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Exploration of Cryptology and BCH Codes. By: Carla Sorrell. Thesis Advisor: Dr. Jennifer Hontz. Introduction. Overview of Cryptology. History of Cryptology. BCH Codes. Algebraic Approach with Maple. New Method for BCH Codes. Application to Education. Further Research and Exploration.

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Exploration of cryptology and bch codes

Exploration of Cryptology

and BCH Codes

By: Carla Sorrell

Thesis Advisor: Dr. Jennifer Hontz


Exploration of cryptology and bch codes

Introduction

  • Overview of Cryptology

  • History of Cryptology

  • BCH Codes

  • Algebraic Approach with Maple

  • New Method for BCH Codes

  • Application to Education

  • Further Research and Exploration


Exploration of cryptology and bch codes

Cryptography

Cryptanalysis

Cryptology

Overview of Cryptology


Exploration of cryptology and bch codes

General History

Interesting Points:

Caesar cipher

History of Cryptology

  • Enigma

  • Jefferson Cylinder


Exploration of cryptology and bch codes

BCH Codes

  • Bose-Chaudhuri-Hocquengham codes

  • Error correcting code

  • Subclass of cyclic codes


Exploration of cryptology and bch codes

BCH Codes: Practical Uses

  • Information transmitted via the Internet

  • Data stored on a computer

  • Encoded music

  • Photograph Transmission

  • Data transmission


Exploration of cryptology and bch codes

BCH Code Example

2-Errors

  • Primitive Polynomial or Not?

  • Generator Element


Exploration of cryptology and bch codes

Parity Check Matrix Construction

  • Cosets

  • Not disjoint

  • Therefore and thus, .

  • So

Ideal:


Exploration of cryptology and bch codes

Word ____________ power of___

100 1 =1

010 x

001

101

111

110 1+x

011

000 0 -

  • The parity check matrix of G: 7X6

=H


Exploration of cryptology and bch codes

Finding The Roots

  • Syndromes:

  • Received word:

  • Multiply received word by matrix H

  • Find syndromes

[1111001][H]=[001100]

=001=

=100= =

=


Exploration of cryptology and bch codes

Finding The Roots continued

  • Where the sum of two words :

  • The roots are at and .

  • Error Polynomial:

  • Corrected Polynomial:


Exploration of cryptology and bch codes

Maple: The Algebraic Solution

  • Using Maple to approach BCH codes

  • Maple Commands


Exploration of cryptology and bch codes

Step-By-Step Decoder

Theorem by Yi-Chang Cheng, Erl Huei Lu, To Chang, and Po-Chiang Lu

For =1 or 2, if and only if

where is the number of errors in received vector, or block length

Cyclic Shift Theorem

For either =1 or 2,

if and only if .


Exploration of cryptology and bch codes

Application to Education

  • Students break the code to learn information for other subjects.

  • Students gain skills in the content and process standards.

  • Start simple with simple cipher, cryptograms, and matching or word search to learn history, and move to more advanced activities.


Exploration of cryptology and bch codes

Conclusion

  • Further Exploration

  • Questions