1. Data Encryption Chris Mraovich

2. Overview Purpose of EncryptionPurpose of Encryption Permutations Bases and FactoradicsPermutations Bases and Factoradics Project SummaryProject Summary

3. Purpose of Encryption

4. Protecting Digital Content DVDs use CSS (Content Scramble System)DVDs use CSS (Content Scramble System) Weak Algorithm -Cracked by Jon Johansen in 1999

5. Protecting Digital Content Uses encryption on program ROM chipsUses encryption on program ROM chips Arcade Printed Circuit Boards Capcom Play System 2 (CPS2)Capcom Play System 2 (CPS2) -Used in mid 1990s for 2D games Graphic ROM chips are not encryptedGraphic ROM chips are not encrypted Cracked by teamCracked by team

6. Protecting Digital Content Why use encryption on an arcade board? 1.ROM chips can be copied to a PC as binary data 2. Program can be written to interpret binary data 3. PC can then run the arcade without the board

7. Permutations, Bases, and Factoradics

8. Permutations Goal is to rearrange bits into a different pattern 1 0 1 1 110 1 Permutation – rearrangement of a set of objects Original Form: Encrypted Form:

9. Factoradic – mixed radix numbering system that uses multiple bases to represent a single number Bases and Factoradics Factoradics provide a way of generating permutationsFactoradics provide a way of generating permutations Why are they important? Generate Factoradic Obtain permutation from factoradic Use permutation to rearrange bits Summary of Encryption Process

10. Order & Total Permutations Suppose there are 4 objects Order – number of objects (N) Total number of permutations for N objects is N! N = 4, so there are 4! or 24 ways to rearrange 4 objects

11. 0{ 0 0 0 0 } ( 0 1 2 3 ) 1 { 0 0 1 0 } ( 0 1 3 2 ) 2 { 0 1 0 0 } ( 0 2 1 3 ) 3 { 0 1 1 0 } ( 0 2 3 1 ) 4 { 0 2 0 0 } ( 0 3 1 2 ) 5 { 0 2 1 0 } ( 0 3 2 1 ) 6 { 1 0 0 0 } ( 1 0 2 3 ) 7 { 1 0 1 0 } ( 1 0 3 2 ) 8{ 1 1 0 0 } ( 1 2 0 3 ) 9{ 1 1 1 0 } ( 1 2 3 0 ) 10 { 1 2 0 0 } ( 1 3 0 2 ) 11 { 1 2 1 0 } ( 1 3 2 0 ) 12 { 2 0 0 0 } ( 2 0 1 3 ) 13 { 2 0 1 0 } ( 2 0 3 1 ) 14 { 2 1 0 0 } ( 2 1 0 3 ) 15 { 2 1 1 0 } ( 2 1 3 0 ) 16 { 2 2 0 0 } ( 2 3 0 1 ) 17 { 2 2 1 0 } ( 2 3 1 0 ) 18 { 3 0 0 0 } ( 3 0 1 2 ) 19 { 3 0 1 0 } ( 3 0 2 1 ) 20 { 3 1 0 0 } ( 3 1 0 2 ) 21 { 3 1 1 0 } ( 3 1 2 0 ) 22 { 3 2 0 0 } ( 3 2 0 1 ) 23 { 3 2 1 0 } ( 3 2 1 0 ) Each factoradic uniquely identifies a particular permutationEach factoradic uniquely identifies a particular permutation Total Permutations of order 4 FactoradicPermutationInt Int is the base 10 representation of the factoradicInt is the base 10 representation of the factoradic Walkthrough of how 20 10 is converted to a permutation of order 4Walkthrough of how 20 10 is converted to a permutation of order 4

12. Bases – Generate Factoradic Write 20 10 in Base 2 2 4 2 3 2 2 2 1 2 0 (16) (8) (4) (2) (1) 1 0 1 0 0 Expand the Binary Number ( x ) + ( x ) + ( x ) + ( x ) + ( x ) = 20 10 (1 2 x 2 4 ) + (0 2 x 2 3 ) + (1 2 x 2 2 ) + (0 2 x 2 1 ) + (0 2 x 2 0 ) = 20 10 Base 10Base 2Multi-Base Factoradic 20101003100

13. From Base 2 to Factoradic ( x ) + ( x ) + ( x ) + ( x ) + ( x ) (E 2 x 2 4 ) + (D 2 x 2 3 ) + (C 2 x 2 2 ) + (B 2 x 2 1 ) + (A 2 x 2 0 ) Factoradic Expansion are all numbers in base 2 (0 or 1) A 2, B 2, C 2, D 2, E 2 are all numbers in base 2 (0 or 1) 2 n are powers of 2 ( x ) + ( x ) + ( x ) + ( x ) + ( x ) (E 5 x 4!) + (D 4 x 3!) + (C 3 x 2!) + (B 2 x 1!) + (A 1 x 0!) 2 n n! The bases of increase from right to left The bases of A 2, B 2, C 2, D 2, E 2 increase from right to left Generalization of Base 2 Expansion What Changes : 1.) 2.) … … (Mixed Radix - multiple bases used)

