ELEC1087: Discrete Mathematics
Lecture 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 functions
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“Whole numbers constitute the backbone of discrete
mathematics, and we often need to convert from
fractions or arbitrary real numbers to integers.”
ÚGraham, Knuth, Patashnik
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Our roadmap:
(notation) floor and ceiling
(notation) division and modulus
prime, and relatively prime
public-key cryptography
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2
Floor and Ceiling Functions
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3
Floor and Ceiling
3
2
1
-4
-3
-2
-1
0
1
2
3
4
1
-1
-2
-3
-4
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4
ceiling
floor
Properties of Floor and Ceiling 1
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5
Properties of Floor and Ceiling 2
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6
Floor/Ceiling Applications (1)
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7
Floor/Ceiling Applications (2)
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Ans: Rosen, p. 145.
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8
Floor/Ceiling Applications (3)
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9
“Real
Real problem #1
#1”:: Josephus problem
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Remember the solution is:
J (1) = 1
J (2n) = 2 J (n) − 1 for
o n ≥1
J (2n + 1) = 2 J (n) + 1 for n ≥ 1
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Can also be written as:
J (1) = 1
J (n) = 2 J ( ⎣n / 2⎦) − (−1) n for
f n >1
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10
“Real
Real problem #2
#2”:: Sorting
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Suppose we are sorting n>1 records
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`
`
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Divide into two approximately equal parts
Sort each part by the same method (recursively)
n = ⎡n / 2⎤ + ⎣n / 2⎦
Merge, with at most n-1 further comparisons
Total number of comparisons is f(n), where
f (1) = 0
f (n) = f ( ⎡n / 2⎤) + f ( ⎣n / 2⎦) + n − 1 for n > 1
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G h et al.l exercise
Graham
i 33.34
34 gives
i
the
h d
derivation
i i ffor f(
f(n):
)
f (n) = nm − 2 m + 1 where m = ⎡log 2 n ⎤
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11
Worth thinking…
thinking
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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12
Integer Division
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13
Integer Functions Theorems
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14
The Division Algorithm
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15
Modular Arithmetic
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In 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
willll hhave to wait llonger at the
h bus
b stop??
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16
Modular Arithmetic Notation
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17
Modular Arithmetic Theorem
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18
Example
Show that if n | m,
m where n > 1,m
1 m > 1 are integers,
integers
and if a ≡ b (mod m), where a and b are integers,
then a ≡ b (mod n)
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19
Using Congruence Arithmetic
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Hash Function
Checksum
Cryptography
yp g p y
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20
Hashing
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A many-to-one mapping between a (relatively) larger set
to a smaller set
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`
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e.g. memory map, database, dictionary
The mapping function is called a hashing function
A simple hash function: h(k) = k mod m
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21
Hashing Example
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`
`
`
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For example, we want to sort students submitted midterm
into 12 boxes according to their Chinese zodiac sign (rat,
(rat ox
ox,
tiger, rabbit, etc)
Solution: First, we label box for zodiac sign rat
rat=0,
0, ox
ox=1,
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 12
F example,
For
l midterm
id
off a student
d
bborn iin year 2000 will
ill bbe
placed at box
b = (2000 - 2008) mod 12 = 4
Therefore, it will be at box for year of dragon
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22
Checksum
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Use to verify a number (code) is valid
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EE.g. HKID card,
d UPC code,
d ISBN code,
d credit
dit card,
d passportt
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,
digits with weights {10
{10, 9,
9 8,
8 … 3,
3 2}
such that
10 a9 + 9 a8 + … + 3 a2 + 2 a1 ≡ -a0 ((mod 11))
In other word, the weighted sum of all digits add up as a
multiple of 11
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PAGE
e.g. ISBN 007-124474-3
007 124474 3
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Cryptography
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Julius Caesar’s Encryption Algorithm
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`
`
Shift each
h lletter
tt iin a message fforward
d iin th
the alphabet
l h b t bby 3
letters (with wrap-around)
E.gg “I LOVE DISCRETE MATH” becomes “L ORYH
GLVFUHWH PDWK”
Algorithm:
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`
Represents
R
t eachh lletter
tt as a number
b “A” = 0,
0 “B”=1,
“B”=1 etc
t
To encrypt a letter p, use the following function:
f ( p) = ( p + 3) mod 26
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To decrypt a letter p, use the inverse function:
f −1 ( p) = ( p − 3) mod 26
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24
Further thoughts
1. Modular arithmetic is “simple” to understand and compute.
2 Th
2.
There’s
’ an inherent
i h
t asymmetry:
t
from
f
the
th remainder
i d we
cannot deduce the original numbers.
p
in several
3. Modular arithmetic turns out to be important
formulas involving prime numbers.
