Euclid’s Algorithm “Backwards”
Theorem 17.1
∀a b ∈ N ∃x y ∈ Z a ∗ x + b ∗ y = gcd(a, b)
Furthermore, x and y are efficiently computable from a and b using Euclid’s Algorithm.
Example 17.2
18 = 1 ∗ 12 + 6
=
gcd(18, 12)
Express gcd(18, 12) in terms of previous values, i.e., 18 and 12.
6
=
18 ∗ 1 + 12 ∗ (−1)
Example 17.3
123
=
42
2 ∗ 42 +
= 1 ∗ 39 +
39
=
3
gcd(123, 42)
Express gcd(a, b) in terms of previous values;
regroup;
3
=
42 ∗ 1
+
39 ∗ (−1)
3
=
42 ∗ 1
+
(123 + 42 ∗ (−2)) ∗ (−1)
3
=
123 ∗ (−1) +
42 ∗ 3
1
repeat.
Example 17.4
13 = 1 ∗ 8
8 = 1∗5
5 = 1∗3
3 = 1∗2
+
+
+
+
5
3
2
1
=
gcd(13, 8)
Express gcd(a, b) in terms of previous values;
3∗1
3∗1
+
+
2 ∗ (−1)
(5 + 3 ∗ (−1)) ∗ (−1)
1 =
1 =
5 ∗ (−1)
5 ∗ (−1)
+
+
3∗2
(8 + 5 ∗ (−1)) ∗ 2
1 =
1 =
8 ∗ (2)
8 ∗ (2)
+
+
5 ∗ −3
(13 + 8 ∗ (−1)) ∗ −3
1
1
1
=
=
=
13 ∗ (−3) +
regroup;
8∗5
Don’t forget to check: 1
=
−39 + 40
2
repeat.
Multipicative Inverses Mod m
Corollary 17.5
Proof:
If gcd(a, b) = 1 then we can efficiently compute the multiplicative inverse of a (mod b).
Compute x, y, s.t.
a ∗ x + b ∗ y = gcd(a, b) = 1.
x is the multiplicative inverse of a (mod b).
a∗x
Corollary 17.6
1−b∗y
=
≡
1 (mod b)
For all integers a, m, s.t. m > 1,
a has a multiplicative inverse mod m
Proof:
a and m are relatively prime.
iff
Recall that a and m are relatively prime iff gcd(a, m) = 1.
Let a and m be arbitrary with m > 1.
Assume: gcd(a, m) = 1
Then by Cor. 17.5,
a has a multiplicative inverse mod m.
Assume: a has a multiplicative inverse mod m, and call it a−1
m .
Thus, a ∗ a−1
m ≡ 1 (mod m).
Thus, a ∗ a−1
m +k∗m = 1
Let
d = gcd(a, m).
Thus d|1.
for some
Thus, d|a
and
k ∈ Z.
d|m.
Therefore, d = 1 = gcd(a, m).
3
∀a, m > 1 (a has mult. inverse mod m iff gcd(a, m) = 1)
× mod 6
0
1
2
3
4
5
0
0
0
0
0
0
0
1
0
1
2
3
4
5
4
2
0
2
4
0
2
4
3
0
3
0
3
0
3
4
0
4
2
0
4
2
5
0
5
4
3
2
1
∀a, m > 1 (a has mult. inverse mod m iff gcd(a, m) = 1)
Definition 17.7
Let the multiplicative group mod m be Z
?
U Zm
a 0 < a < m ∧ gcd(a, m) = 1
=
?
U Zm is the set of natural numbers less than m that are relatively prime to m, with operation multiplication mod
m.
Z?6
1
5
?
6
U Z = {1, 5}
Z?5
1
2
3
4
?
U Z5 = {1, 2, 3, 4}
5
1
1
5
1
1
2
3
4
5
5
1
2
2
4
1
3
3
3
1
4
2
4
4
3
2
1
Definition 17.8 A group is a world of vocabulary Σgroup = (; e, ∗2 [infix],−1 1 [postfix]) that satisfies Γgroup .
Γgroup = ∀x (x ∗ e = x ∧ e ∗ x = x)
∧
∀x (x ∗ x−1 = e ∧ x−1 ∗ x = e)
∧
∀x y z x ∗ (y ∗ z) = (x ∗ y) ∗ z
A group is a set with a an associative operation, an identity and inverses.
Theorem 17.9 For all m > 1,
∗
Zm
is a group.
For m > 1, define
Definition 17.10 (Euler’s phi function)
ϕ(m)
Example 17.11 ϕ(6) = 2;
Proposition 17.12
=
∗
|Zm
|
=
number of numbers less than m and relatively prime to m
ϕ(5) = 4.
For all p,
if p is prime,
then ϕ(p) = p − 1.
6