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9.7. EXERCISES 183
10. Just as we combined addition and multiplication to form Linear Ciphers,
we can form Linear Hill Ciphers by first multiplying by a matrix, and then
adding another matrix to the result. That is, the pairs m of the message
are enciphered to Am + B, where A and B are the multiplier and addend
matrices, respectively.
(a) Use A =
( 13 5
8 13
)
and B =
( ) 8 10
to encipher linear.
12 14
(b) Use the same matrices to decipher CSKKI SEB. (Remember to subtract
before multiplying by the inverse.)
11. The matrices we used for Hill Ciphers were involutions – they are their own
inverses. Limiting ourselves to just( these)
matrices is rather restrictive. 10
a b
In general, the inverse of a matrix exists only when gcd(26, ad −
c d
( ) ed −eb
bc) = 1. If this it so, then the inverse matrix is
, where e
−ec ea
is the inverse of ad − bc modulo 26 found from the Euclidean Algorithm.
(The value ad − bc is called the determinant, since it determines whether
or not the matrix has an inverse.)
(a) Which of the following matrices have inverses modulo 26
( ) ( ) ( )
8 4 5 10 11 6
(i) . (ii) . (iii) .
13 9 16 23 7 4
( ) 2 5
(b) Find the inverse of .
3 9
(c) Encipher determinants with
( ) 2 5
.
3 9
(d) Decipher JWOSY QGKVO. It was enciphered with
( ) 2 5
.
3 9
12. Modern cipher systems all encipher blocks of text at a time. (The block
size in this chapter has generally been two, meaning two letters at a time.)
Here we look at a cipher with block size two.
To do this, first, rather than converting the message blocks to 2-12-15-
3-11-19 before multiplying, we convert it to 0212-1503-1119. Notice the
zeros. With them aa=0101=101 is distinct from l=11; without it is not.
Then performing a Decimation Cipher with a block size of two differs from
our previous Decimation Cipher only in that the modulus must be chosen
to be larger than 2525. That is, choose a number N larger than 2525,
and an enciphering key e. Use the Euclidean Algorithm to check that
gcd(e, N) = 1 and to find the deciphering key d. Encipher by multiplying
10 Recall that for Decimation ciphers there are 12 proper key pairs modulo 26, but only 2 in
which the enciphering and deciphering key are the same (i.e., involutions). In the Hill ciphers
there are 157, 248 proper pairs of 2 × 2 matrices, but only 312 of these are involutions.