Cryptography:

An Application

of Vectors

MatricesDiana Cheng

Towson University

HSN.VM.CPerform operations on matrices & use matrices in applications.

6 Use matrices to represent & manipulate data, e.g., to represent

payoffs or incidence relationships in a network.

7 Multiply matrices by scalars to produce new matrices, e.g., as

when all of the payoffs in a game are doubled.

8 Add, subtract, & multiply matrices of appropriate dimensions.

9 Understand that, unlike multiplication of numbers, matrix

multiplication for square matrices is not a commutative operation,

but still satisfies the associative & distributive properties.

10 Understand that the zero & identity matrices play a role in

matrix addition & multiplication similar to the role of 0 & 1 in the

real numbers. The determinant of a square matrix is nonzero if &

only if the matrix has a multiplicative inverse.

11 Multiply a vector (regarded as a matrix with one column) by a

matrix of suitable dimensions to produce another vector. Work with

matrices as transformations of vectors.

The goal of cryptography…

Is to hide a message’s meaning, & not

necessarily hide the existence of a

message.

If a first message is hidden inside a second

message, the second message can be

made public, yet a person seeing the

second message may not be able to

understand the meaning of the first

message.

Try to crack this cipher:

Try to crack this cipher:

7 0 21 4 0 13 8 2 4 3 0 24

Encryption / Decryption

Uses of cryptography

Warfare

Julius Caesar’s cipher

Used for military purposes & its use is

documented in his Gallic Wars.

For the first letter in the plaintext, the first letter

in the ciphertext is the letter a fixed number n

letters higher in the alphabet; repeat this

process for each letter of the plaintext.

E.g., plaintext ABCD could be shifted three

(n=3) letters, to become ciphertext, CDEF.

Shift Cipher:

Matrix Addition by a constant

How many keys would

you need to try, if you

knew that a message

was encoded using our

coding scheme and the

shift cipher?

Stretch Cipher

(Scalar Multiplication)

Combination Cipher

(Matrix Addition & Scalar

Multiplication)

Polyalphabetic cipher…

Was developed since the monoalphabetic

substitution ciphers were not sufficiently

keeping messages hidden anymore.

Blaise de Vigenère, (French diplomat born in

1523) created the idea of switching between

cipher alphabets. Within 1 message, multiple

cipher alphabets are used.

Vigenère’s cipher is equivalent to the Caesar

shifts of 1 through 26 (Hamilton & Yankosky,

2004; Singh, 1999).

Matrix Addition with a Key

Matrix

Vigenere Cipher: Matrix Addition with a Keyword

WEATHERISNICE

Practice using Vigenere

Cipher

Discussion Question: Using the Vigenere cipher, in how many ways could

you encrypt the word “THE” using keyword “SUPER”?

Matrix multiplication

“Operations with Matrices”

How would the students who produced

the work on question 1 respond to

question 2? How can you help them with

matrix multiplication?

From:

Tobey, C. & Arline, C. (2014). Uncovering Student Thinking

about Mathematics in the Common Core: High School.

Corwin: Thousand Oaks, CA.

Matrix Multiplication (non-

scalar)

Activity Adapted from “Produce Intrigue

with Crypto!” article

[A] x [B] = [C]

To solve for [B], what do you need to

know?

Encode two plaintext messages & decode

two plaintext messages!

Discussion Questions

What are the benefits and shortcomings

of each of these methods of encryption?

What is the role of the choice of the

coding scheme?

How can we improve our encryption

methods?

SmP?

Make sense of problems & persevere in

solving them

Reason abstractly and quantitatively

Construct viable arguments and critique

the reasoning of others

Model with mathematics

Use appropriate tools strategically

Attend to precision

Look for and make use of structure

Look for and express regularity in

repeated reasoning

References• Avila, C. & Ortiz, E. (2012). Produce intrigue with Crypto!

Mathematics Teaching in the Middle School, 18(4), 212-220.

• Chua, B. (2008). Harry Potter and the coding of secrets.

Mathematics Teaching in the Middle School, 14(2), 114-121.

• FBI website –

• http://www.fbi.gov/news/stories/2009/december/code_122409

• Garfunkel, S., Gobold, L., & Pollak, H. (1998). Mathematics:

Modeling our world, Course 1 (Annotated Teacher's Edition ed.).

Lexington, MA: Consortium for Mathematics and Its Applications.

• Hamilton, M., & Yankosky, B. (2004). The Vigenere cipher with the

TI-83. Mathematics and Computer Education, 38(1), 19-31.

• NCTM (2006). Rock Around the Clock. Navigating through Number

and Operations in Grades 9-12. Reston, VA.

• Nykamp, D. Introduction to matrices. From Math Insight.

• http://mathinsight.org/matrix_introduction

• Singh, S. (1999). The code book. New York: First Anchor Books.