ELEC1087: Discrete Mathematicswork1807/lec03-prime.pdfELEC1087: Discrete Mathematics ... illustrations from Rosen, Graham et al., and wikipedia) Integer functions ... `Remember the

ELEC1087: Discrete Mathematics

Lecture 3, 4, 5: Numbers & PrimesLecture 3, 4, 5: Numbers & Primes

Spring 2010 / Edmund Lam(based on notes by Dr Hayden So; illustrations from Rosen, Graham et al., and wikipedia)

Integer functionsInteger functions“Whole numbers constitute the backbone of discrete mathematics, and we often need to convert from fractions or arbitrary real numbers to integers.”

Graham, Knuth, Patashnik

Our roadmap:(notation) floor and ceiling(notation) division and modulusprime, and relatively primepublic-key cryptography

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Floor and Ceiling FunctionsFloor and Ceiling Functions

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Floor and CeilingFloor and Ceiling

3

22

1

01 2 3 4-4 -3 -2 -1

-11

-2

floor

ceiling-3

-4

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Properties of Floor and Ceiling 1Properties of Floor and Ceiling 1

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Properties of Floor and Ceiling 2Properties of Floor and Ceiling 2

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Floor/Ceiling Applications (1)Floor/Ceiling Applications (1)

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Floor/Ceiling Applications (2)Floor/Ceiling Applications (2)

Ans: Rosen, p. 145.

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Floor/Ceiling Applications (3)Floor/Ceiling Applications (3)

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“Real problem #1”: Josephus problemReal problem #1 : Josephus problemRemember the solution is:

1for 1)(2)2(

1)1(

≥−==

nnJnJ

J

1for 1)(2)12(

o)()(

≥+=+ nnJnJ

nnJnJ

Can also be written as:

⎣ ⎦ 1f)1()2/(2)(

1)1( =

JJ

Jn⎣ ⎦ 1for )1()2/(2)( >−−= nnJnJ n

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“Real problem #2”: SortingReal problem #2 : SortingSuppose we are sorting n>1 records

Divide into two approximately equal partsSort each part by the same method (recursively)

Merge, with at most n-1 further comparisons⎡ ⎤ ⎣ ⎦2/2/ nnn +=

Total number of comparisons is f(n), where

0)1( =f

G h l i 3 34 i h d i i f f( )

⎡ ⎤ ⎣ ⎦ 1for 1)2/()2/()( >−++= nnnfnfnf

Graham et al. exercise 3.34 gives the derivation for f(n):

⎡ ⎤nmnmnf m2log where12)( =+−=

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Worth thinkingWorth thinking…

Floor and ceiling are more than just a handy Floor and ceiling are more than just a handy notation; they help simplify the

representation and manipulation of practical problems, i.e. they provide good abstractions.

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Integer DivisionInteger Division

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Integer Functions TheoremsInteger Functions Theorems

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The Division AlgorithmThe Division Algorithm

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Modular ArithmeticModular ArithmeticIn many discrete math problems, we only care about the remainder of an integer division.Example: Assume a bus arrives at a bus stop every 10 minutes starting at 11:00am. It takes Albert 7 minutes to get to the bus stop while it takes Betty 13 minutes. Who

ll h l h b ?will have to wait longer at the bus stop?

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Modular Arithmetic NotationModular Arithmetic Notation

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Modular Arithmetic TheoremModular Arithmetic Theorem

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ExampleExample

Show that if n | m where n >1 m >1 are integersShow that if n | m, where n >1,m >1 are integers,

and if a ≡ b (modm), where a and b are integers,

then a ≡ b (modn)

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Using Congruence ArithmeticUsing Congruence ArithmeticHash FunctionChecksumCryptographyyp g p y

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HashingHashingA many-to-one mapping between a (relatively) larger set to a smaller set

e.g. memory map, database, dictionary

The mapping function is called a hashing functionA simple hash function: h(k) = k mod m

