11 integers and division
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Integers and DivisionCS/APMA 202Rosen section 2.4Aaron Bloomfield
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Rosen, chapter 2We are only doing 2 or 3 of the sections in chapter 22.4: integers and division2.6: applications of number theoryAnd only parts of that section2.7: matrices (maybe)
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Quick surveyHave you seen matrices before?Lots and lots and lotsA fair amountJust a littleIs that kinda like the movie?
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Why prime numbers?Prime numbers are not well understood
Basis for todays cryptography
Unless otherwise indicated, we are only talking about positive integers for this chapter
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The divides operatorNew notation: 3 | 12To specify when an integer evenly divides another integerRead as 3 divides 12
The not-divides operator: 5 | 12To specify when an integer does not evenly divide another integerRead as 5 does not divide 12
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Theorem on the divides operatorIf a | b and a | c, then a | (b+c)Example: if 5 | 25 and 5 | 30, then 5 | (25+30)
If a | b, then a | bc for all integers cExample: if 5 | 25, then 5 | 25*c for all ints c
If a | b and b | c, then a | cExample: if 5 | 25 and 25 | 100, then 5 | 100
The book calls this Theorem 1
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Prime numbersA positive integer p is prime if the only positive factors of p are 1 and pIf there are other factors, it is compositeNote that 1 is not prime!Its not composite either its 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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Fundamental theorem of arithmeticEvery 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
Examples100 = 2 * 2 * 5 * 5182 = 2 * 7 * 1329820 = 2 * 2 * 3 * 5 * 7 * 71
The book calls this Theorem 2
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Composite factorsIf n is a composite integer, then n has a prime divisor less than or equal to the square root of n
Direct proofSince n is composite, it has a factor a such that 1 n)Thus, n has a divisor not exceeding nThis divisor is either prime or a compositeIf the latter, then it has a prime factorIn either case, n has a prime factor less than n
The book calls this Theorem 3
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Showing a number is primeShow that 113 is prime (Rosen, question 8c, 2.4)
SolutionThe only prime factors less than 113 = 10.63 are 2, 3, 5, and 7Neither of these divide 113 evenlyThus, by the fundamental theorem of arithmetic, 113 must be prime
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Showing a number is compositeShow that 899 is prime (Rosen, question 8c, 2.4)
SolutionDivide 899 by successively larger primes, starting with 2We find that 29 and 31 divide 899
On a unix system, enter factor 899aaron@orion:~.16> factor 899899: 29 31
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Primes are infiniteTheorem (by Euclid): There are infinitely many prime numbersThe book calls this Theorem 4
Proof by contradictionAssume there are a finite number of primesList them as follows: p1, p2 , pn.Consider the number q = p1p2 pn + 1This number is not divisible by any of the listed primesIf we divided pi into q, there would result a remainder of 1We must conclude that q is a prime number, not among the primes listed aboveThis contradicts our assumption that all primes are in the list p1, p2 , pn.
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End of lecture on 17 Febrary 2005
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Mersenne numbersMersenne nubmer: any number of the form 2n-1Mersenne prime: any prime of the form 2p-1, where p is also a primeExample: 25-1 = 31 is a Mersenne primeExample: 211-1 = 2047 is not a prime (23*89)Largest Mersenne prime: 224,036,583-1, which has 7,235,733 digitsIf M is a Mersenne prime, then M(M+1)/2 is a perfect numberA perfect number equals the sum of its divisorsExample: 23-1 = 7 is a Mersenne prime, thus 7*8/2 = 28 is a perfect number28 = 1+2+4+7+14Example: 25-1 = 31 is a Merenne prime, thus 31*32/2 = 496 is a perfect number
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Merenne primesReference for Mersenne primes:http://mathworld.wolfram.com/MersennePrime.htmlFinding Mersenne primesGIMPS Great Internet Mersenne Prime Searchhttp://www.mersenne.org/prime.htmA new one was just discovered (last week): http://mathworld.wolfram.com/news/2005-02-18/mersenne/This is only the 42nd such prime discovered
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The prime number theoremThe radio of the number of primes not exceeding x and x/ln(x) approaches 1 as x grows without boundRephrased: 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)
Consider 200 digit prime numbersln (10200) 460The chance of a 200 digit number being prime is 1/460If we only choose odd numbers, the chance is 2/460 = 1/230This result will be used in the next lecture!
