Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take < space > = 0,A = 1,B = 2, and so on. Thus, for example, "ABORT MISSION" becomes
[1 2 15 18 20 0 13 9 19 19 9 15 14].
To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us takeA to be the 2 × 2 matrix
We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left byA.
Encrypted Matrix |
= |
|
|
= |
|
11 |
117 |
60 |
75 |
133 |
87 |
42 |
|
4 |
48 |
40 |
35 |
57 |
33 |
28 |
, |
which we can also write as
[11 4 117 48 60 40 75 35 133 57 87 33 42 28].
To decipher the encoded message, multiply the encrypted matrix by
A−1.
The following question uses the above matrixA for encoding and decoding.
Decode the following message, which was encrypted using the matrixA. (Include any appropriate spaces in your answer.)
[69 21 126 54 27 13 60 40 59 16 149 61 87 28]
space >