1. Chapter 12 Cryptography Explained

2. Search Problems Specified by an algorithm C Two inputs ◦ I is the instance. ◦ S is the solution. ◦ Must complete in polynomial time I. S is a solution to I if and only if C(I,S) is True.

3. NP-Complete Problems A class of search problems ◦ Traveling salesman problem  Time limited. ◦ Rudrata: Knight’s Tour on a chess board.  Cover all 64 squares? ◦ Euler: Graph Theory  Cross a bridge only once. ◦ Knapsack  Add maximum items below a limit.

4. Traveling Salesman Problem

5. Knight’s Tour

6. Knapsack Problem

7. Graph Theory

8. Goals Complexity ◦ Difficult to solve. ◦ Number of possible solutions large. ◦ Brute force solution expected to be infeasible. Satisfiable ◦ Assign values to a formula so that it is true. ◦ (V1) && (v2 || v3) && (!v3 || !v1) Solvable ◦ Simple approach to solve problem.

9. Figure 12-1 Clique Subgraphs in a Graph. Clique: every vertex connected to every other vertex. v1, v2, v7, v8 form clique size = 4.

10. Figure 12-2 Simulating Nondeterminism.

11. Figure 12-3 Hierarchies of Complexity Classes. Problem space. Some solvable in polynomial time (P). Some are beyond Polynomial time (EXP). Class NP between P and EXP.

12. Diffusion, Confusion, Substitution, Permutation Diffusion ◦ Spread the effect of a change to plaintext throughout the cipher text. Confusion ◦ Relationship between plain and cipher text should be as random and not apparent. Substitution (Confusion) S-Boxes ◦ Replace one character with another. Permutation (transposition) P-Boxes ◦ Provide confusion by rearranging the characters in the text.

13. Figure 12-4 Substitutions and Permutations. Substitutions Permutations

14. Figure 12-5 Key Distribution in Pieces.

15. Figure 12-6 Distribution Center for Encrypted Information. Key Clearinghouse, centralize key distribution.

16. Figure 12-7 Cycles of Substitution and Permutation DES: strength from repeating substitution and permutations.

17. Figure 12-8 Product Ciphers. Two weak but complementary ciphers can be made more secure by being applied together, the product of the two ciphers.

18. Figure 12-9 A Cycle in the DES.

19. Figure 12-10 Types of Permutations.

20. Figure 12-11 Details of a Cycle.

21. Figure 12-12 Pattern of Expansion Permutation.

22. Figure 12-13 Structure of the AES.

23. Figure 12-14 Knapsack for Encryption.

24. Figure 12-15 Example of Solving a Simple Knapsack.

25. Figure 12-16 Graph of Change of Merkle–Hellman Knapsack Function.

26. Figure 12-17 Coinciding Discontinuities.

27. Elliptical Curve Cryptography

28. Offers considerably greater security for a given key size The smaller key size also makes possible much more compact implementations for a given level of security, which means faster cryptographic operations, running on smaller chips or more compact software. This means less heat production and less power consumption — all of which is of particular advantage in constrained devices, but of some advantage anywhere. There are extremely efficient, compact hardware implementations available for ECC exponentiation operations, offering potential reductions in implementation footprint even beyond those due to the smaller key length alone.

29. Quantum Cryptography

30. Instead of depending on the computational difficulty of cracking one-way functions, quantum encryption creates uncrackable codes that employ the laws of physics to guarantee security. Different quantum states, such as photon polarization, can be used to represent 1s and 0s in a manner that cannot be observed without the receiver's discovering it. For instance, if hackers observe a polarized photon, then 50 percent of the time they will scramble the result, making it impossible to hide the eavesdropping attempt from the receiver.

31. Figure 12-18 Transmission of Photons.

32. Figure 12-19 Results Interpreted Through Filters.

33. Figure 12-20 Filters Used.

34. Figure 12-21 Correct Filters.

35. Figure 12-22 Correct Results.