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25
Prime Numbers
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`
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Prime numbers have been studied extensively since
ancient time
An extremely important class of numbers in modern
math,
h especially
i ll with
i h regard
d to cryptography
h
A positive integer p greater than 1 is called prime if
the only positive factors of p are 1 and p
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PAGE
Otherwise, it is composite
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Division Algorithm Example
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`
`
`
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Example A: 100 is divided by 7.
Solution: We have 100 = 14(7) + 2.
Therefore
quotient
i
= 100 di
div 7 = 14
remainder = 100 mod 7 = 2
E
Example
l B
B: -13
13 iis di
divided
id d by
b 5.
5
Solution: We have -13 = -3 (5) + 2.
Th f
Therefore
quotient = -13 div 5 = -3
remainder = -13
13 mod 5 = 2
Note: remainder must be positive and smaller than
the divisor.
divisor
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27
Fundamental Theorem of Arithmetic
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`
Every positive integer greater than 1 can be written
uniquely
l as a prime or as the
h product
d
off two or more
primes where the prime factors are written in order
of nondecreasing size.
size
Examples:
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`
`
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PAGE
10 = 2 × 5
12 = 2 × 2 × 3
242 = 2 × 11 × 11
105 = 3 × 5 × 7
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Prime factors
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`
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There are infinite number of primes.
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Proof: [Rosen, p. 211]
Proof: [Rosen,
[Rosen p.
p 212]
What is the largest known prime then?
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29
Mersenne Primes
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Mersenne primes are prime numbers of the form
2p -1 where p is prime
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`
`
`
`
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30
Useful to find extremely large primes that are useful for
modern cryptographic applications
Much easier to verify its primality using LucasLehmer test than other primes
L
Largest
t kknown prime
i
number:
b 243112609-1
1
` 12,978,189 digits!
316,470,269,330,255,923,143,453,723,949,337,516,054,106,188,475,264,644,14
0,304,176,732,811,247,493,069,368,692,043,.. …
791,908,398,130,223,304,824,083,119,093,195,998,014,562,456,347,941,202,19
5,900,928,079,670,729,447,921,616,491,887,478,265,780,022,181,166,697,152,
511
http://prime.isthe.com/chongo/tech/math/prime/m4311260
9/prime-c.html
Lucas Lehmer test
Lucas-Lehmer
(only for odd p)
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Some values:
s0 = 4
p=3
s1 = 4 − 2 = 14(= 7 × 2)
2
M2 = 3
M3 = 7
s2 = 14 − 2 = 194
2
p=5
PAGE
s3 = 194 2 − 2 = 37634(= 31× 1214)
31
Twin Prime
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Examples:
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`
`
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PAGE
3&5
5&7
…
4967 & 4969
…
65,516,468,355×2333,333±1 (100,355 digits)
32
M 5 = 31
Greatest Common Divisor
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`
`
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The greatest integer d that divides two non-zero integers
a, b is
i called
ll d the
h greatest common di
divisor
i
off a and
db
Denoted gcd(a,b)
T integers
Two
i
a and
d b are relatively
l ti l prime
i
if gcd(a,b)
d( b) = 1
Example
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`
PAGE
gcd(12,21)
d(12 21) = 3
gcd(15,28) = 1 [relatively prime]
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Least Common Multiple
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`
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The least common multiple (LCM) of the integers a and b
is the smallest positive integer that is divisible by both a
and 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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34
The Euclidean Algorithm (1)
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To find the gcd of 2 integers
Based on the following observation:
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Proof:
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35
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The Euclidean Algorithm (2)
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`
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To find gcd(a,b):
Let x := max(a,b), y := min(a,b)
while y ≠ 0
b i
begin
r := x mod y
x := y
y := r
end
In other words, divide the larger number x with the
smaller one y to get remainder r. Then set y to be
the bigger number, and r be the smaller number, and
repeat
p
the division until the remainder is 0
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36
Euclidean Algorithm Example
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`
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Find gcd(15,72)
Step 1: 72 = 4 * 15 + 12
Stepp 2: 15 = 1 * 12 + 3
Step 3: 12 = 4 * 3 + 0
Therefore gcd(15,72)
Therefore,
gcd(15 72) = 3
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37
Bringing them together
Public-key
P
bli k cryptography:
t
h Can
C
we
communicate a secret message in the open?
Yes with the discrete math we know!
Yes—with
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38
Public key: an example
Public-key:
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Green = 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
Eve
Alice and Bob know 2, but Eve doesn’t!
Use it as the private key
thereafter
Magic?
Alice
PAGE
Bob
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Public key: an example
Public-key:
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Requirements:
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Calculations:
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p is a prime
g is a primitive root mod p (we did not discuss what this means, but
you can look up the web)
A = ga mod p
B = gb mod p
s = Ab mod p = gab mod p = gba mod p = Ba mod p
Key insight—the Discrete Logarithm Problem:
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PAGE
Given A, g, p: difficult to calculate a.
Need big prime number p (~ several hundred digits), and big a,b
g needs not be big, e.g. 2 or 5
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