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1 Integers and Division CS/APMA 202 Rosen section 2.4 Aaron Bloomfield

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2 Rosen, chapter 2 We are only doing 2 or 3 of the sections in chapter 2 2.4: integers and division 2.4: integers and division 2.6: applications of number theory 2.6: applications of number theory And only parts of that section 2.7: matrices (maybe) 2.7: matrices (maybe)

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3 Quick survey Have you seen matrices before? Have you seen matrices before? a) Lots and lots and lots b) A fair amount c) Just a little d) Is that kinda like the movie?

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4 Why prime numbers? Prime numbers are not well understood Basis for today’s cryptography Unless otherwise indicated, we are only talking about positive integers for this chapter

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5 The divides operator New notation: 3 | 12 To specify when an integer evenly divides another integer To specify when an integer evenly divides another integer Read as “3 divides 12” Read as “3 divides 12” The not-divides operator: 5 | 12 To specify when an integer does not evenly divide another integer To specify when an integer does not evenly divide another integer Read as “5 does not divide 12” Read as “5 does not divide 12”

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6 Theorem on the divides operator If a | b and a | c, then a | (b+c) Example: if 5 | 25 and 5 | 30, then 5 | (25+30) Example: if 5 | 25 and 5 | 30, then 5 | (25+30) If a | b, then a | bc for all integers c Example: if 5 | 25, then 5 | 25*c for all ints c Example: if 5 | 25, then 5 | 25*c for all ints c If a | b and b | c, then a | c Example: if 5 | 25 and 25 | 100, then 5 | 100 Example: if 5 | 25 and 25 | 100, then 5 | 100 The book calls this Theorem 1

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7 Prime numbers A positive integer p is prime if the only positive factors of p are 1 and p If there are other factors, it is composite If there are other factors, it is composite Note that 1 is not prime! Note that 1 is not prime! It’s not composite either – it’s in its own class An integer n is composite if and only if there exists an integer a such that a | n and 1 < a < n

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8 Fundamental theorem of arithmetic Every positive integer greater than 1 can be uniquely written as a prime or as the product of two or more primes where the prime factors are written in order of non-decreasing size Examples 100 = 2 * 2 * 5 * 5 100 = 2 * 2 * 5 * 5 182 = 2 * 7 * 13 182 = 2 * 7 * 13 29820 = 2 * 2 * 3 * 5 * 7 * 71 29820 = 2 * 2 * 3 * 5 * 7 * 71 The book calls this Theorem 2

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9 Composite factors If n is a composite integer, then n has a prime divisor less than or equal to the square root of n Direct proof Since n is composite, it has a factor a such that 1<a<n Since n is composite, it has a factor a such that 1<a<n Thus, n = ab, where a and b are positive integers greater than 1 Thus, n = ab, where a and b are positive integers greater than 1 Either a≤ n or b≤ n (Otherwise, ab > n* n > n) Either a≤ n or b≤ n (Otherwise, ab > n* n > n) Thus, n has a divisor not exceeding n Thus, n has a divisor not exceeding n This divisor is either prime or a composite This divisor is either prime or a composite If the latter, then it has a prime factor In either case, n has a prime factor less than n In either case, n has a prime factor less than n The book calls this Theorem 3

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10 Showing a number is prime Show that 113 is prime (Rosen, question 8c, §2.4) Solution The only prime factors less than 113 = 10.63 are 2, 3, 5, and 7 The only prime factors less than 113 = 10.63 are 2, 3, 5, and 7 Neither of these divide 113 evenly Neither of these divide 113 evenly Thus, by the fundamental theorem of arithmetic, 113 must be prime Thus, by the fundamental theorem of arithmetic, 113 must be prime

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11 Showing a number is composite Show that 899 is prime (Rosen, question 8c, §2.4) Solution Divide 899 by successively larger primes, starting with 2 Divide 899 by successively larger primes, starting with 2 We find that 29 and 31 divide 899 We find that 29 and 31 divide 899 On a unix system, enter “factor 899” aaron@orion:~.16> factor 899 899: 29 31

