Cryptography and Matrices
Part 1: Using 2 x 2 Matrices
Cryptology is a real-world application of matrix inverses. At the web site 2 x 2 Matrices,
Determinants, Inverses, scroll to the section Cryptology and read how matrices are used
to encode and decode secret messages. Summarize the procedures for encoding and
decoding messages in the chart below.
Procedures to Encode a Message
a.
Procedures to Decode a Message
Write text message as 2 x 1 text Receive encoded message, alphabet
matrices.
assignment, and secret matrix separately.
Use alphabet assignment to
b. write as numerical 2 x 1
matrices.
If written as a text phrase, use alphabet
assignment to write as numerical matrices.
Multiply numerical matrices by
Compute ______________________
c. secret matrix resulting in new
matrix.
numerical matrices.
d.
Use alphabet assignment to
write as text matrices.
e. Write as text phrase.
f. Send encoded message.
Alphabet Assignment
For the problems in Part 1, use the alphabet assignment in the table below. Each letter is
assigned a number based on its position and 27 is used for a blank space.
A B C D E F G H I
J
K L M
1 2 3 4 5 6 7 8 9 10 11 12 13
N O P Q R S T U V W X Y Z
14 15 16 17 18 19 20 21 22 23 24 25 26
Blank = 27
Encode a Message
1. Write the message ALGEBRA IS FUN as 2 x 1 matrices, similar to the MATH
ROCKS example at the web site.
2. Rewrite these matrices using the above alphabet assignment.
3. Using the following matrix, perform the necessary multiplications to encode the
given message. Do this yourself, do not use technology.
4. Use the same alphabet assignment to write the encoded message as 2 x 1 text
matrices.
Notes:
o
o
o
o
For numbers greater than 27, the alphabet simply repeats.
Example: 28 = A, 37 = J
For negative numbers, the alphabet is reversed.
Example: 0 = space, –1 = z, –2 = y
For numbers less than 0 or larger than 27, add additional rows to the
alphabet assignment table or do one of the following two steps.
a. If the number is positive and larger than 27, simply subtract 27
(repeating if necessary) until your new number is less than 27 and
in the original table.
Example: 50 = 50 – 27 = 23 = W
b. If the number is negative, add 27 (repeating if necessary) until your
new number is positive or in the original table. Example: –65 = –
65 + 27 + 27 + 27 = 16 = P
Note the similarity to modulo 27 arithmetic.
5. Write the last set of 2 x 1 matrices from problem 4 as a horizontal text message.
This is the way the encoded message would appear if sent.
Decode a Message
6. Suppose you had previously received the alphabet assignment and the encoding
matrix. You now receive the coded message from problem 5. Begin to decode it
by writing the horizontal text message as 2 x 1 matrices and use the alphabet
assignment to rewrite as 2 x 1 numerical matrices.
7. To decode the message, find the inverse of the encoding matrix used in problem
3. If necessary, review information on finding inverses at the web site.
8. Now multiply the inverse matrix times each of the encoded 2 x 1 matrices you
wrote in problem 6. Do this yourself, do not use technology.
9. Use the alphabet assignment and write the letter for each number and the message
as horizontal text. For numbers less than 0 and greater than 27, see the notes in
problem 4. Did your original message return?
Part 2: Using 3 x 3 Matrices
It is beneficial to use large matrices in cryptography because they are more difficult to
break. Along with the secret encrypting matrix, a unique alphabet assignment, known
only to the parties sending and receiving the message, can provide additional security.
10. Two agencies have agreed on the following alphabet assignment: Use it to
complete the placement of the message WAR OF NUMBERS into a horizontal
list of numbers and then complete the sequence of 3 x 1 matrices. Note that the
number 1 can represent a blank space or any needed fillers.
A B C D E F G H I
J
K L M
4 6 8 10 12 14 16 18 20 22 24 26 28
N O P Q R S T U V W X Y Z
5 7 9 11 13 15 17 19 21 23 25 27 29
Blank = 1
11.
W A R
O F
N U M B E R S
23 4 13 1 7 14
12.
13. Encode the message above using the matrix below. Leave the message in matrix
form. You may wish to write the individual 3 x 1 matrices from problem 10 as a
single 3 x 5 matrix before completing the calculations. Note: Either the matrix
calculator at Matrix Calculator or Matrix Algebra Tool may be used for
calculations required in the remaining worksheet problems.
14. This site suggests sending or transmitting the encoded message as a list of
numbers separated by commas. Write the original message using a similar linear
list of numbers.
15. Use the same encoding matrix to calculate the inverse decoding matrix that the
second agency would use. You may use technology to do this. When appropriate,
request results as fractions not as decimals.
16. Using the inverse matrix from problem 13, decode the following message and
write as a text message.
63, –33, –9, 88, –62, 16, 93, –91, –75, 57, –27, –33, 43, –11, –29