Acc. Math 2
Cryptology TASK
Name ___________________________
Cryptology is the science of creating and deciphering codes. Matrices can be used to encrypt
and decode messages.
Assign a numeric code to each letter of the alphabet. For this task use:
A=1
G=7
M = 13
S = 19
Y = 25
B=2
H=8
N = 14
T = 20
Z = 26
C=3
I=9
O = 15
U = 21
Space = 27
D=4
J = 10
P = 16
V = 22
E=5
K = 11
Q = 17
W = 23
F=6
L = 12
R = 18
X = 24
The encryption matrix must be an invertible (nonsingular) matrix. Remember the two criteria for
an invertible matrix are (1) must be square, and (2) no row can be a multiple of another row.
Example:
Message:
MATH IS FUN
−2 4
Encryption matrix: [
] 13 1 20 8 0 9 19 0 6 21 14
5 −3
Convert message to numeric code: 13 1 20 8 27 9 19 27 6 21 14
Since our encryption matrix is a 2 X 2 matrix, group numeric code into groups of 2:
[13 1][20 8][27 9][19 27][6 21][14 𝟐𝟕] Note: Used space to fill out last group.
Transpose those row matrices into column matrices and write as one 2 X 6 matrix.
[
13 20
1
8
27 19
9 27
6 14
]
21 27
Multiply the encryption matrix by the code matrix.
[
−2 4 13 20
]∙[
5 −3 1
8
27 19
9 27
6
21
14
−22 −8 −18 70
]=[
62 76 108 14
27
72
80
]
−33 −11
The encrypted message would be written as:
-22 62 -8 76 -18 108 70 14 72 -33 80 -11
The message could be decoded by anyone who had the letter assignments and the
encryption matrix by multiplying the inverse of the encryption matrix by the encrypted
message in a 2 X 6 matrix.
3
2
[14
5
7
1]
14
7
−22
∙[
62
−8 −18 70
76 108 14
72
−33
80
13
]=[
−11
1
20
8
27 19 6
9 27 21
14
]
27
Rewritten as 13 1 20 8 27 9 19 27 6 21 14 27, use the letter-number code to
convert to MATH IS FUN.
1
Acc. Math 2—Encryption Task
1. Using the same encryption matrix, decode the following:
90 -36 50 -34 10 87 24 45 -26 93 26 75 78 -6 16 -5 -50 132 10 87 54 -37 -8 76 62 -29
2. Find the encrypted Malcolm Forbes quote if the matrix used to encrypt it was
1 2 0 


5 0 3 

2 1 3
.

-4 57 -38 30 114 -57 49 52 -10 35 152 -68 -15 114 -48 33 82 -48 -9 153 -63
19 150 -46 -3 150 -57 35 47 -26 -15 189 -102 11 156 -98 31 126 -79
3. Encrypt a message and have a classmate discover your encrypted message.
--------------------------------------------------------------------------------------------------------------Another approach
𝐶= [
13 1 20 8 29 9
]
19 29 6 21 14 29
Since the message has an odd number of letters, the second row is completed
with a 29, which represents an empty space added to the message.
2
Acc. Math 2
Cryptology TASK
Name ___________________________
To encode the message, multiply the matrix C on the left by the encoding
matrix A:
𝐷 = 𝐴𝐶
𝐷= [
= [
3 1 13 1 20 8 29 9
][
]
2 1 19 29 6 21 14 29
58 32 66 45 101 56
]
45 31 46 37 72 47
The entries of D are the coded message:
58 32 66 45 101 56 45 31 46 37 72 47
The receiver can now use the decoding matrix B to reverse the steps.
𝐵𝐷 = 𝐵𝐴𝐶 = 𝐼𝐶 = 𝐶
𝐵𝐷 = [
= [
1 −1 58 32 66 45 101 56
][
]
−2 3 45 31 46 37 72 47
13 1 20 8 29 9
]
19 29 6 21 14 29
The result is the original matrix.
Use matrix A to encode the following messages.
1.
T E X T
M E
A T
L U N C H
3
Acc. Math 2—Encryption Task
2.
I
L O V E
T O
S I N G
4