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12 Primes are infinite Theorem (by Euclid): There are infinitely many prime numbers The book calls this Theorem 4 The book calls this Theorem 4 Proof by contradiction Assume there are a finite number of primes List them as follows: p 1, p 2 …, p n. Consider the number q = p 1 p 2 … p n + 1 This number is not divisible by any of the listed primes This number is not divisible by any of the listed primes If we divided p i into q, there would result a remainder of 1 We must conclude that q is a prime number, not among the primes listed above We must conclude that q is a prime number, not among the primes listed above This contradicts our assumption that all primes are in the list p 1, p 2 …, p n.

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13 End of lecture on 17 Febrary 2005

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14 Mersenne numbers Mersenne nubmer: any number of the form 2 n -1 Mersenne prime: any prime of the form 2 p -1, where p is also a prime Example: 2 5 -1 = 31 is a Mersenne prime Example: 2 5 -1 = 31 is a Mersenne prime Example: 2 11 -1 = 2047 is not a prime (23*89) Example: 2 11 -1 = 2047 is not a prime (23*89) Largest Mersenne prime: 2 24,036,583 -1, which has 7,235,733 digits If M is a Mersenne prime, then M(M+1)/2 is a perfect number A perfect number equals the sum of its divisors A perfect number equals the sum of its divisors Example: 2 3 -1 = 7 is a Mersenne prime, thus 7*8/2 = 28 is a perfect number Example: 2 3 -1 = 7 is a Mersenne prime, thus 7*8/2 = 28 is a perfect number 28 = 1+2+4+7+14 Example: 2 5 -1 = 31 is a Merenne prime, thus 31*32/2 = 496 is a perfect number Example: 2 5 -1 = 31 is a Merenne prime, thus 31*32/2 = 496 is a perfect number

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15 Merenne primes Reference for Mersenne primes: http://mathworld.wolfram.com/MersennePrime.html http://mathworld.wolfram.com/MersennePrime.html http://mathworld.wolfram.com/MersennePrime.html http://mathworld.wolfram.com/MersennePrime.html Finding Mersenne primes GIMPS – Great Internet Mersenne Prime Search GIMPS – Great Internet Mersenne Prime Search http://www.mersenne.org/prime.htm http://www.mersenne.org/prime.htm http://www.mersenne.org/prime.htm A new one was just discovered (last week): http://mathworld.wolfram.com/news/2005-02- 18/mersenne/ http://mathworld.wolfram.com/news/2005-02- 18/mersenne/ http://mathworld.wolfram.com/news/2005-02- 18/mersenne/ This is only the 42 nd such prime discovered This is only the 42 nd such prime discovered

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16 The prime number theorem The radio of the number of primes not exceeding x and x/ln(x) approaches 1 as x grows without bound Rephrased: the number of prime numbers less than x is approximately x/ln(x) Rephrased: the number of prime numbers less than x is approximately x/ln(x) Rephrased: the chance of an number x being a prime number is 1 / ln(x) Rephrased: the chance of an number x being a prime number is 1 / ln(x) Consider 200 digit prime numbers ln (10 200 ) 460 ln (10 200 ) 460 The chance of a 200 digit number being prime is 1/460 The chance of a 200 digit number being prime is 1/460 If we only choose odd numbers, the chance is 2/460 = 1/230 If we only choose odd numbers, the chance is 2/460 = 1/230 This result will be used in the next lecture! This result will be used in the next lecture! The book calls this Theorem 5

