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184 CHAPTER 9. DIGRAPHIC CIPHERS
pairs of letters by e modulo N, and decipher by multiplying by d. (The
resulting ciphertexts will be numbers, rather than letters.)
(a) If the modulus N is chosen to be 2999, show that e = 100 is a proper
enciphering key. Find the corresponding deciphering key d.
(b) Use e = 100 and N = 2999 to encipher blocks.
(c) Use the deciphering key found in part (a) to decipher 2543-1513-
1460.
(d) If N = 3001 and e = 29, find d.
(e) Use e = 29 and N = 3001 to encipher number.
(f) Use the deciphering key found in part (d) to decipher 218-2644-17-
2037.
13. (Continuing the ideas of Exercise 12.) A problem with the cipher from
Exercise 12 is that the ciphertext is numbers not letters. If we want
a bigram-decimation cipher, perhaps we should use 676 = 26 2 as the
modulus, rather than 26. Here is one way to try to do this:
Choose a key k with 1 ≤ k < 676, relatively prime to 26. For each pair of
plaintext letters p 1 p 2 , first convert the letters into their numerical equivalent,
then compute the cipher-number c = ( k ∗ (p 1 ∗ 26 + p 2 ) ) %676. The
ciphertext is then C 1 C 2 , where C 1 and C 2 are the quotient and remainder
when c is divided by 26.
For example, let k = 19. Then to = 20 15 has ciphernumber c = ( 19 ∗
(20 ∗ 26 + 15) ) %676 = (19 ∗ 535)%676 = 25. Since 25 ÷ 26 = 0 = Z and
25%26 = 25 = Y, to is enciphered to ZY.
To decipher, first find the inverse of 19 modulo 676, which is 427. Then
multiply ZY = 25 = 00225 by 427%676 to receive 535. Finally 535 ÷ 26 =
20 = t and 535%26 = 15 = o.
(a) Use k = 101 to encipher stat.
(b) Use k = 219 to encipher data.
(c) Find the inverse of k = 87 modulo 676, and use it to decipher BONO.
(d) Carefully examine the ciphertexts from this Exercise. What appears
to be happening What does this say about the value of this cipher
In particular, how “digraphic” is this digraphic ciphers
14. It was hinted in the text that Hill Ciphers are simply a fancy Decimation
Cipher. In this Exercise we see why this is so.
(a) Find the form the enciphering matrix takes if we choose a = 0.
(b) Encipher the “message” p 1 p 2 p 3 p 4 p 5 p 6 using the matrix you found in
part (a).
(c) Is the cipher performed in part (b) a Decimation Cipher If not, how
close is it