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This chapter delves into cryptography, highlighting its essential algorithms, search problems, and NP-Complete issues like the Traveling Salesman Problem, Knight's Tour, and the Knapsack Problem. It discusses complexity classes, the feasibility of finding solutions, and the importance of polynomial time in problem-solving. Additionally, it explores advanced topics such as Elliptical Curve Cryptography and Quantum Cryptography, emphasizing their advantages in security and efficiency. Ultimately, readers gain insights into the intricacies of cryptographic methods and the computational challenges they involve.
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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.
