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Exploration of Cryptology and BCH Codes

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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  1. Exploration of Cryptology and BCH Codes By: Carla Sorrell Thesis Advisor: Dr. Jennifer Hontz

  2. 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

  3. Cryptography Cryptanalysis Cryptology Overview of Cryptology

  4. General History Interesting Points: Caesar cipher History of Cryptology • Enigma • Jefferson Cylinder

  5. BCH Codes • Bose-Chaudhuri-Hocquengham codes • Error correcting code • Subclass of cyclic codes

  6. BCH Codes: Practical Uses • Information transmitted via the Internet • Data stored on a computer • Encoded music • Photograph Transmission • Data transmission

  7. BCH Code Example 2-Errors • Primitive Polynomial or Not? • Generator Element

  8. Parity Check Matrix Construction • Cosets • Not disjoint • Therefore and thus, . • So Ideal:

  9. 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

  10. Finding The Roots • Syndromes: • Received word: • Multiply received word by matrix H • Find syndromes [1111001][H]=[001100] =001= =100= = =

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

  12. Maple: The Algebraic Solution • Using Maple to approach BCH codes • Maple Commands

  13. 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 .

  14. 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.

  15. Conclusion • Further Exploration • Questions

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