Exploration of Cryptology and BCH Codes

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# Exploration of Cryptology and BCH Codes - PowerPoint PPT Presentation

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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Presentation Transcript

Exploration of Cryptology

and BCH Codes

By: Carla Sorrell

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
Cryptography

Cryptanalysis

Cryptology

Overview of Cryptology

General History

Interesting Points:

Caesar cipher

History of Cryptology

• Enigma
• Jefferson Cylinder

BCH Codes

• Bose-Chaudhuri-Hocquengham codes
• Error correcting code
• Subclass of cyclic codes

BCH Codes: Practical Uses

• Information transmitted via the Internet
• Data stored on a computer
• Encoded music
• Photograph Transmission
• Data transmission

BCH Code Example

2-Errors

• Primitive Polynomial or Not?
• Generator Element

Parity Check Matrix Construction

• Cosets
• Not disjoint
• Therefore and thus, .
• So

Ideal:

Process continues to construct parity check matrix

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

Finding The Roots

• Syndromes:
• Multiply received word by matrix H
• Find syndromes

[1111001][H]=[001100]

=001=

=100= =

=

Finding The Roots continued

• Where the sum of two words :
• The roots are at and .
• Error Polynomial:
• Corrected Polynomial:

Maple: The Algebraic Solution

• Using Maple to approach BCH codes
• Maple Commands

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 .

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.

Conclusion

• Further Exploration
• Questions