1. The Fundamentals: Algorithms, Integers, and Matrices CSC-2259 Discrete Structures Konstantin Busch - LSU1

2. Integers and Division Konstantin Busch - LSU2 divides : Examples: Integers factor

3. Konstantin Busch - LSU3 Number of positive integers divisible by and not exceeding :

4. Konstantin Busch - LSU4 integers if then

5. Konstantin Busch - LSU5 integers if and then

6. Konstantin Busch - LSU6 integers if and then

7. Konstantin Busch - LSU7 integers if and then

8. Konstantin Busch - LSU8 The division “algorithm” There are unique such that: divisor quotientremainder

9. Konstantin Busch - LSU9 Examples:

10. Konstantin Busch - LSU10 Division_algorithm( ) { while ( ) { } if ( and ) { } return, } //a is negative //adjust r //adjust q

11. Konstantin Busch - LSU11 Time complexity of division alg.: There is a better algorithm: (based on binary search)

12. “ is congruent to modulo ” Konstantin Busch - LSU12 Modular Arithmetic Examples:

13. Konstantin Busch - LSU13 Equivalent definitions

14. Konstantin Busch - LSU14 3 Length of line represents number

15. Konstantin Busch - LSU15 11 Length of helix line represents number

16. Konstantin Busch - LSU16 19 Length of helix line represents number

17. Konstantin Busch - LSU17 3 1119 Helix lines terminate in same number

18. Konstantin Busch - LSU18 Congruence class of modulo : There are congruence classes:

19. Konstantin Busch - LSU19

20. Konstantin Busch - LSU20

21. Konstantin Busch - LSU21

22. Konstantin Busch - LSU22 Follows from previous results by using:

23. Konstantin Busch - LSU23 Modular exponentiation Compute efficiently using small numbers Binary expansion of

24. Konstantin Busch - LSU24 Example:

25. Konstantin Busch - LSU25 Compute all the powers of 3 efficiently Use the powers of 3 to get result efficiently

26. Konstantin Busch - LSU26 Modular_Exponentiation( ) { for to { if ( ) } return } Time complexity: bit operations

27. Konstantin Busch - LSU27 Congruent application: Hashing functions Example: Employer idFolder# collision

28. Konstantin Busch - LSU28 Application: Pseudorandom numbers Linear congruential method: Sequence of pseudorandom numbers seed Example: 3,7,8,6,1,2,0,4,5,3,7,8,6,1,2,0,4,5,3… seed

29. Konstantin Busch - LSU29 Application: Cryptology “MEET YOU IN THE PARK” “PHHW BRX LQ WKH SDUN” Shift cipher: Affine transformation: encryptiondecryption

30. Primes and Greatest Common Divisor Konstantin Busch - LSU30 Prime :Positive integer greater than 1, only positive factors are Non-prime = composite Primes:2,3,5,7,11,13,17,19,23,29,31,37,41,…

31. Konstantin Busch - LSU31 Fundamental theorem of arithmetic Every positive integer is either prime or a unique product of primes Examples: prime Prime factorization:

32. Konstantin Busch - LSU32 Theorem: If is composite then it has prime divisor Proof: is composite since otherwise From fundamental theorem of arithmetic is either prime or has a prime divisor End of Proof

33. Konstantin Busch - LSU33 Prime_factorization( ) { while ( and ) { if ( divides ) { is a factor of } else next prime after } return all prime factors found } //first prime

34. Konstantin Busch - LSU34 do not divide 7007 does not divide 143

35. Konstantin Busch - LSU35 Theorem:There are infinitely many primes Proof:Suppose finite primes Let If some prime Since impossible No prime dividesis prime Contradiction! (From fundamental theorem of arithmetic) End of Proof

36. Konstantin Busch - LSU36 Largest prime known (as of 2006) Mersenne primes have the form:

37. Konstantin Busch - LSU37 Prime number theorem The number of primes less or equal to approaches to:

38. Konstantin Busch - LSU38 Goldbach’s conjecture: Every integer is the sum of two primes Twin prime conjecture: There are infinitely many twin primes Twin primes differ by 2:

39. Konstantin Busch - LSU39 Greatest common divisor largest integer such that and Examples: Common divisors of 24, 36:1, 2, 3, 4, 6, 12 Common divisors of 17, 22:1

40. Konstantin Busch - LSU40 Trivial cases:

41. Konstantin Busch - LSU41 Theorem: If then Proof: Thus, and have the same set of common divisors End of proof

42. Konstantin Busch - LSU42 first zeroresult divisions remainder

43. Konstantin Busch - LSU43 result

44. Konstantin Busch - LSU44 Euclidian Algorithm gcd( ) { while ( ) { } return } Time complexity:divisions

45. and have no common factors in their prime factorization Konstantin Busch - LSU45 Relatively prime numbers If then are relatively prime Example: 21, 22 are relatively prime

46. Konstantin Busch - LSU46 Least common multiple smallest positive integer such that and Examples:

47. Applications of Number Theory Konstantin Busch - LSU47 Linear combination: if then there are such that Example:

48. Konstantin Busch - LSU48 The linear combination can be found by reversing the Euclidian algorithm steps

49. Konstantin Busch - LSU49 Linear congruences We want to solve the equation for

50. Konstantin Busch - LSU50 Inverse of :

51. Konstantin Busch - LSU51 If and are relatively prime then the inverse modulo exists Theorem: Proof: End of proof

52. Konstantin Busch - LSU52 Example: solve equation Inverse of 3:

53. Konstantin Busch - LSU53 Chinese remainder problem :pairwise relatively prime Has unique solution for modulo

54. Konstantin Busch - LSU54 Solution: :inverse of modulo

55. Konstantin Busch - LSU55 Explanation::inverse of modulo Similar for any

56. Konstantin Busch - LSU56 Example:

57. Konstantin Busch - LSU57 Solution is unique modulo, since for any other solution it holds iff

58. Konstantin Busch - LSU58 Application of Chinese remainder problem Perform arithmetic with large numbers using arithmetic modulo small numbers Example:relatively prime numbers Any number smaller than has unique representation

59. Konstantin Busch - LSU59 + We obtain this by using the Chinese remainder problem solution ++++

60. Konstantin Busch - LSU60 Fermat’s little theorem For any prime and integer not divisible by ( ): Example:

61. Konstantin Busch - LSU61 RSA cryptosystem “MEET YOU IN THE PARK” “9383772909383637467” encryptiondecryption Large primes are public keys are private keys

62. Konstantin Busch - LSU62 Message to encrypt: “STOP” Encryption example: Translate to equivalent numbers “18 19 14 15” Group into blocks of two numbers

63. Konstantin Busch - LSU63 “1819 1415” “2081 2182” Apply encryption function to each block Encrypted message:

64. Konstantin Busch - LSU 64 Message decryption :an original block of the message :respective encrypted block “1819 1415” “2081 2182” We want to find by knowing

65. Konstantin Busch - LSU65 :inverse of modulo Inverse exists because by definition of congruent

66. Konstantin Busch - LSU66

67. Konstantin Busch - LSU67 Very likely it holds (because is a large prime and is small) By Fermat’s little theorem

68. Konstantin Busch - LSU68

69. Konstantin Busch - LSU69 By symmetry, when replacing with : We showed: By the Chinese remainder problem:

70. Konstantin Busch - LSU70 We showed: In other words:

71. Konstantin Busch - LSU71 Decryption example: “2081 2182” It can be shown that: “1819 1415” “18 19 14 15” = “STOP”