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9.2. HILL CIPHERS 171
th 2.96 es 1.34 at 1.16
in 1.88 on 1.31 ed 1.14
er 1.75 st 1.24 ti 1.10
an 1.56 nt 1.21 nd 1.07
re 1.45 en 1.20 to 1.04
Figure 9.3: 18 Most Frequent Bigrams, in percent
Monthly, the undergraduate journal of the American Mathematical Society. He
eventually received U.S. patent 1,845,947 for an apparatus that mechanically
performed his cipher. Much of Hill’s work involved the use of mathematics
in communications, for example, methods for splicing telephone cables. His
cipher provides us with an easy example of a polygraphic cipher, but is more
important because it shows that by the middle of the first half of the 20th
century cryptology was being done primarily by mathematicians.
To explain the cipher we need to introduce matrices (plural of matrix).
Matrices are simply rectangular arrays of numbers and are quite important
in mathematics, chemistry ( ) and physics. We will deal only with two-by-two
3 0
matrices, such as . Matrices are very easy to add and subtract –
−1 4
simply perform the operation entry-by-entry. For example
( ) 4 3
+
−2 7
( ) 0 5
=
2 4
( ) 4 8
0 11
and
( ) 1 9
−
0 2
( ) −2 3
=
5 1
( ) 3 6
.
−5 1
Multiplication is a bit more complicated as each row of the first matrix is
multiplied by the column entries in the second. (Rows go across and columns
go down.) For example,
( ) ( ( ) ( )
3 5 2 3 × 2 + 5 × 9 6 + 45
× =
=
=
14 23 9)
14 × 2 + 23 × 9 28 + 209
( )
51
,
235
and
( ) 3 5
×
14 23
( ) ( ) ( )
15 3 × 15 + 5 × 19 45 + 95
=
=
=
19 14 × 15 + 23 × 19 210 + 437
( ) a b
The general form for multiplication is
c d
( e
× =
f)
( ) 140
.
647
( )
a × e + b × f
.
c × e + d × f