Questions/Complaints About
Homework?
Here’s the procedure for homework questions/complaints:
1. Read the solutions Frst.
2. Talk to the person who graded it (check initials)
3. If (1) and (2) don’t work, talk to me.
±urther comments:
•
There’s no statute of limitations on grade changes
◦
although asking questions right away is a good
strategy
•
Remember that 10/12 homeworks count.
Each one
is roughly worth 50 points, and homework is 35% of
your Fnal grade.
◦
16 homework points = 1% on your Fnal grade
•
Remember we’re grading about 100 homeworks and
graders are not expected to be mind readers.
It’s
your
problem to write clearly.
•
Don’t forget to staple your homework pages together,
add the cover sheet, and put your name on clearly.
◦
I’ll deduct 2 points if that’s not the case
1
Algorithmic number theory
Number theory used to be viewed as the purest branch
of pure mathematics.
•
Now it’s the basis for most modern cryptography.
•
Absolutely critical for ecommerce
◦
How do you know your credit card number is safe?
Goal:
•
To give you a basic understanding of the mathematics
behind the RSA cryptosystem
◦
Need to understand how prime numbers work
2
Division
±or
a, b
∈
Z
,
a
6
= 0,
a
divides
b
if there is some
c
∈
Z
such that
b
=
ac
.
•
Notation:
a

b
•
Examples: 3

9, 3
6 
7
If
a

b
, then
a
is a
factor
of
b
,
b
is a
multiple
of
a
.
Theorem 1:
If
a, b, c
∈
Z
, then
1. if
a

b
and
a

c
then
a

(
b
+
c
).
2. If
a

b
then
a

(
bc
)
3. If
a

b
and
b

c
then
a

c
(divisibility is transitive).
Proof:
How do you prove this? Use the deFnition!
•
E.g., if
a

b
and
a

c
, then, for some
d
1
and
d
2
,
b
=
ad
1
and
c
=
ad
2
.
•
That means
b
+
c
=
a
(
d
1
+
d
2
)
•
So
a

(
b
+
c
).
Other parts: homework.
Corollary 1:
If
a

b
and
a

c
, then
a

(
mb
+
nc
) for
any integers
m
and
n
.
3
The division algorithm
Theorem 2:
±or
a
∈
Z
and
d
∈
N
,
d >
0, there exist
unique
q, r
∈
Z
such that
a
=
q
·
d
+
r
and 0
≤
r < d
.
•
r
is the remainder when
a
is divided by
d
Notation:
r
≡
a
(mod
d
);
a
mod
d
=
r
Examples:
•
Dividing 101 by 11 gives a quotient of 9 and a remain
der of 2 (101
≡
2 (mod 11); 101 mod 11 = 2).
•
Dividing 18 by 6 gives a quotient of 3 and a remainder
of 0 (18
≡
0 (mod 6); 18 mod 6 = 0).
Proof:
Let
q
=
b
a/d
c
and deFne
r
=
a

q
·
d
.
•
So
a
=
q
·
d
+
r
with
q
∈
Z
and 0
≤
r < d
(since
q
·
d
≤
a
).
But why are
q
and
d
unique?
•
Suppose
q
·
d
+
r
=
q
0
·
d
+
r
0
with
q
0
, r
0
∈
Z
and
0
≤
r
0
< d
.
•
Then (
q
0

q
)
d
= (
r

r
0
) with

d < r

r
0
< d
.
•
The lhs is divisible by
d
so
r
=
r
0
and we’re done.
4
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View Full DocumentPrimes
•
If
p
∈
N
,
p >
1 is
prime
if its only positive factors
are 1 and
p
.
•
n
∈
N
is
composite
if
n >
1 and
n
is not prime.
◦
If
n
is composite then
a

n
for some
a
∈
N
with
1
< a < n
◦
Can assume that
a
≤
√
n
.
*
Proof:
By contradiction:
Suppose
n
=
bc
,
b >
√
n
,
c >
√
n
.
But then
bc > n
, a contradiction.
Primes: 2
,
3
,
5
,
7
,
11
,
13
, . . .
This is the end of the preview.
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 Spring '07
 SELMAN
 Number Theory, Prime number, gcd, denom

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