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17 Showing a number is prime or not Consider showing that 2 650 -1 is prime That number has about 200 digits That number has about 200 digits There are approximately 10 193 prime numbers less than 2 650 -1 By theorem 5 (x/ln(x), where x = 2 650 -1) By theorem 5 (x/ln(x), where x = 2 650 -1) How long would that take to test each of those prime numbers? Assume a computer can do 1 billion (10 9 ) per second Assume a computer can do 1 billion (10 9 ) per second It would take 10 193 /10 9 = 10 184 seconds It would take 10 193 /10 9 = 10 184 seconds That’s 3.2 * 10 176 years! That’s 3.2 * 10 176 years! There are quicker methods to show a number is prime, but not to find the factors if the number is found to be composite There are quicker methods to show a number is prime, but not to find the factors if the number is found to be composite We will use this in the next lecture

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18 The division “algorithm” Let a be an integer and d be a positive integer. Then there are unique integers q and r, with 0 ≤ r < d, such that a = dq+r We then define two operators: q = a div d q = a div d r = a mod d r = a mod d The book calls this Theorem 6

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19 Greatest common divisor The greatest common divisor of two integers a and b is the largest integer d such that d | a and d | b Denoted by gcd(a,b) Denoted by gcd(a,b)Examples gcd (24, 36) = 12 gcd (24, 36) = 12 gcd (17, 22) = 1 gcd (17, 22) = 1 gcd (100, 17) = 1 gcd (100, 17) = 1

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20 Relative primes Two numbers are relatively prime if they don’t have any common factors (other than 1) Rephrased: a and b are relatively prime if gcd (a,b) = 1 Rephrased: a and b are relatively prime if gcd (a,b) = 1 gcd (25, 39) = 1, so 25 and 39 are relatively prime

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21 Pairwise relative prime A set of integers a 1, a 2, … a n are pairwise relatively prime if, for all pairs of numbers, they are relatively prime Formally: The integers a 1, a 2, … a n are pairwise relatively prime if gcd(a i, a j ) = 1 whenever 1 ≤ i < j ≤ n. Formally: The integers a 1, a 2, … a n are pairwise relatively prime if gcd(a i, a j ) = 1 whenever 1 ≤ i < j ≤ n. Example: are 10, 17, and 21 pairwise relatively prime? gcd(10,17) = 1, gcd (17, 21) = 1, and gcd (21, 10) = 1 gcd(10,17) = 1, gcd (17, 21) = 1, and gcd (21, 10) = 1 Thus, they are pairwise relatively prime Thus, they are pairwise relatively prime Example: are 10, 19, and 24 pairwise relatively prime? Since gcd(10,24) ≠ 1, they are not Since gcd(10,24) ≠ 1, they are not

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22 More on gcd’s Given two numbers a and b, rewrite them as: Example: gcd (120, 500) Example: gcd (120, 500) 120 = 2 3 *3*5 = 2 3 *3 1 *5 1 500 = 2 2 *5 3 = 2 2 *3 0 *5 3 Then compute the gcd by the following formula: Example: gcd(120,500) = 2 min(3,2) 3 min(1,0) 5 min(1,3) = 2 2 3 0 5 1 = 20 Example: gcd(120,500) = 2 min(3,2) 3 min(1,0) 5 min(1,3) = 2 2 3 0 5 1 = 20

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23 Least common multiple The least common multiple of the positive integers a and b is the smallest positive integer that is divisible by both a and b. Denoted by lcm (a, b) Denoted by lcm (a, b) Example: lcm(10, 25) = 50 What is lcm (95256, 432)? 95256 = 2 3 3 5 7 2, 432=2 4 3 3 95256 = 2 3 3 5 7 2, 432=2 4 3 3 lcm (2 3 3 5 7 2, 2 4 3 3 ) = 2 max(3,4) 3 max(5,3) 7 max(2,0) = 2 4 3 5 7 2 = 190512 lcm (2 3 3 5 7 2, 2 4 3 3 ) = 2 max(3,4) 3 max(5,3) 7 max(2,0) = 2 4 3 5 7 2 = 190512

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24 lcm and gcd theorem Let a and b be positive integers. Then a*b = gcd(a,b) * lcm (a, b) Example: gcd (10,25) = 5, lcm (10,25) = 50 10*25 = 5*50 10*25 = 5*50 Example: gcd (95256, 432) = 216, lcm (95256, 432) = 190512 95256*432 = 216*190512 95256*432 = 216*190512 The book calls this Theorem 7

