By: Carla Sorrell

Thesis Advisor: Dr. Jennifer Hontz

•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

General History

Interesting Points:

• Caesar cipher

•Jefferson Cylinder •Enigma

•Bose-Chaudhuri-Hocquengham codes

•Error correcting code

•Subclass of cyclic codes

•Information transmitted via the Internet

•Data stored on a computer

•Encoded music

•Photograph Transmission

•Data transmission

•Primitive Polynomial or Not?123 xx

03)1(01)0(

1)( 23

pp

xxxp

)1)(1()( 2323 xxxxxg

•Generator Element

•Cosets

•Not disjoint

•Therefore and thus, .

•So

...},1,1,,1,,1,0{][ 22222 xxxxxxxxx

,...}1)1)(1(,)1(,1,0{)1( 242334232323 xxxxxxxxxxxxxxxx

,...}1,,1,{ 2344233 xxxxxxxxx ,...}1,,,1{)1( 2344322 xxxxxxxxx

23 1 xx 23 1 xx

34 xxx 21 xx

325 xxxx

xxxx

11 22

Parity Check Matrix Construction

Ideal:

Word ____________ power of___ 100 1 =1010 x001101111110 1+x011000 0 -

)(mod xpx i 0

12x

221 x

321 xx

45

2xx 6

•Process continues to construct parity check matrix

•The parity check matrix of G: 7X6

4186

155

5124

293

62

3

11

111011010110110111001101011001101010100100

=H

•Syndromes:

•Received word:

•Multiply received word by matrix H

•Find syndromes

Finding The Roots

313 ,)](),([ sswwwH

6321)( xxxxxr

[1111001][H]=[001100]

=001= =100= = =

1s )(2

3s10 )( 3 6323

1 )( s 03s

245222

0

001111110)(

•Where the sum of two words : 2

110101011001111110001110111110011101101100001000010010101001100

36

245

254

63

72

11

20

•The roots are at and .

•Error Polynomial:

•Corrected Polynomial:

5454)( xxxe

654321)()()( xxxxxxxcxexr

Finding The Roots continued

•Using Maple to approach BCH codes

•Maple Commands

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

)}({ xew 11

ne

)}({ xew12 mn

Cyclic Shift Theorem

For either =1 or 2,

if and only if .

)}({ xew 3~

13

~

)(SS

1ke

3~

13

~

)(SS

•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.

•Further Exploration

•Questions