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Hashing ExampleHashing ExampleFor example, we want to sort students submitted midterm into 12 boxes according to their Chinese zodiac sign (rat ox into 12 boxes according to their Chinese zodiac sign (rat, ox, tiger, rabbit, etc)Solution: First, we label box for zodiac sign rat=0, ox=1, Solution: First, we label box for zodiac sign rat 0, ox 1, tiger=2…Then the box number (b) to hold the midterm for a student born in year (y) midterm will be calculated as:

b = (y - 2008) mod 12F l id f d b i 2000 ill b For example, midterm of a student born in year 2000 will be placed at box

b = (2000 - 2008) mod 12 = 4b (2000 2008) mod 12 4Therefore, it will be at box for year of dragon

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ChecksumChecksumUse to verify a number (code) is valid

E HKID d UPC d ISBN d dit d t E.g. HKID card, UPC code, ISBN code, credit card, passport number, etc

ISBN-10 code has 10 digits g{a9 a8 a7 … a1 a0}

Last digit is a check digit, computed against the weighted sum of previous 9 digits with weights {10 9 8 3 2} sum of previous 9 digits, with weights {10, 9, 8, … 3, 2} such that10 a9 + 9 a8 + … + 3 a2 + 2 a1 ≡ -a0 (mod 11)9 8 2 1 0 ( )In other word, the weighted sum of all digits add up as a multiple of 11

ISBN 007 124474 3e.g. ISBN 007-124474-3

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CryptographyCryptographyJulius Caesar’s Encryption Algorithm

Shift h l tt i f d i th l h b t b 3 Shift each letter in a message forward in the alphabet by 3 letters (with wrap-around)E.g “I LOVE DISCRETE MATH” becomes “L ORYH gGLVFUHWH PDWK”

Algorithm:R t h l tt b “A” = 0 “B”=1 t Represents each letter as a number “A” = 0, “B”=1, etc To encrypt a letter p, use the following function:

f (p) (p + 3) mod 26To decrypt a letter p, use the inverse function:

f (p) = (p + 3) mod 26

f −1(p) = (p − 3) mod 26

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Further thoughtsFurther thoughts

1. Modular arithmetic is “simple” to understand and compute.

2 Th ’ i h t t f th i d 2. There’s an inherent asymmetry: from the remainder we cannot deduce the original numbers.

3. Modular arithmetic turns out to be important in several pformulas involving prime numbers.

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Prime NumbersPrime NumbersPrime numbers have been studied extensively since

ancient timeAn extremely important class of numbers in modern

h i ll i h d hmath, especially with regard to cryptography

A positive integer p greater than 1 is called prime if the only positive factors of p are 1 and p

Otherwise, it is composite

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Division Algorithm ExampleDivision Algorithm ExampleExample A: 100 is divided by 7.Solution: We have 100 = 14(7) + 2.Therefore

i 100 di 7 14quotient = 100 div 7 = 14remainder = 100 mod 7 = 2

E l B 13 i di id d b 5Example B: -13 is divided by 5.Solution: We have -13 = -3 (5) + 2.Th fThereforequotient = -13 div 5 = -3remainder = 13 mod 5 = 2remainder = -13 mod 5 = 2

Note: remainder must be positive and smaller than the divisorthe divisor.

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Fundamental Theorem of ArithmeticFundamental Theorem of ArithmeticEvery positive integer greater than 1 can be written

l h d f uniquely as a prime or as the product of two or more primes where the prime factors are written in order of nondecreasing sizeof nondecreasing size.Examples:

10 = 2 × 510 = 2 × 512 = 2 × 2 × 3242 = 2 × 11 × 11242 = 2 × 11 × 11105 = 3 × 5 × 7

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Prime factorsPrime factors

Proof: [Rosen, p. 211]

There are infinite number of primes.Proof: [Rosen p 212]Proof: [Rosen, p. 212]

What is the largest known prime then?