14. Factoradic Number System Factoradic Expansion ( x ) + ( x ) + ( x ) + ( x ) + ( x ) (E 5 x 4!) + (D 4 x 3!) + (C 3 x 2!) + (B 2 x 1!) + (A 1 x 0!) Simplify Factorials ( x ) + ( x ) + ( x ) + ( x ) + ( x ) (E 5 x 24) + (D 4 x 6) + (C 3 x 2) + (B 2 x 1) + (A 1 x 1) 00000 1111 222 33 4 Since,,,, and have different bases, they have different ranges of valid values Since A, B, C, D, and E have different bases, they have different ranges of valid values …

15. ( x ) + ( x ) + ( x ) + ( x ) + ( x ) (E 5 x 24) + (D 4 x 6) + (C 3 x 2) + (B 2 x 1) + (A 1 x 1) 00000 1111 222 33 4 Factoradic Number System Write 20 10 in Factoradic notation ( x ) + ( x ) + ( x ) + ( x ) = 20 (3 x 6) + (1 x 2) + (0 x 1) + (0 x 1) = 20 ( x ) + ( x ) + ( x ) + ( x ) = (3 x 3!) + (1 x 2!) + (0 x 1!) + (0 x 0!) = Final Factoradic for 20 10 : Final Factoradic for 20 10 : 3 1 0 0

16. Obtain Permutation from Factoradic Initial Factoradic: Initial Factoradic: 3 1 0 0

17. 1) Increment every digit by 1 4 2 1 1 Obtain Permutation from Factoradic

18. Initial Factoradic: Initial Factoradic: 3 1 0 0 1) Increment every digit by 1 4 2 1 1 2) Replace right-most digit with a 1 4 2 1 1 Obtain Permutation from Factoradic

19. Initial Factoradic: Initial Factoradic: 3 1 0 0 1) Increment every digit by 1 4 2 1 1 2) Replace right-most digit with a 1 4 2 1 11 3)This is the “new value” (N) 3)This 1 is the “new value” (N) If any red value to the right of N is >= N, it gets incremented by 1 1 Obtain Permutation from Factoradic

20. Initial Factoradic: Initial Factoradic: 3 1 0 0 1) Increment every digit by 1 4 2 1 1 2) Replace right-most digit with a 1 4 2 1 1 21 3)This is the “new value” (N) 3)This 1 is the “new value” (N) If any red value to the right of N is >= N, it gets incremented by 1 Obtain Permutation from Factoradic

21. Initial Factoradic: Initial Factoradic: 3 1 0 0 1) Increment every digit by 1 4 2 1 1 2) Replace right-most digit with a 1 4 2 1 1 21 3)This is the “new value” (N) 3)This 1 is the “new value” (N) If any red value to the right of N is >= N, it gets incremented by 1 4) Repeat step 3 until all red numbers have been used 21 2 Obtain Permutation from Factoradic

22. Initial Factoradic: Initial Factoradic: 3 1 0 0 1) Increment every digit by 1 4 2 1 1 2) Replace right-most digit with a 1 4 2 1 1 21 3)This is the “new value” (N) 3)This 1 is the “new value” (N) If any red value to the right of N is >= N, it gets incremented by 1 4) Repeat step 3 until all red numbers have been used 312 4 312 Obtain Permutation from Factoradic

23. Initial Factoradic: Initial Factoradic: 3 1 0 0 1) Increment every digit by 1 4 2 1 1 2) Replace right-most digit with a 1 4 2 1 1 21 3)This is the “new value” (N) 3)This 1 is the “new value” (N) If any red value to the right of N is >= N, it gets incremented by 1 4) Repeat step 3 until all red numbers have been used 312 4312 4) Decrement all numbers by 1 3 1 0 2 Obtain Permutation from Factoradic

24. 3 1 0 2 Original Binary Data: Use Permutation to swap bits Obtained Permutation: 1 0 1 0 Encrypted Bit Array Data: 01 23

25. 3 1 0 2 Original Binary Data: Use Permutation to swap bits Obtained Permutation: 1 0 1 0 Encrypted Bit Array Data: 1 01 23

26. 3 1 0 2 Original Binary Data: Use Permutation to swap bits Obtained Permutation: 1 0 1 0 Encrypted Bit Array Data: 10 01 23

27. 3 1 0 2 Original Binary Data: Use Permutation to swap bits Obtained Permutation: 1 0 1 0 Encrypted Bit Array Data: 10 1 01 23

28. 3 1 0 2 Original Binary Data: Use Permutation to swap bits Obtained Permutation: 1 0 1 0 Encrypted Bit Array Data: 10 1 0 01 23

29. Project Summary Encrypt/Decrypt any binary file on theEncrypt/Decrypt any binary file on the Windows platform Generate keys to decrypt filesGenerate keys to decrypt files Like a really long password stored in a text file Use the principles of factoradics to: