T U ciphertext V W X Y Z S L E P A B C D The ciphertext letter F corresponds to the plaintext letter A. Continuing this: F G E T T I W D J Z A B U L L D O Z E R What do you call a sleeping bull? A bulldozer! 2

3 5. To turn the ciphertext into plaintext, we must do the Route cipher backwards. With 6 columns and 4 rows, the grid will have the following layout: Fill the grid with the ciphertext, starting in the lower left corner and spiralling counterclockwise. O O E F H C N R E M C L E E K A I E M W O T R S Begin reading the plaintext out of the grid vertically, starting in the upper left corner. ONEMOREWEEKOFMATHCIRCLES Now look for recognizable words within the plaintext. The plaintext is One more week of math circles. 6. To turn the ciphertext into plaintext, we must do the columnar transposition cipher backwards. Putting our keyword FLAKE into alphabetical order gives AEFKL. Thus the first group of letter in the ciphertext (NSNQl) corresponds with the letter A in the keyword, EOTIT corresponds with the letter E, MRMOC corresponds with the letter F, TCGUY corresponds with the letter K, and IIIOK corresponds with the letter L. Rearranging the groups of letters so that the keyword becomes FLAKE again gives: WRMOC IIIOK NSNQL TCGUY EOTIT Now, arrange the letters into columns, where each column is a group of letters from above. 3

4 W I N T E R I S C O M I N G T O O Q U I C K L Y T Beginning reading the rows of plaintext out, starting with the top row. WINTERISCOMINGTOOQUICKLYT Now look for recognizable words within the plaintext. The plaintext is Winter is coming too quickly(t). Note that the last T in the plaintext is just a place holder, or a null, meant to complete the final column above. 7. To turn the ciphertext into plaintext, we must do the Polybius Square cipher backwards. Every pair of numbers in the ciphertext is a coordinate for a plaintext letter in the Polybius Square H A K U N A M A T A T A Now look for recognizable words within the plaintext. The plaintext is Hakuna matata, from the Lion King. 8. To turn the ciphertext into plaintext, we must do the ADFGX cipher backwards. Putting our keyword EARTH into alphabetical order gives AEHRT. Thus the first group of letter in the ciphertext (GFFAGAD) corresponds with the letter A in the keyword, ADDFFDG corresponds with the letter E, AADGXGG corresponds with the letter H, XGAFAFA corresponds with the letter R, and FGDGDAF corresponds with the letter T. Rearranging the groups of letters so that the keyword becomes EARTH again gives: ADDFFDG GFFAGAD XGAFAFA FGDGDAF AADGXGG 4

5 Now, arrange the letters into columns, where each column is a group of letters from above. A G X F A D F G G A D F A D D F A F G G F G A D X D A F A G G D A F G Beginning reading the rows of letters out, starting with the top row. AGXFADFGGADFADDFAFGGFGADXDAFAGGDAFG Every pair of letters above is a coordinate for a plaintext letter in the modified Polybius Square used in the ADFGX cipher. (Note: There are 35 letters above, but encrypting with the Polybius Square can only result in an even amount of letters/numbers. Thus the last letter G must be a null.) AG XF AD FG GA DF AD DF AF GG FG AD XD AF AG GD AF L E T S G O T O A U S T R A L (J or I) A Now look for recognizable words within the plaintext. The plaintext is either Let s go to Australja or Let s go to Australia. The first option doesn t really make sense, so it must be the second one. 9. * You must decrypt in the opposite order of encryption. That is, first do the Polybius Square cipher backwards, and then decrypt again using the Atbash cipher. Every pair of numbers in the ciphertext is a coordinate for a letter in the Polybius Square S R H G L I/J B L U X I/J B K G L T I/J Z K S B Now decrypt using the Atbash cipher. The ciphertext letter S corresponds to the plaintext letter H. Continuing this: 5

6 S R H G L I/J B L U X I/J B K G L T I/J Z K S B H I S T O R/Q Y O F C R/Q Y P T O G R/Q A P H Y The plaintext is History of Cryptography. 10. * You must decrypt in the opposite order of encryption. That is, first do the Route cipher backwards, and then decrypt again using the mixed alphabet cipher. With 5 columns and 4 rows, the grid will have the following layout: Fill the grid with the ciphertext, starting in the upper left corner and spiralling clockwise. O J Z M H C G W T D T D N A I C R Z J B Begin reading the letters out of the grid vertically, starting in the upper left corner. OCTCJGDRZWNZMTAJHDIB A mixed alphabet cipher is then used with keyword PARTY and keyletter B. ciphertext Z P A R T Y B C D E F G H ciphertext I J K L M N O Q S U V W X The ciphertext letter O corresponds to the plaintext letter T. Continuing this: O C T C J G D R Z W N Z M T A J H D I B T H E H O L I D A Y S A R E C O M I N G 6

7 The plaintext is The holidays are coming. 11. ** You must decrypt in the opposite order of encryption. That is, first do the ADFGX cipher backwards, then decrypt again using the Atbash cipher, and then decrypt again using the Caesar0 cipher with a shift of 3. To begin, we must do the ADFGX cipher backwards. Putting our keyword WATER into alphabetical order gives AERTW. Thus the first group of letter in the ciphertext (DDXGGGGGGX) corresponds with the letter A in the keyword, FGGADGXFDD corresponds with the letter E, GGDXD- FGFXD corresponds with the letter R, FXXGGDXAXF corresponds with the letter T, and GGDFXFFFGX corresponds with the letter W. Rearranging the groups of letters so that the keyword becomes EARTH again gives: GGDFXFFFGX DDXGGGGGGX FXXGGDXAXF FGGADGXFDD GGDXDFGFXD Now, arrange the letters into columns, where each column is a group of letters from above. G D F F G G D X G G D X X G D F G G A X X G G D D F G D G F F G X X G F G A F F G G X D X X X F D D Beginning reading the rows of letters out, starting with the top row. GDFFGGDXGGDXXGDFGGAXXGGDDFGDGFFGXXGFGAFFGGXDXXXFDD Every pair of letters above is a coordinate for a plaintext letter in the modified Polybius Square used in the ADFGX cipher. 7

8 GD FF GG DX GG DX XG DF GG AX XG GD DF J V U K U K W O U P W J O GD GF FG XX GF GA FF GG XD XX XF DD J C S Y C G V U R Y E H The Atbash cipher is then used. The ciphertext letter J corresponds to the plaintext letter Q. Continuing this: J V U K U K W O U P W J O J C S Y C G V U R Y E H Q E F P F P D L F K D Q L Q X H B X T E F I B V S A Caesar cipher with a shift of three is then used. ciphertext X Y Z A B C D E F G H I J ciphertext K L M N O P Q R S T U V W The ciphertext letter Q corresponds to the plaintext letter T. Continuing this: Q E F P F P D L F K D Q L Q X H B X T E F I B V S T H I S I S G O I N G T O T A K E A W H I L E Y V The plaintext is This is going to take a while. The final two letters are nulls. 8

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