Transcript THE GROUP CONCEPT IN ARITHMETIC

PRODUCTS OF GROUPS
If F and H are groups then their product
F x H is the group defined as follows:
F  H  {( x, y) : x  F , y  H }
( x, y)  (a, b)  ( x  a, y  b)

(0,0) (1,0) (0,1) (1,1)
Example
If F = H (0,0) (0,0) (1,0) (0,1) (1,1)
= {0,1} (1,0) (1,0) (0,0) (1,1) (0,1)
with + (0,1) (0,1) (1,1) (0,0) (1,0)
mod 2
(1,1) (1,1) (0,1) (1,0) (0,0)
PERMUTATIONS
A permutation of a set S is a function
f : S --> S that is one-to-one and onto
The set of all permutation of a set S forms
a group, denoted by P(S), under the binary
operation, called composition, defined by
f  h ( x)  f (h( x)), x  S
EXAMPLES
P({a})={(a>a)} has one element
P({a,b})={(a>a, b>b), (a>b,b>a)} has
two elements
P({a,b,c})={(a>a,b>b,c>c), (a>b,b>c,c>a),
(a>c,b>a,c>b), (a>a,b>c,c>b),
(a>c,b>b,c>a), (a>b,b>a,c>c)}
has six elements
P({1,…,N}) has N! (N factorial) elements
EXAMPLES
The set of rigid transformations that
map a geometric object into itself form
a group under composition
P({a,b,c}) describes the group of rigid
transformations of an isosceles triangle
P({a,b,c,d}) does not describe the group
of rigid transformations of a square
DEFINITIONS
A group is abelian if r  s  s  r
The groups we constructed from numbers
are abelian, the permutation and geometric
transformation groups are generally not
The order of a group is a positive number
or infinity that counts its elements
An element r in a finite group
k
generates a cyclic subgroup {r : k  1}
MORE DEFINITIONS
Two groups G and H are isomorphic if
there exists a function f: G-->H that is
one-to-one and onto and satisfies
f ( x)  f ( y)  f ( x  y) x, y  G
 mod3 1 2
 mod2 0 1
1
1 2
0
0 1
2
2 1
1
1 0
Here the isomorphism f = (0>1,1>2)
INTERESTING EXAMPLES
Chinese Remainder Theorem:
and Z 2  Z3 with isomorphism
0>(0,0), 1>(1,1),
2>(0,2),
3>(1,0), 4>(0,1),
Theorem (Fermat):
5>(1,2)
*
Z5 and Z4 with isomorphism
Z6
2>1, 4>2, 3>3,
1>4
SUBGROUPS
H is a subgroup of a group G if it is a
group under the binary operation on G
Z 4 has 3 subgroups {0}, {0,2}, Z 4
Z 2  Z 2 has 5 subgroups {(0,0)},
{(0,0),{0,1}}, {(0,0}, (0,1)},
{(0,0),(1,0)}, Z 2  Z 2
A coset of H in G is a subset of the form
r  H  {r  x : x  H } for somer  G
LAGRANGE’s THEOREM
The order of a finite group is a multiple
of the order of any subgroup
Proof. This follows from three facts:
1. G = union of all the cosets of H in G
2. every coset has the same # elements
3. distinct cosets are disjoint
To prove 3 assume that x  u  H  v  H
 there exist a, b  H with u  a  v  b
1
 v  u  c where c  a  b  H
 v  H  u  {c  H }  u  H
CONSEQUENCES
If G is a finite group with order m then
m
x  I , x G
If p and q are prime numbers then
   ( p 1)(q 1)
oZ
L
 x  x,
*
pq
*
x  Zpq , L  (p  1)(q  1)  1
If p = 3 mod 5 and q = 3 mod 5
then L=5K for some integer K
PUBLIC KEY ENCRYPTION
Rivest, Shamir, Adelman (~ to 1978 alg.)
1. You generate huge primes p =3, q=3
mod 5 (by Dirichlet’s Theorem) then
distribute N=pq to the public and keep
K = ((p-1)(q-1)+1)/5 secret
2. Mr Public generates message x  Z
then computes & publicly sends encrypted
5
message y  x modN
K
3. You decrypt message x  y mod N
*
N
PUBLIC KEY ENCRYPTION
You are a private encrypter and hold the
private key K
Mr Public can in theory compute K from
the public key N but it will require
factorizing N=pq, presumably intractible
This algorithm revolutionized secret
communications and in particular
enabled e-commerce
MIND MENDING EXERCISES
Problem 5. Prove all previous assertions
Problem 6. Find all subgroups of the
group Z of integers and all the cosets of
each subgroup
Problem 7. Find all subgroups and
associated subgroups of S({a,b,c})
Problem 8. Develop a tractible method to
K
compute x  y mod N