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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

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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___

1001 =1

010x

001

101

111

110 1+x

011

0000 -

• 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