Cryptography: An Application of Vectors Matrices Diana Cheng Towson University
Jul 16, 2015
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.