The book calls this Theorem 5
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Showing a number is prime or notConsider showing that 2650-1 is primeThat number has about 200 digitsThere are approximately 10193 prime numbers less than 2650-1By theorem 5 (x/ln(x), where x = 2650-1)How long would that take to test each of those prime numbers?Assume a computer can do 1 billion (109) per secondIt would take 10193/109 = 10184 secondsThats 3.2 * 10176 years!There are quicker methods to show a number is prime, but not to find the factors if the number is found to be compositeWe will use this in the next lecture
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The division algorithmLet 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 dr = a mod d
The book calls this Theorem 6
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Greatest common divisorThe greatest common divisor of two integers a and b is the largest integer d such that d | a and d | bDenoted by gcd(a,b)
Examplesgcd (24, 36) = 12gcd (17, 22) = 1gcd (100, 17) = 1
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Relative primesTwo numbers are relatively prime if they dont have any common factors (other than 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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Pairwise relative primeA set of integers a1, a2, an are pairwise relatively prime if, for all pairs of numbers, they are relatively primeFormally: The integers a1, a2, an are pairwise relatively prime if gcd(ai, aj) = 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) = 1Thus, they are pairwise relatively primeExample: are 10, 19, and 24 pairwise relatively prime?Since gcd(10,24) 1, they are not
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More on gcdsGiven two numbers a and b, rewrite them as:Example: gcd (120, 500)120 = 23*3*5 = 23*31*51500 = 22*53 = 22*30*53Then compute the gcd by the following formula:Example: gcd(120,500) = 2min(3,2)3min(1,0)5min(1,3) = 223051 = 20
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Least common multipleThe 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) Example: lcm(10, 25) = 50What is lcm (95256, 432)?95256 = 233572, 432=2433lcm (233572, 2433) = 2max(3,4)3max(5,3)7max(2,0) = 243572 = 190512
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lcm and gcd theoremLet a and b be positive integers. Then a*b = gcd(a,b) * lcm (a, b)
Example: gcd (10,25) = 5, lcm (10,25) = 5010*25 = 5*50
Example: gcd (95256, 432) = 216, lcm (95256, 432) = 19051295256*432 = 216*190512
The book calls this Theorem 7
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Modular arithmeticIf a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a-bNotation: a b (mod m)Rephrased: m | a-bRephrased: a mod m = bIf they are not congruent: a b (mod m)
Example: Is 17 congruent to 5 modulo 6?Rephrased: 17 5 (mod 6)As 6 divides 17-5, they are congruentExample: Is 24 congruent to 14 modulo 6?Rephrased: 24 14 (mod 6)As 6 does not divide 24-14 = 10, they are not congruent
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More on congruenceLet 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 mThe book calls this Theorem 8Example: Is 17 congruent to 5 modulo 6?Rephrased: does 17 5 (mod 6)?17 mod 6 = 5 mod 6Example: Is 24 congruent to 14 modulo 6?Rephrased: 24 14 (mod 6)24 mod 6 14 mod 6
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Even more on congruenceLet 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 + kmThe book calls this Theorem 9
Example: 17 and 5 are congruent modulo 617 = 5 + 2*65 = 17 -2*6
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Even even more on congruenceLet 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
ExampleWe 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 77 2 (mod 5)
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Uses of congruencesHashing functions
aaron@orion:~/ISOs/dvd.39> md5sum debian-31-i386-binary.iso 96c8bba5a784c2f48137c22e99cd5491 debian-31-i386-binary.iso
md5 (file) = mod 2128Not really this is a simplification
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Todays demotivators
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Pseudorandom numbersComputers cannot generate truly random numbers!
Algorithm for random numbers: choose 4 integersSeed x0: starting valueModulus m: maximum possible valueMultiplier a: such that 2 a < m Increment c: between 0 and m
Formula: xn+1 = (axn + c) mod m
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Pseudorandom numbersFormula: xn+1 = (axn + c) mod mLet x0 = 3, m = 9, a = 7, and c = 4
x1 = 7x0+4 = 7*3+4 = 25 mod 9 = 7x2 = 7x1+4 = 7*7+4 = 53 mod 9 = 8x3 = 7x2+4 = 7*8+4 = 60 mod 9 = 6x4 = 7x3+4 = 7*6+4 = 46 mod 9 = 1x5 = 7x4+4 = 7*1+4 = 46 mod 9 = 2x6 = 7x5+4 = 7*2+4 = 46 mod 9 = 0x7 = 7x6+4 = 7*0+4 = 46 mod 9 = 4x8 = 7x7+4 = 7*4+4 = 46 mod 9 = 5
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Pseudorandom numbersFormula: xn+1 = (axn + c) mod mLet x0 = 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, 3Note that it repeats!But it selects all the possible numbers before doing so
The common algorithms today use m = 232-1You have to choose 4 billion numbers before it repeats
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The Caesar cipherJulius Caesar used this to encrypt messages
A function f to encrypt a letter is defined as: f(p) = (p+3) mod 26Where 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 cipherYou are substituting one letter with another
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The Caesar cipherEncrypt go cavaliersTranslate to numbers: g = 6, o = 14, etc.Full sequence: 6, 14, 2, 0, 21, 0, 11, 8, 4, 17, 18Apply the cipher to each number: f(6) = 9, f(14) = 17, etc.Full sequence: 9, 17, 5, 3, 24, 3, 14, 11, 7, 20, 21Convert the numbers back to letters 9 = j, 17 = r, etc.Full sequence: jr wfdydolhuv
Decrypt jr wfdydolhuvTranslate to numbers: j = 9, r = 17, etc. Full sequence: 9, 17, 5, 3, 24, 3, 14, 11, 7, 20, 21Apply 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, 18Convert the numbers back to letters 6 = g, 14 = 0, etc. Full sequence: go cavaliers
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Rot13 encodingA Caesar cipher, but translates letters by 13 instead of 3Then, apply the same function to decrypt it, as 13+13=26Rot13 stands for rotate by 13
Example:aaron@orion:~.4> echo Hello World | rot13Uryyb Jbeyqaaron@orion:~.5> echo Uryyb Jbeyq | rot13Hello Worldaaron@orion:~.6>
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Quick surveyI felt I understood the material in this slide setVery wellWith some review, Ill be goodNot reallyNot at all
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Quick surveyThe pace of the lecture for this slide set wasFastAbout rightA little slowToo slow
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Quick surveyHow interesting was the material in this slide set? Be honest!Wow! That was SOOOOOO cool!Somewhat interestingRather bortingZzzzzzzzzzz