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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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Presentation Transcript
slide2

Exploration of Cryptology

and BCH Codes

By: Carla Sorrell

Thesis Advisor: Dr. Jennifer Hontz

slide3

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
slide4
Cryptography

Cryptanalysis

Cryptology

Overview of Cryptology

slide5
General History

Interesting Points:

Caesar cipher

History of Cryptology

  • Enigma
  • Jefferson Cylinder
slide6

BCH Codes

  • Bose-Chaudhuri-Hocquengham codes
  • Error correcting code
  • Subclass of cyclic codes
slide7

BCH Codes: Practical Uses

  • Information transmitted via the Internet
  • Data stored on a computer
  • Encoded music
  • Photograph Transmission
  • Data transmission
slide8

BCH Code Example

2-Errors

  • Primitive Polynomial or Not?
  • Generator Element
slide9

Parity Check Matrix Construction

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

Ideal:

slide10

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

slide11

Finding The Roots

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

[1111001][H]=[001100]

=001=

=100= =

=

slide12

Finding The Roots continued

  • Where the sum of two words :
  • The roots are at and .
  • Error Polynomial:
  • Corrected Polynomial:
slide13

Maple: The Algebraic Solution

  • Using Maple to approach BCH codes
  • Maple Commands
slide14

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 .

slide15

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.
slide16

Conclusion

  • Further Exploration
  • Questions
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