Chapter 12 Cryptography Explained

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.

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.

Traveling Salesman Problem

Knight’s Tour

Knapsack Problem

Graph Theory

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.

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

Figure 12-2 Simulating Nondeterminism.

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.

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.

Figure 12-4 Substitutions and Permutations. Substitutions Permutations

Figure 12-5 Key Distribution in Pieces.

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

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

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.

Figure 12-9 A Cycle in the DES.

Figure Types of Permutations.

Figure Details of a Cycle.

Figure Pattern of Expansion Permutation.

Figure Structure of the AES.

Figure Knapsack for Encryption.

Figure Example of Solving a Simple Knapsack.

Figure Graph of Change of Merkle–Hellman Knapsack Function.

Figure Coinciding Discontinuities.

Elliptical Curve Cryptography

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.

Quantum Cryptography

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.

Figure Transmission of Photons.

Figure Results Interpreted Through Filters.

Figure Filters Used.

Figure Correct Filters.

Figure Correct Results.