Transcript Excel Definitions

390 Codes, Ciphers,
and Cryptography
Polygraphic Substitution
Ciphers – Hill’s System
Hill’s System
We now look at the system for enciphering
blocks of text developed by Lester Hill.
 Matrices form the basis of this substitution
cipher!
 We’ll work with blocks of size two letters –
the idea can be generalized to larger
blocks.
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Steps to Encipher a Message
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1. Choose a 2 x 2 matrix
with entries in Z26 for a key.
Make sure that (ad – bc)-1 (mod 26) exists, i.e.
(ad – bc) = 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, or
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This will guarantee that A-1 exists (mod 26).
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Steps to Encipher a Message
2. Split the plaintext into pairs and assign
numbers to each plaintext letter, with a =
1, b = 2, … , z = 26 = 0 (mod 26).
 Plaintext: p1p2|p3p4| … |pn-1pn
 If necessary, append an extra character to
the plaintext to get an even number of
plaintext characters.
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Steps to Encipher a Message
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Example 8
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Use Hill’s scheme to encipher the
message: “Meet me at the usual place at
ten rather than eight o’clock.”
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Example 8
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Solution: For the key,
choose a 2 x 2 matrix,
with entries in Z26.
Note that
(ad – bc) (mod 26)
= (9*7 – 4*5) (mod 26)
= (63 – 20) (mod 26)
= 43 (mod 26)
= 17 (mod 26)
and 17-1 (mod 26) exists!
 More on this later …
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Example 8
Next convert the plaintext into pairs of
numbers from Z26:
 me | et | me | at … cl | oc | kz.
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13,5 | 5,20 | 13,5 | 1,20 | … 3,12 | 15,3 | 11,0
Now convert the plaintext to numbers to
ciphertext numbers, using (*) above.
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Example 8
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Example 8
Thus, “me” is encrypted as “GV”.
 Try the next pair!
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Example 8
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Example 8
Thus, “et” is encrypted as “UI”.
 HW – Finish encrypting message!
 Note that for the word “meet”, the first “e”
is encrypted as “G” and the second “e” is
encrypted as “U”.
 Frequency analysis won’t work for this
scheme!
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Deciphering a Message
To decipher a message encrypted with
Hill’s Scheme, we can use the idea of
matrix inverses!
 Since ciphertext (ck,ck+1) is obtained from
plaintext (pk,pk+1) by multiplying key matrix
A by plaintext (pk,pk+1), all we need to do is
multiply matrix A-1 by ciphertext (ck,ck+1).
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Deciphering a Message
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Deciphering a Message
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Deciphering a Message
The same idea will work for matrices of
numbers from Z26!
 Matrix A will be invertible, provided that
(ad-bc)-1 (mod 26) exists!
 The only difference is that instead of
1/(ad-bc), we need to use (ad-bc)-1.
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Deciphering a Message
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Deciphering a Message
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Deciphering a Message
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Example 9
Decipher the ciphertext found above in
Example 8!
 Write ciphertext as pairs of numbers in Z26:
 GV | UI
 7,22 | 21,9
 Use the inverse of the key matrix to
decipher!
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Example 9
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Example 9
Thus, “GV” is deciphered as “me”.
 Repeat with “UI”.
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Example 9
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Example 9
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Thus, “UI” is deciphered as “et”.
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References
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Cryptological Mathematics by Robert
Edward Lewand (section on matrices).
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