Cryptography Reference
In-Depth Information
or
IISHT
IVRNS
AXXNA
Of course, we must ensure that transpositions are reversible. It seems natural to think that
they are, but how do we do this in the setting of matrices? Of course, what we seek is an
inverse
A
of the transposition matrix
A
so that
P
=
A
C.
Transposition matrices are easily invertible using Gauss-Jordan elimination with the
augmented matrix
. Since a transposition matrix is chosen so that each row and column
contains a single 1, and nothing else, any such matrix can be reduced to an identity matrix
simply by swapping rows! Thus, an inverse
A
|
I
A
of any transposition matrix
A
always exists.
E
XAMPLE
.
Let
A
be defined as the same matrix used in our transposition cipher example; that
is
00100
00001
00010
01000
10000
A
=
.
To find an inverse
A
of
A
, first form the augmented matrix
A
|
I
.
0010010000
0000101000
0001000100
0100000010
1000000001
By simply swapping the rows so that we obtain the identity matrix on the left hand side,
we get
1000000001
0100000010
0010010000
0001000100
0000101000
.
Thus, the inverse
A
of
A
that we seek is
00001
00010
10000
00100
01000
A
=
.
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