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Teaching the Group Theory of Permutation Ciphers

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One of the first topics often taught in an abstract algebra class is permutations, since they provide good examples of non-commutative finite groups which the students can manipulate and visualize. This visualization is often done through symmetry groups. For students who are less geometrically inclined, however, the use of permutation ciphers provides another good way of motivating permutations. They can easily be used to illustrate composition, non-commutativity, inverses, and the order of group elements, which are fundamental topics in group theory. We will give examples of how this can be done and suggest other courses besides abstract algebra in which this could also prove useful.

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Teaching the Group Theory of Permutation Ciphers

  1. 1. Teaching the Group Theory of Permutation Ciphers Joshua Holden Rose-Hulman Institute of Technology http://www.rose-hulman.edu/~holden Joshua Holden (RHIT) Group Theory of Permutation Ciphers 1 / 13
  2. 2. A historical example Taj ad-Din Ali ibn ad-Duraihim (1312–1361), worked and taught in Damascus and Cairo. In a book that was considered lost until the late 20th century, ibn ad-Duraihim described 24 variations of transposition ciphers. Example plaintext: dr in kt ot he ro se ciphertext: RD NI TK TO EH OR ES Joshua Holden (RHIT) Group Theory of Permutation Ciphers 2 / 13
  3. 3. So what? We are seeing the first explicit example of a permutation cipher. Definition A permutation is a bijection from a finite set to itself. Example plaintext: dr in kt ot he ro se ciphertext: RD NI TK TO EH OR ES Ibn ad-Duraihim’s permutation is 1 2 2 1 : Joshua Holden (RHIT) Group Theory of Permutation Ciphers 3 / 13
  4. 4. Another example Example plaintext: ruby wine ciphertext: UYBR IENW This is a cipher based on the permutation 1 2 3 4 2 4 3 1 : Note Some people prefer to use a notation based on where the letters go. We will use one based on where they come from. Joshua Holden (RHIT) Group Theory of Permutation Ciphers 4 / 13
  5. 5. Keys and keywords Clearly the key to a permutation cipher is the permutation. We can choose and remember a permutation by a keyword. Example TALE is a keyword for the permutation 1 2 3 4 2 4 3 1 : plaintext: thep aper andt hepe nllu Joshua Holden (RHIT) Group Theory of Permutation Ciphers 5 / 13
  6. 6. Keys and keywords Clearly the key to a permutation cipher is the permutation. We can choose and remember a permutation by a keyword. Example TALE is a keyword for the permutation 1 2 3 4 2 4 3 1 : keyword: TALE TALE TALE TALE TALE plaintext: thep aper andt hepe nllu Joshua Holden (RHIT) Group Theory of Permutation Ciphers 5 / 13
  7. 7. Keys and keywords Clearly the key to a permutation cipher is the permutation. We can choose and remember a permutation by a keyword. Example TALE is a keyword for the permutation 1 2 3 4 2 4 3 1 : 4132 4132 4132 4132 4132 keyword: TALE TALE TALE TALE TALE plaintext: thep aper andt hepe nllu Joshua Holden (RHIT) Group Theory of Permutation Ciphers 5 / 13
  8. 8. Keys and keywords Clearly the key to a permutation cipher is the permutation. We can choose and remember a permutation by a keyword. Example TALE is a keyword for the permutation 1 2 3 4 2 4 3 1 : 4132 4132 4132 4132 4132 keyword: TALE TALE TALE TALE TALE plaintext: thep aper andt hepe nllu ciphertext: HPET PREA NTDA EEPH LULN Joshua Holden (RHIT) Group Theory of Permutation Ciphers 5 / 13
  9. 9. Keys and keywords Clearly the key to a permutation cipher is the permutation. We can choose and remember a permutation by a keyword. Example TALE is a keyword for the permutation 1 2 3 4 2 4 3 1 : 4132 4132 4132 4132 4132 keyword: TALE TALE TALE TALE TALE plaintext: thep aper andt hepe nllu ciphertext: HPET PREA NTDA EEPH LULN HPETP REANT DAEEP HLULN Joshua Holden (RHIT) Group Theory of Permutation Ciphers 5 / 13
  10. 10. Deciphering In order to decipher, you need to take the inverse of the permutation. The inverse of 1 2 3 4 2 4 3 1 is 1 2 3 4 4 1 3 2 . Note We saw the numbers 4132 earlier! Equivalently, alphabetize the keyword and read off the plaintext in original keyword order: Example ciphertext: HBET TLTA ADNE HSET ODRW Joshua Holden (RHIT) Group Theory of Permutation Ciphers 6 / 13
  11. 11. Deciphering In order to decipher, you need to take the inverse of the permutation. The inverse of 1 2 3 4 2 4 3 1 is 1 2 3 4 4 1 3 2 . Note We saw the numbers 4132 earlier! Equivalently, alphabetize the keyword and read off the plaintext in original keyword order: Example keyword: AELT AELT AELT AELT AELT ciphertext: HBET TLTA ADNE HSET ODRW Joshua Holden (RHIT) Group Theory of Permutation Ciphers 6 / 13
