Substitution Transposed

7943 /www/nrich/html/content/id/7943/ 1 2011-12-21T10:40:26 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">This problem follows on from <a href="/4957">Substitution Cipher</a> and <a href="/7940">Transposition Cipher</a> Below is a secret message. It's been double encrypted, first using a substitution cipher and then using a transposition cipher. Can you decipher it? <pre> whhujnjwuzlwvvdhgdlkqwdhvdkelqhldwurlhhwklwwuhdwdgqjkhzqqbrhowkgbqkq ywhuovwfzebzlhhzfwrlkwrhddhhhkdqsqhkkuohkrfqqdujdbjdvulhgoidpwggvhlg vwqdhollduwzudhwqqhhrwqmguqzrzlljykrzyaprgxuidlqdwfrhhwlhfdppvhduwwv kerukuwuhvldwqwbkgfkhgbrdkwsnpgrdkufrdrogpxllkwhhduqhhrrowohrvogrdoc pggugouodvdghzrxvbrhlwderdhqiohddqjdqrwqkquxwggurujbwkphoxdhgluldfhi qhfvwhdrvubpfhokrqzfqkxddqjvbornghdofkrwlseqgldvlwijgogklqvosrvffxxy kuosozzwyvgrxrrdgehhrddddlhhduvwirovxguqvvewqsqb </pre> If you want to work on a computer, you can download the ciphertext as a text file <a href="/content/id/7943/subtrans.txt">here</a>. You can then use our <a href="http://nrich.maths.org/7983">Cipher Challenge Toolkit</a> to help you decrypt it. If you are interested in code breaking you might enjoy the <a href="/7934">Secondary Cipher Challenge</a>.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Well done to Holly from Hymers School who cracked this problem. She described her method as follows: The way I solved this problem was looking at the frequency graph on the Cipher Toolkit. I realised that it would line up with the normal English letter frequency if every letter was moved three to the right (e.g. h-e, d-a), so I moved them all three to the right and came up with a code, which I then transposed. I discovered the factors of 456 are 1, 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, 228 and 456. The first and last two wouldn't be likely, so I kept trying the different factors until I came up with 19 being the correct number of columns. Writing the message in a 24 by 19 grid and decoding the Caesar shift of +3 gives: <pre> t e e r g k g t r w i t s s a e d a i </pre> <pre> h n t a e s a h b i n e i a t r o i e e t h i t t r e a t a d n g h e w n n y o e l t h d y n h n v t e r l s t c w b y w i e e w c t o i h t o e a a e e e h a n p n e h h r l e h o c n n a r g a y g a s r i e d l f a m t d d s e i d s t n a e l i i a r t w r a e t n n e e o t n j d r n w o w i i g v h o w v x m o d u r f a i n a t c o e e t i e c a m m s e a r t t s h b o r h r t r e s i a t n t y h d c h e d y o a h t p k m d o a h r c o a o l d m u i i h t e e a r n e e o o l t l e o s l d o a l z m d d r d l r l a s a d e w o u s y o e i t a b o a e n f l e a a n g a n o t n h n r u t d d r o r g y t h m e l u a e d i r i a c e f n e c s t e a o s r y m c e l h o n w c n h u a a n g s y l o k d e a l c h o t i p b n d i a s i t f g d l d h i n s l p o s c c u u v h r l p l w w t v s d o u o o a d b e e o a a a a i e e a r s t f o l s u d r n s s b t n p n y </pre> <pre> </pre> Reading down the columns gives the plaintext: <pre> theywerenotrailwaychildrentobeginwithidontsupposetheyhadeverthoughta boutrailwaysexceptasameansofgettingtomaskelyneandcooksthepantomimezo ologicalgardensandmadametussaudstheywerejustordinarysuburbanchildren andtheylivedwiththeirfatherandmotherinanordinaryredbrickfrontedvilla withcolouredglassinthefrontdooratiledpassagethatwascalledahallabathr oomwithhotandcoldwaterelectricbellsfrenchwindowsandagooddealofwhitep aintandeverymodernconvenienceasthehouseagentssay </pre> Here is the text that Holly managed to decipher: They were not railway children to begin with. I don't suppose they had ever thought about railways except as a means of getting to Maskelyne and Cook's, the Pantomime, Zoological Gardens, and Madame Tussaud's. They were just ordinary suburban children, and they lived with their Father and Mother in an ordinary red-brick-fronted villa, with coloured glass in the front door, a tiled passage that was called a hall, a bath-room with hot and cold water, electric bells, French windows, and a good deal of white paint, and 'every modern convenience', as the house-agents say. It's the first paragraph from "The Railway Children" by E. Nesbit</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem?</h3> Together with the <a href="/7934">Secondary Cipher Challenge</a>, this problem provides a challenging follow-up for students who have worked on <a href="/4957">Substitution Cipher</a> and <a href="/7940">Transposition Cipher</a>. <h3> </h3> <h3>Possible approach</h3> As this problem is quite challenging, it could be offered as an extension for those students who have sailed through the easier codebreaking problems. If a whole class is going to work on the problem, they could work in small groups and share out responsibility for checking different transposition arrangements for each likely substitution. If a computer room is available, introduce students to the <a href="/7983">Cipher Challenge Toolkit</a> and give them plenty of time to explore it and learn how to use the tools. <h3> </h3> <h3>Possible extension</h3> The <a href="/7081">Stage 5 Cipher Challenge</a> offers a set of seven very difficult linked ciphers to crack. <h3>Possible support</h3> Spend plenty of time working on and discussing the separate Substitution and Transposition problems before trying this one.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 456 has 14 factors. The two methods used to encrypt the message are commutable, so it doesn't matter which method you decipher first. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">It's the first paragraph from "The Railway Children" by E. Nesbit encrypted with a matrix transposition of size 19 by 24 and a Caesar shift of +3. Writing the message in a 24 by 19 grid and decoding the Caesar shift gives: <pre> t e e r g k g t r w i t s s a e d a i </pre> <pre> h n t a e s a h b i n e i a t r o i e e t h i t t r e a t a d n g h e w n n y o e l t h d y n h n v t e r l s t c w b y w i e e w c t o i h t o e a a e e e h a n p n e h h r l e h o c n n a r g a y g a s r i e d l f a m t d d s e i d s t n a e l i i a r t w r a e t n n e e o t n j d r n w o w i i g v h o w v x m o d u r f a i n a t c o e e t i e c a m m s e a r t t s h b o r h r t r e s i a t n t y h d c h e d y o a h t p k m d o a h r c o a o l d m u i i h t e e a r n e e o o l t l e o s l d o a l z m d d r d l r l a s a d e w o u s y o e i t a b o a e n f l e a a n g a n o t n h n r u t d d r o r g y t h m e l u a e d i r i a c e f n e c s t e a o s r y m c e l h o n w c n h u a a n g s y l o k d e a l c h o t i p b n d i a s i t f g d l d h i n s l p o s c c u u v h r l p l w w t v s d o u o o a d b e e o a a a a i e e a r s t f o l s u d r n s s b t n p n y </pre> <pre> </pre> Reading down the columns gives the plaintext: <pre> theywerenotrailwaychildrentobeginwithidontsupposetheyhadeverthoughta boutrailwaysexceptasameansofgettingtomaskelyneandcooksthepantomimezo ologicalgardensandmadametussaudstheywerejustordinarysuburbanchildren andtheylivedwiththeirfatherandmotherinanordinaryredbrickfrontedvilla withcolouredglassinthefrontdooratiledpassagethatwascalledahallabathr oomwithhotandcoldwaterelectricbellsfrenchwindowsandagooddealofwhitep aintandeverymodernconvenienceasthehouseagentssay </pre> <pre> </pre> <pre> </pre> <pre> </pre> <pre> </pre></mdoxml> 2 5 0 0 1 1 0 Substitution Transposed Substitution and Transposition all in one! How fiendish can these codes get? Applications Codes and cryptography Numbers and the Number System Factors and multiples Handling, Processing and Representing Data Comparing data Handling, Processing and Representing Data Frequency tables and diagrams