1. Tuesday, 27 April Number-Theoretic Algorithms Chapter 31
UMass Lowell Computer Science Analysis of Algorithms Prof. Karen Daniels Spring, 2010
Tuesday, 27 April
Number-Theoretic Algorithms
Chapter 31

2. Chapter Dependencies
Math: Number Theory
Ch 31
Number-Theoretic Algorithms
RSA
You’re responsible for material in this chapter that we discuss in lecture. (Note that this does not include sections 31.8 or 31.9.)

3. Overview Motivation: RSA Basics Euclid’s GCD Algorithm
Chinese Remainder Theorem
Powers of an Element
RSA Details

4. Motivation: RSA

5. RSA Encryption
31.5
source: textbook Cormen et al.

6. RSA Digital Signature
31.6 ?
assume Alice also sends her name so Bob knows whose public key to use
source: textbook Cormen et al.

7. RSA Cryptosystem + EXAMPLE encode decode to be explained later….
(31.19)*
(31.26)
source: textbook Cormen et al., 3rd edition
to be explained later….
(31.20)
(31.35)
Assume M < n
(31.36)
encode
need efficient ways to compute P(M), S(C)
decode
+ EXAMPLE

8. RSA Dependence Correctness: Efficiency: Security: Euler’s f Function
Fermat’s Theorem
Chinese Remainder Theorem
Efficiency:
Modular Exponentiation
Primality Testing
Security:
Difficulty of Factoring Large Integers
Need to show:
see chart of result dependencies on next slide (courtesy of Mark Micire)

9. with thanks to Mark Micire
EUCLID GCD
EXTENDED-EUCLID
(Eqn )
2002
with thanks to Mark Micire

10. Notes on Primality Testing
Efficient primality testing has been goal for > 2,000 years.
Early attempts required exponential time.
Miller-Rabin (Section 31.8) primality test is a randomized polynomial-time algorithm (1980’s).
Agrawal, Kayal, Saxena provided a deterministic polynomial-time algorithm (2002).

11. Basic Concepts
* Indicates that result is on chart of result dependencies

12. Division & Remainders + EXAMPLE * 31.1
(3.8)
source: textbook Cormen et al.

13. Equivalence Class Modulo n
(31.1)
(31.2)
+ EXAMPLE
source: textbook Cormen et al.

14. Common Divisors + EXAMPLE * * (31.3) (31.4) (31.5)
source: textbook Cormen et al.

15. Greatest Common Divisor
(31.6)
(31.7)
(31.8)
(31.9) *
(31.10) *
31.2
(3.8)
+ EXAMPLE
(31.4)
source: textbook Cormen et al.

16. Greatest Common Divisor
31.3 *
(31.4)
31.2
31.4
+ EXAMPLE
source: textbook Cormen et al.

17. Relatively Prime Integers*
31.6
31.2
31.2
+ EXAMPLE
source: textbook Cormen et al.

18. Relatively Prime Integers
31.7
31.6 *
31.1-6
+ EXAMPLE
source: textbook Cormen et al.

19. Greatest Common Divisor*
31.9
(31.5)
(3.8)
(31.4)
(31.3)
(31.14)
(31.15)
+ EXAMPLE
source: textbook Cormen et al.

20. Euclid’s GCD Algorithm

21. Euclid’s GCD Algorithm*
+ EXAMPLE
Also see Java code on course web site
source: textbook Cormen et al.

22. Extended Euclid + EXAMPLE * * (31.16)
source: textbook Cormen et al.

23. Chinese Remainder Theorem

24. Modular Arithmetic
source: textbook Cormen et al.

25. Finite Groups Additive group mod 6 Multiplicative group mod 15 31.2
size of this group is 6
size of this group is 8
source: textbook Cormen et al.
elements relatively prime to n

26. Finite Groups
31.12
source: textbook Cormen et al.

27. Finite Groups
31.13
31.6
31.12
31.26
source: textbook Cormen et al.

28. Euler’s Phi Function + EXAMPLE * (31.19)
source: textbook Cormen et al.

29. Lagrange’s Theorem + EXAMPLE * 31.15
source: textbook Cormen et al.

30. Finite Groups + EXAMPLE * * additive subgroup generated by a 31.17
source: textbook Cormen et al.
31.18
31.19 *
where k
+ EXAMPLE

31. Solving Modular Linear Eq*
31.20
+ EXAMPLE
(31.4)
source: textbook Cormen et al.

32. Solving Modular Linear Eq
source: textbook Cormen et al.
31.22
31.18
31.24 *
+ EXAMPLE

33. Solving Modular Linear Eq*
+ EXAMPLE
31.26 *
source: textbook Cormen et al.

34. Chinese Remainder Theorem
31.27 *
(31.23)
+ EXAMPLE
(31.23)
(31.24)
(31.25)
(31.26)
source: textbook Cormen et al.

35. Chinese Remainder Theorem
Corollary If n1, n2, …, nk are pairwise relatively prime and n = n1n2…nk, then, for any integers a1, a2, …, ak, the set of simultaneous equations for i = 1, 2, …, k, has a unique solution modulo n for the unknown x.
31.29 *
source: textbook Cormen et al.

36. NumTheory
Example.
Given the two equations what is a mod 65? Note that 65 = 5•13.
The table of moduli wrt 5 and 13 for all integers in Z65.
source: textbook Cormen et al. & Prof. Pecelli
Table can be generated diagonally.
1/1/2019

37. NumTheory Knowing that find a mod 65. We have
source: textbook Cormen et al. & Prof. Pecelli
Knowing that find a mod 65.
We have
a1 = 2, n1 = 5 , m1 = n/n1 = 13,
a2 = 3, n2 = 13, m2 = n/n2 = 5.
We can compute:
1/1/2019

38. Powers of an Element

39. Theorems of Euler & Fermat
31.30 *
31.31 *
31.20
source: textbook Cormen et al.

40. Modular Exponentiation*
+ EXAMPLE
Also see Java code on course web site
source: textbook Cormen et al.

41. RSA Details

42. RSA Encryption
31.5
source: textbook Cormen et al.

43. RSA Digital Signature
31.6 ?
assume Alice also sends her name so Bob knows whose public key to use
source: textbook Cormen et al.

44. RSA Cryptosystem encode decode
(31.19)
(31.26)
source: textbook Cormen et al., 3rd edition
(31.20)
(31.35)
(31.36)
encode
decode
need efficient ways to compute P(M), S(C)

45. RSA Correctness p q by Thm 31.31 (Fermat)
(31.37)
(31.38)
31.31) p
by Thm (Fermat) q
31.29
source: textbook Cormen et al. 3rd edition