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Mersenne PrimesMersenne PrimesMersenne primes are prime numbers of the form 2p -1 where p is prime2p -1 where p is prime

Useful to find extremely large primes that are useful for modern cryptographic applications

Much easier to verify its primality using Lucas-Lehmer test than other primesL t k i b 243112609 1Largest known prime number: 243112609-1

12,978,189 digits!316,470,269,330,255,923,143,453,723,949,337,516,054,106,188,475,264,644,14316,470,269,330,255,923,143,453,723,949,337,516,054,106,188,475,264,644,140,304,176,732,811,247,493,069,368,692,043,.. … 791,908,398,130,223,304,824,083,119,093,195,998,014,562,456,347,941,202,195,900,928,079,670,729,447,921,616,491,887,478,265,780,022,181,166,697,152,511511

http://prime.isthe.com/chongo/tech/math/prime/m43112609/prime-c.html

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Lucas Lehmer testLucas-Lehmer test

(only for odd p)

Some values:40 =s 32 =M

)121431(376342194

194214

)27(1424

2

22

21

=−=

×==−=

s

s

31

73 =

M

M

5

3=p

)121431(37634219423 ×==−=s 315 =M5=p

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Twin PrimeTwin PrimeExamples:

3 & 55 & 7…4967 & 4969…65,516,468,355×2333,333±1 (100,355 digits)

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Greatest Common DivisorGreatest Common DivisorThe greatest integer d that divides two non-zero integers

b i ll d h di i f d ba, b is called the greatest common divisor of a and bDenoted gcd(a,b)T i d b l ti l i if d( b) 1Two integers a and b are relatively prime if gcd(a,b) = 1Example

d(12 21) 3gcd(12,21) = 3gcd(15,28) = 1 [relatively prime]

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Least Common MultipleLeast Common MultipleThe least common multiple (LCM) of the integers a and bis the smallest positive integer that is divisible by both aand b.For example, lcm(12,15) = 60

For two positive integers a and b,ab = gcd(a,b) × lcm(a,b)

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The Euclidean Algorithm (1)The Euclidean Algorithm (1)To find the gcd of 2 integersBased on the following observation:Based on the following observation:

Proof:

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The Euclidean Algorithm (2)The Euclidean Algorithm (2)To find gcd(a,b):Let x := max(a,b), y := min(a,b)while y ≠ 0b ibegin

r := x mod yx := yx := yy := r

endendIn other words, divide the larger number x with the smaller one y to get remainder r. Then set y to be smaller one y to get remainder r. Then set y to be the bigger number, and r be the smaller number, and repeat the division until the remainder is 0p

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Euclidean Algorithm ExampleEuclidean Algorithm ExampleFind gcd(15,72)Step 1: 72 = 4 * 15 + 12Step 2: 15 = 1 * 12 + 3pStep 3: 12 = 4 * 3 + 0Therefore gcd(15 72) = 3 Therefore, gcd(15,72) 3

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Bringing them togetherBringing them together

P bli k t h C Public-key cryptography: Can we communicate a secret message in the open?

Yes with the discrete math we know!Yes—with the discrete math we know!

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Public key: an examplePublic-key: an exampleGreen = public Red = secret

Alice and Bob agrees prime p=23 and base g=5

Alice’s secret integer: a=6. Alice sends Bob: A=56 mod 23=8

Bob’s secret integer: b=15. Bob sends Alice: B=515 mod 23=19

Alice computes s=196 mod 23=2

Bob computes s=815 mod 23=2

Alice and Bob know 2, but Eve doesn’t!Eve

Use it as the private keythereafter

Magic?

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Alice Bob

Public key: an examplePublic-key: an exampleRequirements:

p is a primeg is a primitive root mod p (we did not discuss what this means, but you can look up the web)you can look up the web)

Calculations:A = ga mod pA g mod pB = gb mod ps = Ab mod p = gab mod p = gba mod p = Ba mod pp g p g p p

Key insight—the Discrete Logarithm Problem:Given A, g, p: difficult to calculate a. Need big prime number p (~ several hundred digits), and big a,bg needs not be big, e.g. 2 or 5

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