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25 Modular arithmetic If a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a-b Notation: a ≡ b (mod m) Notation: a ≡ b (mod m) Rephrased: m | a-b Rephrased: m | a-b Rephrased: a mod m = b Rephrased: a mod m = b If they are not congruent: a ≡ b (mod m) If they are not congruent: a ≡ b (mod m) Example: Is 17 congruent to 5 modulo 6? Rephrased: 17 ≡ 5 (mod 6) Rephrased: 17 ≡ 5 (mod 6) As 6 divides 17-5, they are congruent As 6 divides 17-5, they are congruent Example: Is 24 congruent to 14 modulo 6? Rephrased: 24 ≡ 14 (mod 6) Rephrased: 24 ≡ 14 (mod 6) As 6 does not divide 24-14 = 10, they are not congruent As 6 does not divide 24-14 = 10, they are not congruent

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26 More on congruence Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only if a mod m = b mod m The book calls this Theorem 8 The book calls this Theorem 8 Example: Is 17 congruent to 5 modulo 6? Rephrased: does 17 ≡ 5 (mod 6)? Rephrased: does 17 ≡ 5 (mod 6)? 17 mod 6 = 5 mod 6 17 mod 6 = 5 mod 6 Example: Is 24 congruent to 14 modulo 6? Rephrased: 24 ≡ 14 (mod 6) Rephrased: 24 ≡ 14 (mod 6) 24 mod 6 ≠ 14 mod 6 24 mod 6 ≠ 14 mod 6

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27 Even more on congruence Let m be a positive integer. The integers a and b are congruent modulo m if and only if there is an integer k such that a = b + km The book calls this Theorem 9 The book calls this Theorem 9 Example: 17 and 5 are congruent modulo 6 17 = 5 + 2*6 17 = 5 + 2*6 5 = 17 -2*6 5 = 17 -2*6

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28 Even even more on congruence Let m be a positive integer. If a ≡ b (mod m) and c ≡ d (mod m), then a+c ≡ (b+d) (mod m) and ac ≡ bd (mod m) The book calls this Theorem 10 The book calls this Theorem 10Example We know that 7 ≡ 2 (mod 5) and 11 ≡ 1 (mod 5) We know that 7 ≡ 2 (mod 5) and 11 ≡ 1 (mod 5) Thus, 7+11 ≡ (2+1) (mod 5), or 18 ≡ 3 (mod 5) Thus, 7+11 ≡ (2+1) (mod 5), or 18 ≡ 3 (mod 5) Thus, 7*11 ≡ 2*1 (mod 5), or 77 ≡ 2 (mod 5) Thus, 7*11 ≡ 2*1 (mod 5), or 77 ≡ 2 (mod 5)

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29 Uses of congruences Hashing functions aaron@orion:~/ISOs/dvd.39> md5sum debian-31-i386-binary.iso 96c8bba5a784c2f48137c22e99cd5491 debian-31-i386-binary.iso md5 (file) = mod 2 128 Not really – this is a simplification Not really – this is a simplification

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30 Today’s demotivators

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31 Pseudorandom numbers Computers cannot generate truly random numbers! Algorithm for “random” numbers: choose 4 integers Seed x 0 : starting value Seed x 0 : starting value Modulus m: maximum possible value Modulus m: maximum possible value Multiplier a: such that 2 ≤ a < m Multiplier a: such that 2 ≤ a < m Increment c: between 0 and m Increment c: between 0 and m Formula: x n+1 = (ax n + c) mod m

