Chapter 3 Encryption Algorithms & Systems

1. Chapter 3Encryption Algorithms & Systems

2. Outline • NP-completeness & Encryption • Symmetric (secret key) vs Asymmetric (public key) Encryptions • Popular Encryption Algorithms • Merkle-Hellman Knapsacks • RSA Encryption • El Gamal Algorithms • DES • Hashing Algorithms • Key Escrow & Clipper csci5233 computer security & integrity (Chap. 3)

3. Review of Concepts • Rate of growth: • Polynomial functions Example: n, 10n, n2, 5n3, … • Exponential functions Example: 2n, 210n, … • The order of functions matters! See next page. csci5233 computer security & integrity (Chap. 3)

4. Order wins out! • From Baase, Computer Algorithms (2nd ed.) • Exercise: How if a computer that runs at 2N ns.? csci5233 computer security & integrity (Chap. 3)

5. Review of Concepts • Polynomial bounded An algorithm is said to be polynomial bounded if its worst-case complexity is bounded by a polynomial function of the input size. That is, there exists a polynomial function p such that the algorithm terminates after at most p(n) steps, where n is the input size. A problem is said to be polynomial bounded if there is a polynomial bounded algorithm for it. • Exercise: Is an algorithm whose order is n10,000,000 polynomial-bounded? csci5233 computer security & integrity (Chap. 3)

6. Nondeterminism • A nondeterministic algorithm has two phases: • The nondeterministic phase (the guessing phase) • A guess at a solution for the problem is proposed. • The deterministic phase (the checking phase) • A deterministic algorithm is executed, with or without the proposed solution from phase 1. csci5233 computer security & integrity (Chap. 3)

7. Complexity Classes • Complexity Classes: Class P P is the class of problems that are polynomial-bounded. Class NP NP is the class of problems for which there is a nondeterministicpolynomial-bounded algorithm. • A theorem: P  NP • Implication: A deterministic algorithm is a special case of a nondeterministic algorithm (with phase 1 ignored). csci5233 computer security & integrity (Chap. 3)

8. Characteristics of Hard Problems • P.72 • Each problem is solvable, and a relatively simple approach solves it (although the approach may be time-consuming). • There are 2n cases to consider if we use the approach of enumerating all possibilities, where n is the input size. • The problems exist in various areas (logic, number theory, graph theory, …). • If it were possible to guess perfectly, each problem could be solved in relatively little time. That is, the verification process is polynomial bounded. csci5233 computer security & integrity (Chap. 3)

9. Sample NP Problems • Satisfiability, p.71 • Knapsack • Subset sum, a simpler version of Knapsack, p.72 • Clique, p.72 • Graph coloring • Job scheduling with penalties • Bin packing • Hamilton paths (or circuits) • Minimum tour (Traveling salesperson problem) • . . . csci5233 computer security & integrity (Chap. 3)

10. A broader Knapsack problem • From Baase, Computer Algorithms (2nd ed.) • Given: • a knapsack of capacity C (a positive integer • N objects with sizes s1, …, sn and “profits” p1, …, pn, wheres1, …, sn and p1, …, pn are positive integers. • Problem: Find the largest total profit of any subset of the objects that fits in the knapsack (and find a subset that achieves the maximum profit). csci5233 computer security & integrity (Chap. 3)

11. The Graph Coloring Problem • A coloring of a graph G = (V, E) is a mapping C: VS, where S is a finite set of “coloring”, such that if vw  E then C(v)  C(w); that is, adjacent vertices are not assigned the same color. • A decision problem: Given G and a positive integer k, is there a coloring of G using at most k colors? (Is G k-colorable?) • An application: Scheduling the final exams at a university without conflicts. Let V be the set of courses. Let E be the pairs of courses whose exams must not be at the same time. Suppose there are 15 time slots (3 exams per day X 5 days). Then the exams can be scheduled in the 15 time slots without conflicts if and only if the graph G = (V, E) is 15-colorable. csci5233 computer security & integrity (Chap. 3)

12. NP-completeness • P.75 • [Cook 71] • A problem is called NP-complete if it can represent the entire class NP. • Example: The ‘satisfiability’ problem (p.71) is a NP-complete problem. (Cook’s theorem) • Implication: If a NP-complete problem can be solved by a deterministic, polynomial time algorithm, then all NP problems can be solved by such algorithms. • The converse implication: “If for even one of the NP-complete problems it could be shown that there was no deterministic algorithm that ran in polynomial time, then no deterministic algorithm could exist for any of them.” csci5233 computer security & integrity (Chap. 3)

