An investigatory project about the application of matrices in cryptography.

1. Application of
Matrices in
Cryptography
Researchers: Laiz, Princess Pauline & Reponte Charmagne
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2. Table of Contents
We will talk about
this first.
We will talk about
this second.
Then we will talk
about this.
01 Introduction 02 Statement of the Problem
03 Definition of terms
After that we will
talk about this.
04 Discussion
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05 Summary, Conclusions, Recommendations
And we will talk
about this last.

3. Cryptography, the discipline of encrypting and
decrypting messages encoded in secret codes, has
played a critical role in information security.
The Hill Algorithm, a form of encryption that
uses matrix multiplication and matrix inverses,
represented the introduction of modern
mathematical theory and methods to the area of
cryptography.
Introduction
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4. This paper aims to discuss the application
of matrices in cryptography. It also aims
to figure out how to produce keys using
Hill Cyphers to encrypt and descrypt
words.
Statement of
the Problem
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5. Specifically, it will answer the following subquestions:
1.
What is the effectiveness of using matrices in cryptography in terms of
the level of security when sending or receiving information that required
confidentiality?
2.
What is the process of producing keys using Hill Cyphers to encrypt
and decrypt words?
3.
What are the determinants that make a matrix qualified to be used as a
key for cyphers?
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6. Definition of
terms
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7. Matrices (plural), is a
rectangular array of numbers,
symbols or expressions that are
arranged in rows and columns and
are frequently written in box
brackets (Lumen Learning, n.d.).
A term that refers to the
process of converting plain text
into unintelligible text, and
vice versa.
It is a technique for storing
and transmitting data in a
specific format that only the
intended receiver can read and
process (Wanjonhi,2021).
Cryptography Matrix
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8. ● Matrices
● Cryptography
● Hill Cypher
● Encryption and Decryption with Hill Cipher
● Cracking the Hill Cypher
Discussion
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9. Matrix is a Latin term that means “womb”. Matrix is an ordered
arrangement of a number or symbols in particular, rows and columns
contained by brackets, subscribed by magnitude of order, and labeled
by capital letters.
― Meyer & Siam (2009)
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Matrices

10. Operations on Matrices
― Chattaroy, Majhi, & Rath (2013)
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Matrices can be added, subtracted and multiplied. Other than this, we
also have matrix transpose, matrix adjoint, inverse, conjugate, trans
conjugate which are also operations on matrices.

11. Determinants that make a matrix
qualified as a key for cyphers
― Abdullah, Khalaf & Riza (2015)
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Hill Cipher uses matrix multiplication and inverse matrix techniques.
The K-matrix n x n where n is the block size, that will generate the
key must be an invertible matrix with inverse ( 𝐾−1 ),because the 𝐾−1
matrix is the decoding key, hence they must have an inverse.

12. Cryptography
― Chua (2006)
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Cryptography is an interdisciplinary topic that incorporates ideas
from other areas. In today’s world, cryptography takes extensive use
of computer science and mathematics, particularly discrete
mathematics.

13. Hill Cipher
― Stallings (2005)
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It was created by Lester Hill, a mathematician. Matrix operations are
at the core of the Hill Cipher. Each character in the Hill Cipher is
given a numerical value in reverse order or in any order that suits
your needs.

14. Encryption with Hill Cipher
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In general, the Hill Cipher Encryption uses the structure of numbers
and letters shown below:

15. Encryption with Hill Cipher
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Let’s have a look at a simple encryption. We’ll encrypt with a 3 x 3 key matrix for
convenience. “PRETTY YOU” is the message we wish to encrypt:
From the word “PRETTY YOU”, after multiplying it by our key
matrix, the encrypted message we got is “CNPYTT EIY”.

16. Decryption with Hill Cipher
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Given the key matrix in the encryption, we got the decoding matrix which is:
We decoded the encrypted message which is “PRETTY YOU”.

17. Cracking the Hill Cipher
― Pugliano and Sehestedt (2017)
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Cracking the cipher was a difficult undertaking when it was initially
devised. Cracking the message by hand took a long time because there
were so many possible key matrices and the cipher hides repeated
letters. The “Plain Text Attack” is a method for cracking the Hill
Cypher that was devised.

18. Summary,
Conclusions
and
Recommendations
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19. Summary
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Upon the consolidation of some existing researches, the usage of
matrices in cryptography was observed particularly in the Hill
Cipher. Research had shown how encryption and decryption with Hill
Cipher works and the matrix operations which is at the core of this
cipher have been fully discussed in this research.

20. Conclusions
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While the Hill Cipher was a breakthrough in encryption when it was
invented in 1929, it does not give the level of security required to
convey anything remotely private or secret in today’s environment.
It’s vulnerable to a variety of attacks, and if the cipher is
cracked, messages can be intercepted and forwarded without either
party knowing.

21. Recommendations
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Future study into different types of ciphers that use matrices in
encryption and decryption should be increased as suggested by the
researchers, because it will widen our understanding of how
beneficial matrices may be in cryptography. It will also be
interesting to see if future researchers investigate the use of
matrices in real life and in other domains.

22. Thank you!
Do you have any questions? Feel free to send
it to the e-mails provided below.
princesspauline.laiz@msugensan.edu.ph
charmagne.reponte@msugensan.edu.ph
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