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32 Pseudorandom numbers Formula: x n+1 = (ax n + c) mod m Let x 0 = 3, m = 9, a = 7, and c = 4 x 1 = 7x 0 +4 = 7*3+4 = 25 mod 9 = 7 x 2 = 7x 1 +4 = 7*7+4 = 53 mod 9 = 8 x 3 = 7x 2 +4 = 7*8+4 = 60 mod 9 = 6 x 4 = 7x 3 +4 = 7*6+4 = 46 mod 9 = 1 x 5 = 7x 4 +4 = 7*1+4 = 46 mod 9 = 2 x 6 = 7x 5 +4 = 7*2+4 = 46 mod 9 = 0 x 7 = 7x 6 +4 = 7*0+4 = 46 mod 9 = 4 x 8 = 7x 7 +4 = 7*4+4 = 46 mod 9 = 5

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33 Pseudorandom numbers Formula: x n+1 = (ax n + c) mod m Let x 0 = 3, m = 9, a = 7, and c = 4 This sequence generates: 3, 7, 8, 6, 1, 2, 0, 4, 5, 3, 7, 8, 6, 1, 2, 0, 4, 5, 3 Note that it repeats! Note that it repeats! But it selects all the possible numbers before doing so But it selects all the possible numbers before doing so The common algorithms today use m = 2 32 -1 You have to choose 4 billion numbers before it repeats You have to choose 4 billion numbers before it repeats

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34 The Caesar cipher Julius Caesar used this to encrypt messages A function f to encrypt a letter is defined as: f(p) = (p+3) mod 26 Where p is a letter (0 is A, 1 is B, 25 is Z, etc.) Where p is a letter (0 is A, 1 is B, 25 is Z, etc.) Decryption: f -1 (p) = (p-3) mod 26 This is called a substitution cipher You are substituting one letter with another You are substituting one letter with another

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35 The Caesar cipher Encrypt “go cavaliers” Translate to numbers: g = 6, o = 14, etc. Translate to numbers: g = 6, o = 14, etc. Full sequence: 6, 14, 2, 0, 21, 0, 11, 8, 4, 17, 18 Apply the cipher to each number: f(6) = 9, f(14) = 17, etc. Apply the cipher to each number: f(6) = 9, f(14) = 17, etc. Full sequence: 9, 17, 5, 3, 24, 3, 14, 11, 7, 20, 21 Convert the numbers back to letters 9 = j, 17 = r, etc. Convert the numbers back to letters 9 = j, 17 = r, etc. Full sequence: jr wfdydolhuv Decrypt “jr wfdydolhuv” Translate to numbers: j = 9, r = 17, etc. Translate to numbers: j = 9, r = 17, etc. Full sequence: 9, 17, 5, 3, 24, 3, 14, 11, 7, 20, 21 Apply the cipher to each number: f -1 (9) = 6, f -1 (17) = 14, etc. Apply the cipher to each number: f -1 (9) = 6, f -1 (17) = 14, etc. Full sequence: 6, 14, 2, 0, 21, 0, 11, 8, 4, 17, 18 Convert the numbers back to letters 6 = g, 14 = 0, etc. Convert the numbers back to letters 6 = g, 14 = 0, etc. Full sequence: go cavaliers

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36 Rot13 encoding A Caesar cipher, but translates letters by 13 instead of 3 Then, apply the same function to decrypt it, as 13+13=26 Then, apply the same function to decrypt it, as 13+13=26 Rot13 stands for “rotate by 13” Example: aaron@orion:~.4> echo Hello World | rot13 Uryyb Jbeyq aaron@orion:~.5> echo Uryyb Jbeyq | rot13 Hello World aaron@orion:~.6>

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37 Quick survey I felt I understood the material in this slide set… I felt I understood the material in this slide set… a) Very well b) With some review, I’ll be good c) Not really d) Not at all

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38 Quick survey The pace of the lecture for this slide set was… The pace of the lecture for this slide set was… a) Fast b) About right c) A little slow d) Too slow

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39 Quick survey How interesting was the material in this slide set? Be honest! How interesting was the material in this slide set? Be honest! a) Wow! That was SOOOOOO cool! b) Somewhat interesting c) Rather borting d) Zzzzzzzzzzz

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