13. Does P = NP or P  NP? • P.76 • NP-complete problems have been studied for a long time by many different groups of people and so far no polynomial, deterministic solution has been found for any one of the problems. • Implication: It is very likely that no polynomial, deterministic solution exists for NP-complete problems. That is, NP  P. • Note: The question ‘Does P = NP?’ is still an open question. csci5233 computer security & integrity (Chap. 3)

14. NP-completeness & Cryptography • Hard-to-solve problems are fundamental to cryptography, because the interceptor would need to work hard to break the encryption. • But, be aware of the fallacies: • An NP-complete problem does not guarantee that there is no solution easier than exponential. • Every NP-complete problem has a deterministic exponential time solution. That is, O(2n). • Continuing advances in hardware make problems of larger size tractable. • The interceptor does not always have to solve the had problem in order to crack the encryption. csci5233 computer security & integrity (Chap. 3)

15. Review of Some Arithmetic Concepts • Inverses A number i is an identity for an operation op if, for every number x, x op i = x and i op x = x. Example: 0 is an identity for +, because x + 0 = 0 + x = x. 1 is an identity for *, because x * 1 = 1 * x = x. The inverse of a number x, x-1, under an operation op, is a number that makes x * x-1 = x-1 * x = i, where i is the identity under op. Example: The inverse of 5, under +, is –5. The inverse of 5, under *, is 1/5. csci5233 computer security & integrity (Chap. 3)

16. Review of Some Arithmetic Concepts • Prime numbers • GCD • Euclidean algorithm for computing GCD • Modular arithmetic Two integers, a and b, are equivalent under modulus n, if a mod n = b mod n. If a is equivalent to b, under modulus n, then (x – y) = k * n for some k. (That is, the difference between x and y is divisible by n.) csci5233 computer security & integrity (Chap. 3)

17. Modular arithmetic • Table on p.79 • Reducibility (a + b) mod n = ((a mod n) + (b mod n)) mod n (a * b) mod n = ((a mod n) * (b mod n)) mod n • Modular operations on negative numbers -13 mod 3 = ? • The inverse under modular operations Example: The inverse of 3 mod 5, under *, is 2 because (3 * 2) mod 5 = 1. (See the multiplication table on p.80.) csci5233 computer security & integrity (Chap. 3)

18. The inverse under modular operations • Exercise: What is the inverse of 4 mod 5 under * ? • Fermat’s Theorem: ap mod p = a (and ap-1 mod p = 1), where p is a prime number and a < p. • Computing the inverse of a number The inverse of a number a = ap-2 mod p (p.81) Example: The inverse of 3 mod 5 = 35-2 mod 5 = 2. Verify: 3 * 2 mod 5 = 1. csci5233 computer security & integrity (Chap. 3)

19. Secret versus Public Key Encryptions • Secret Key Encryption Algorithms Also called single key, symmetric, private key, or conventional algorithms. Both the encryptor and the decryptor use the same key. Example: DES • Public Key Encryption Algorithms The sender uses the receiver’s public key to encrypt the message. The receiver uses his own private key to decrypt the encrypted message. Examples: Merkle-Hellman Knapsacks, RSA Encryption, El Gamal Algorithms csci5233 computer security & integrity (Chap. 3)

20. Secret Key Encryptions • Advantages The symmetry of the key provides a two-way channel. Both ends can both encrypt and decrypt information. Authentication of incoming messages: An incoming ciphertext is authenticated because only legitimate sender can produce the ciphertext. csci5233 computer security & integrity (Chap. 3)

21. Secret Key Encryptions • Disadvantages • A single-point failure: A compromised key allows the interceptor not only to decrypt all the ciphertext encrypted by the key, but also to fabricate “bogus” ciphertext to be sent to legitimate receivers. • Distribution of keys can be a problem. (See Fig. 3-10, p.101: key distribution in pieces) • The number of keys increases rapidly, roughly half of the square of the number of people. (See Fig. 3-11, p.101: distribution center) • Adding a new user into an existing network of N users requires the addition of N new keys. • The symmetric key encryptions can be relatively weak, unless based on hard problems. csci5233 computer security & integrity (Chap. 3)

22. Public Key Encryptions • Proposed by Diffie and Hellman in 1976. • Each user has two keys: a public key and a private key, which operate as inverses. P = C (KPRIV, E (KPUB, P ) ) • Advantage: No need for a shared key between every two users. csci5233 computer security & integrity (Chap. 3)

23. Next • Popular Encryption Algorithms • Merkle-Hellman Knapsacks • RSA Encryption • El Gamal Algorithms • DES • Hashing Algorithms • Key Escrow & Clipper csci5233 computer security & integrity (Chap. 3)