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Number Theory

Number Theory. Importance of Number Theory. Before the dawn of computers, many viewed number theory as last bastion of “ pure math ” which could not be useful and must be enjoyed only for its aesthetic beauty.

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Number Theory

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  1. Number Theory

  2. Importance of Number Theory • Before the dawn of computers, many viewed number theory as last bastion of “pure math” which could not be useful and must be enjoyed only for its aesthetic beauty. • No longer the case. Number theory is crucial for encryption algorithms. Of utmost importance to everyone from Bill Gates, to the CIA, to Osama Bin Laden.

  3. Divisors • DEF: Let a, b and c be integers such that • a = b ·c . • Then b and c are said to divide (or are factors) of a, while a is said to be a multiple of b (as well as of c). The pipe symbol “|” denotes “divides” so the situation is summarized by: • b | a  c | a .

  4. Divisors.Examples A: • 77 | 7: false bigger number can’t divide smaller positive number • 7 | 77: true because 77 = 7 ·11 • 24 | 24: true because 24 = 24 ·1 • 0 | 24: false, only 0 is divisible by 0 • 24 | 0: true, 0 is divisible by every number (0 = 24 ·0)

  5. Formula for Number of Multiples up to given n • How many positive multiples of 15 are less than 100? 15, 30, 45, 60, 75, 80, 95. Therefore the answer is 6. Q: How many positive multiples of 15 are less than 1,000,000?

  6. Formula for Number of Multiples up to Given n In general: The number of d-multiples less than N is given by: |{m  Z+ |d |m and m  N }| = N/d

  7. Prime Numbers DEF: A number n 2 prime if it is only divisible by 1 and itself. A number n 2 which isn’t prime is called composite. Q: Which of the following are prime? 0,1,2,3,4,5,6,7,8,9,10

  8. Primality Testing • Prime numbers are very important in encryption schemes. Essential to be able to verify if a number is prime or not. It turns out that this is quite a difficult problem. First try: • boolean isPrime(integer n) • if ( n < 2 ) return false • for(i = 2 to n /2) • if( i |n ) // “divides”! not disjunction • return false • return true

  9. Primality Testing LEMMA: If n is a composite, then its smallest prime factor is  Proof (by contradiction). Suppose the smallest prime factor is > . Then by the fundamental theorem of arithmetic we can decompose n = pqx where p and q are primes > and x is some integer. Therefore implying that n>n, which is impossible showing that the original supposition was false and the theorem is correct. 

  10. Greatest Common DivisorRelatively Prime DEF Let a,b be integers, not both zero. The greatest common divisor of a and b (or gcd(a,b) ) is the biggest number d which divides both a and b. Equivalently: gcd(a,b)is smallest number which divisibly by any x dividing both a and b. DEF: a and b are said to be relatively primeif gcd(a,b) = 1, so no prime common divisors.

  11. Greatest Common DivisorRelatively Prime A: • gcd(11,77) = 11 • gcd(33,77) = 11 • gcd(24,36) = 12 • gcd(24,25) = 1. Therefore 24 and 25 are relatively prime. NOTE: A prime number are relatively prime to all other numbers which it doesn’t divide.

  12. Euclidean Algorithm Euclidean Algorithm m , n gcd(m,n) integer euclid(pos. integer m, pos. integer n) x = m, y = n while(y > 0) r = x mod y x = y y = r return x

  13. Euclidean Algorithm.Example gcd(244,117): By definition  244 and 117 are rel. prime.

  14. Modular Arithmetic There are two types of “mod” (confusing): • the modfunction • Inputs a number a and a base b • Outputs a mod b a number between 0 and b –1 inclusive • This is the remainder of ab • Similar to Java’s % operator. • the (mod) congruence • Relates two numbers a, a’ to each other relative some base b • a  a’(mod b) means that a and a’ have the same remainder when dividing by b

  15. mod function Similar to Java’s “%” operator except that answer is always positive. E.G. -10 mod 3 = 2, but in Java –10%3 = -1. Q: Compute • 113 mod 24 • -29 mod 7

  16. mod function A: Compute • 113 mod 24: • -29 mod 7

  17. Modular arithmeticharder examples Q: Compute the following. • 3071001 mod 102 • (-45 · 77) mod 17

  18. Modular arithmeticharder examples A: Use the previous identities to help simplify: • Using multiplication rules, before multiplying (or exponentiating) can reduce modulo 102: 3071001 mod 102  3071001(mod 102)  11001(mod 102)  1(mod 102). Therefore, 3071001 mod 102 = 1.