  12. 12. Deciphering In order to decipher, you need to take the inverse of the permutation. The inverse of 1 2 3 4 2 4 3 1 is 1 2 3 4 4 1 3 2 . Note We saw the numbers 4132 earlier! Equivalently, alphabetize the keyword and read off the plaintext in original keyword order: Example 2431 2431 2431 2431 2431 keyword: AELT AELT AELT AELT AELT ciphertext: HBET TLTA ADNE HSET ODRW Joshua Holden (RHIT) Group Theory of Permutation Ciphers 6 / 13
  13. 13. Deciphering In order to decipher, you need to take the inverse of the permutation. The inverse of 1 2 3 4 2 4 3 1 is 1 2 3 4 4 1 3 2 . Note We saw the numbers 4132 earlier! Equivalently, alphabetize the keyword and read off the plaintext in original keyword order: Example 2431 2431 2431 2431 2431 keyword: AELT AELT AELT AELT AELT ciphertext: HBET TLTA ADNE HSET ODRW plaintext: theb attl eand thes word Joshua Holden (RHIT) Group Theory of Permutation Ciphers 6 / 13
  14. 14. How many keys? The number of keys for a permutation cipher on n letters is the number of permutations, n!. But one of them gives the trivial cipher: Example plaintext: ruby wine Joshua Holden (RHIT) Group Theory of Permutation Ciphers 7 / 13
  15. 15. How many keys? The number of keys for a permutation cipher on n letters is the number of permutations, n!. But one of them gives the trivial cipher: Example keyword: ABCD ABCD plaintext: ruby wine Joshua Holden (RHIT) Group Theory of Permutation Ciphers 7 / 13
  16. 16. How many keys? The number of keys for a permutation cipher on n letters is the number of permutations, n!. But one of them gives the trivial cipher: Example 1234 1234 keyword: ABCD ABCD plaintext: ruby wine Joshua Holden (RHIT) Group Theory of Permutation Ciphers 7 / 13
  17. 17. How many keys? The number of keys for a permutation cipher on n letters is the number of permutations, n!. But one of them gives the trivial cipher: Example 1234 1234 keyword: ABCD ABCD plaintext: ruby wine ciphertext: RUBY WINE Joshua Holden (RHIT) Group Theory of Permutation Ciphers 7 / 13
  18. 18. Double encipherment? Can we improve the security of a permutation cipher by using two different keys? Example 4132 4132 4132 4132 4132 keyword: TALE TALE TALE TALE TALE plaintext: thep aper andt hepe nllu first ciphertext: HPET PREA NTDA EEPH LULN 4312 4312 4312 4312 4312 keyword: POEM POEM POEM POEM POEM first ciphertext: hpet prea ntda eeph luln second ciphertext: ETPH EARP DATN PHEE LNUL Joshua Holden (RHIT) Group Theory of Permutation Ciphers 8 / 13
  19. 19. But look: Example plaintext: thep aper andt hepe nllu second ciphertext: ETPH EARP DATN PHEE LNUL This is the same as if you had just used the key . 1 2 3 4 3 1 4 2 The combination of two ciphers is called a product cipher. In fact: 1 2 3 4 2 4 3 1 1 2 3 4 3 4 2 1 = : 1 2 3 4 3 1 4 2 Note Not everyone writes permutation products in the same order, either. Joshua Holden (RHIT) Group Theory of Permutation Ciphers 9 / 13
  20. 20. Noncommutativity Note that 1 2 3 4 2 4 3 1 1 2 3 4 3 4 2 1 is not the same as 1 2 3 4 3 4 2 1 1 2 3 4 2 4 3 1 : (I.e., permutation products are not commutative.) If you don’t believe it, try encrypting our plaintext using the keyword POEM first and then the keyword TALE. Joshua Holden (RHIT) Group Theory of Permutation Ciphers 10 / 13
  21. 21. Bad keys Some ciphers have bad keys — they don’t decrypt properly. Example 1 2 3 4 looks like a permutation, but... 4 1 1 3 plaintext: garb agei ngar bage outx Joshua Holden (RHIT) Group Theory of Permutation Ciphers 11 / 13
  22. 22. Bad keys Some ciphers have bad keys — they don’t decrypt properly. Example 1 2 3 4 looks like a permutation, but... 4 1 1 3 plaintext: garb agei ngar bage outx ciphertext: BGGR IAAE RNNA EBBG XOOT Joshua Holden (RHIT) Group Theory of Permutation Ciphers 11 / 13
  23. 23. Bad keys Some ciphers have bad keys — they don’t decrypt properly. Example 1 2 3 4 looks like a permutation, but... 4 1 1 3 plaintext: garb agei ngar bage outx ciphertext: BGGR IAAE RNNA EBBG XOOT plaintext: g?rb a?ei n?ar b?ge o?tx It’s a function but not a permutation — it doesn’t have an inverse. Joshua Holden (RHIT) Group Theory of Permutation Ciphers 11 / 13
  24. 24. Group theory Okay, what about the math? So far we’ve actually covered: An example of a group, the permutation (cipher)s on n letters. (Actually, infinitely many groups!) The order of the group (number of keys). The group identity, the trivial permutation (cipher). Inverses in the group, i.e. decryption. The group operation, permutation (cipher) products. Noncommutativity — permutations are not commutative, and neither are permutation ciphers. A set which contains a group, but is not a group, because it is not closed under inverses, i.e. functions. Joshua Holden (RHIT) Group Theory of Permutation Ciphers 12 / 13
  25. 25. HNAT SOFK LSIR EINT GZXN NOJET EHYET EMNSG IEOUA Joshua Holden (RHIT) Group Theory of Permutation Ciphers 13 / 13

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