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Lecture 8 Overview

Lecture 8 Overview. Analysis of Algorithms. Algorithms Time Complexity Space Complexity An algorithm whose time complexity is bounded by a polynomial is called a polynomial-time algorithm. An algorithm is considered to be efficient if it runs in polynomial time. Time and Space.

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Lecture 8 Overview

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  1. Lecture 8 Overview

  2. Analysis of Algorithms • Algorithms • Time Complexity • Space Complexity • An algorithm whose time complexity is bounded by a polynomial is called a polynomial-time algorithm. • An algorithm is considered to be efficient if it runs in polynomial time. CS 450/650 Lecture 8: Algorithm Background

  3. Time and Space • Should be calculated as function of problem size (n) • Sorting an array of size n, • Searching a list of size n, • Multiplication of two matrices of size n by n • T(n) = function of n (time) • S(n) = function of n (space) • relative rates of growth • 1000n vs. n2 CS 450/650 Lecture 8: Algorithm Background

  4. Definitions • T(n) = O(f(n)): T is bounded above by f The growth rate of T(n) <= growth rate of f(n) • T(n) = W (g(n)): T is bounded below by g The growth rate of T(n) >= growth rate of g(n) • T(n) = Q(h(n)): T is bounded both above and below by h The growth rate of T(n) = growth rate of h(n) • T(n) = o(p(n)): T is dominated by p The growth rate of T(n) < growth rate of p(n) CS 450/650 Lecture 8: Algorithm Background

  5. Time Complexity • C • O(n) • O(log n) • O(nlogn) • O(n2) • … • O(nk) • O(2n) • O(kn) • O(nn) Polynomial O(2log n) Exponential CS 450/650 Lecture 8: Algorithm Background

  6. P, NP, NP-hard, NP-complete • A problem belongs to the class P if the problem can be solved by a polynomial-time algorithm • A problem belongs to the class NP if the correctness of the problem’s solution can be verified by a polynomial-time algorithm • A problem is NP-hard if it is as hard as any problem in NP • Existence of a polynomial-time algorithm for an NP-hard problem implies the existence of polynomial solutions for every problem in NP • NP-complete problems are the NP-hard problems that are also in NP CS 450/650 Lecture 8: Algorithm Background

  7. Relationships between different classes NP NP-hard NP-complete P CS 450/650 Lecture 8: Algorithm Background

  8. Lecture 9Rivest-Shamir-Adelman (RSA) CS 450/650 Fundamentals of Integrated Computer Security Slides are modified from Hesham El-Rewini

  9. RSA • Invented by Cocks (GCHQ), independently, by Rivest, Shamir and Adleman (MIT) • Two keys e and d used for Encryption and Decryption • The keys are interchangeable • M = D(d, E(e, M) ) = D(e, E(d, M) ) • Public key encryption • Based on problem of factoring large numbers • Not in NP-complete • Best known algorithm is exponential CS 450/650 Lecture 9: RSA

  10. RSA • To encrypt message M compute • c = Me mod N • To decrypt ciphertext c compute • M = cd mod N CS 450/650 Lecture 9: RSA

  11. Key Choice • Let p and q be two large prime numbers • Let N = pq • Choose e relatively prime to (p1)(q1) • a prime number larger than p-1 and q-1 • Find d such that ed mod (p1)(q1) = 1 CS 450/650 Lecture 9: RSA

  12. RSA • Recall that e and N are public • If attacker can factor N, he can use e to easily find d • since ed mod (p1)(q1) = 1 • Factoring the modulus breaks RSA • It is not known whether factoring is the only way to break RSA CS 450/650 Lecture 9: RSA

  13. Does RSA Really Work? • Given c = Me mod N we must show • M = cd mod N = Med mod N • We’ll use Euler’s Theorem • If x is relatively prime to N then x(N) mod N =1 • (n): number of positive integers less than n that are relatively prime to n. • If p is prime then, (p) = p-1 CS 450/650 Lecture 9: RSA

  14. Does RSA Really Work? • Facts: • ed mod (p  1)(q  1) = 1 • ed = k(p  1)(q  1) + 1 by definition of mod • (N) = (p  1)(q  1) • Then ed  1 = k(p  1)(q  1) = k(N) • Med = M(ed-1)+1 = MMed-1 = MMk(N) = M(M(N)) k mod N = M1 k mod N = M mod N CS 450/650 Lecture 9: RSA

  15. Example • Select primes p=11, q=3. • N = p* q = 11*3 = 33 • Choose e = 3 • check gcd(e, p-1) = gcd(3, 10) = 1 • i.e. 3 and 10 have no common factors except 1 • check gcd(e, q-1) = gcd(3, 2) = 1 • therefore gcd(e, (p-1)(q-1)) = gcd(3, 20) = 1 CS 450/650 Lecture 9: RSA

  16. Example (cont.) • p-1 * q-1 = 10 * 2 = 20 • Compute d such that e * d mod (p-1)*(q-1) = 1 3 * d mod 20 = 1 d = 7 Public key = (N, e) = (33, 3) Private key = (N, d) = (33, 7) CS 450/650 Lecture 9: RSA

  17. Example (cont.) • Now say we want to encrypt message m = 7 • c = Me mod N = 73 mod 33 = 343 mod 33 = 13 • Hence the ciphertext c = 13 • To check decryption, we compute M' = cd mod N = 137 mod 33 = 7 CS 450/650 Lecture 9: RSA

  18. More Efficient RSA • Modular exponentiation example • 520 = 95367431640625 = 25 mod 35 • A better way: repeated squaring • Note that 20 = 2  10, 10 = 2  5, 5 = 2  2 + 1, 2 = 1 2 • 51= 5 mod 35 • 52= (51) 2 = 52 = 25 mod 35 • 55= (52) 2 51 = 252 5 = 3125 = 10 mod 35 • 510 = (55) 2 = 102 = 100 = 30 mod 35 • 520 = (510) 2 = 302 = 900 = 25 mod 35 • No huge numbers and it’s efficient! CS 450/650 Lecture 9: RSA

  19. RSA key-length strength • RSA has challenges for different key-lengths • RSA-140 • Factored in 1 month using 200 machines in 1999 • RSA-155 (512-bit) • Factored in 3.7 months using 300 machines in 1999 • RSA-160 • Factored in 20 days in 2003 • RSA-200 • Factored in 18 month in 2005 • RSA-210, RSA-220, RSA-232, … RSA-2048 CS 450/650 Lecture 9: RSA

  20. Symmetric vs Asymmetric CS 450/650 Fundamentals of Integrated Computer Security

  21. Group Work Find keys d and e for the RSA cryptosystem with p = 7 and q = 11 Solution • p*q = 77 • (p-1) * (q-1) = 60 • e = 37 • d = 13 • n = 13 * 37 = 481 = 1 mod 60 CS 450/650 Lecture 9: RSA

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