  19. Modular arithmeticharder examples A: Use the previous identities to help simplify: • Repeatedly reduce after each multiplication: (-45·77) mod 17  (-45·77)(mod 17) (6·9)(mod 17)  54(mod 17)  3(mod 17). Therefore (-45·77) mod 17 = 3.

  20. Modular arithmeticharder examples A: Use the previous identities to help simplify: • Similarly, before taking sum can simplify modulo 11: Therefore, the answer is 0.

  21. Pseudo Random number generation • Linear Congruential method • xn+1 = (axn + c) mod m

  22. Simple Encryption Variations on the following have been used to encrypt messages for thousands of years. • Convert a message to capitals. • Think of each letter as a number between 1 and 26. • Apply an invertible modular function to each number. • Convert back to letters (0 becomes 26).

  23. Letter  Number Conversion Table

  24. Encryption example Let the encryption function be f (a) = (3a + 9) mod 26 Encrypt “Stop Thief” • STOP THIEF (capitals) • 19,20,15,16 20,8,9,5,6 • 14,17,2,5 17,7,10,24,1 • NQBE QGJXA

  25. Decryption example Decryption works the same, except that you apply the inverse function. EG: Find the inverse of f (a) = (3a + 9) mod 26 If we didn’t have to deal with mod 26, inverse would be g (a) = 3-1 (a - 9) We’ll see that since gcd(3,26) = 1, the inverse of 3 is actually well defined modulo 26 and is the number 9. This gives: g (a) = 9(a - 9) mod 26 = (9a – 3) mod 26

  26. Public Key Cryptography • How can Alice send a secret message to Bob if Bob cannot send a secret key to Alice? ALICE BOB My public key is: 13890580304018329082310291802198210923810830129823019128092183021398301292381320498068029809347849394598178479388287398457923893848928823748283829929384020010924380915809283290823823

  27. RSA • Rivest – Shamir – Adelman • n = pq. p, q are large primes • Choose e relatively prime to (p-1)(q-1) • Find d, k such that de + k(p-1)(q-1) = 1 by Euclid’s Algorithm • Publish e as the encryption key, d is kept private as the decryption key

  28. Message protocol • Bob • Precompute p, q, n, e, d • Publish e, n • Alice • Read e, n from Bob’s public site • To send message M, compute C = Me mod n • Send C to Bob • Bob • Compute Cd to decode message M

  29. RSA Cryptography m emod N N = 4559, e = 13. Smiley Transmits: “Last name Smiley” • L A S T N A M E S M I L E Y • 1201 1920 0014 0113 0500 1913 0912 0525 • 120113mod 4559, 192013mod 4559, … • 2853 0116 1478 2150 3906 4256 1445 2462

  30. RSA Cryptography N = 4559, d = 3397 • 2853 0116 1478 2150 3906 4256 1445 2462 • 28533397mod 4559, 01163397mod 4559, … • 1201 1920 0014 0113 0500 1913 0912 0525 • L A S T N A M E S M I L E Y

  31. Decryption • de = 1 + k(p-1)(q-1) • Cd (Me)d= Mde = M1 + k(p-1)(q-1) (mod n) • Cd M (Mp-1)k(q-1) M (mod p) • Cd M (Mq-1)k(p-1) M (mod q) • Hence Cd M (mod pq)

  32. Representations of Integers • Example:What is the base 8 expansion of (12345)10 ? • First, divide 12345 by 8: • 12345 = 81543 + 1 • 1543 = 8192 + 7 • 192 = 824 + 0 • 24 = 83 + 0 • 3 = 80 + 3 • The result is: (12345)10 = (30071)8.

  33. Number Systems EG: Convert 646 to Oggian (base-5):

  34. ( )2 Addition of Integers carry 1 1 1 • How do we (humans) add two integers? • Example: 7583 + 4932 1 2 5 1 5 carry 1 1 Binary expansions: (1011)2 +(1010)2 1 0 1 0 1

  35. 2’s Complement • Fixes the non-uniqueness of zero problem • Adding mixed signs still easy • No cycle overflow (pre-computed) • Java’s approach (under the hood) • Same fixed length k , sign convention, and definition of positive numbers as with 1’s complement • Represent numbers with -2(k-1)x < 2(k-1) • EG. Java’s byte ranges from -128 to +127

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