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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
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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.
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 12-3 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.
Substitutions Permutations Figure 12-4 Substitutions and Permutations.
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 12-16 Graph of Change of Merkle–Hellman Knapsack Function.
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.
