Introduction
Modular Arithmetic
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Modular Arithmetic II
332
Previous Lecture
Relatively prime, Euler’s totient function
Congruences
Residues
Residues and arithmetic operations
Discrete Mathematics – Modular Arithmetic II
333
Residues
(cntd)
The
residue
of an integer
a
modulo
m
is such a number
b
that
a
≡
b (mod m)
and
0
≤
b < m
In other words the residue of
a
modulo
m
is the remainder of
a
when divided by
m
Let
denote the set
{0,1,2,…,n – 1}.
This is the set of all
possible remainders of integers when divided by
n
It is called the
set of residues
,
and its members are called
residues
n
Z
Discrete Mathematics – Modular Arithmetic II
334
Modular Arithmetic
We define addition, subtraction, and multiplication of residues:
Let
a,b
∈
.
Then
a + b (mod n)
is the element
c
∈
such that
c
≡
a + b (mod n)
a – b (mod n)
is the element
c
∈
such that
c
≡
a – b (mod n)
a
⋅
b (mod n)
is the element
c
∈
such that
c
≡
a
⋅
b (mod n)
Example.
Construct operation tables for
n
Z
n
Z
n
Z
n
Z
5
Z
+
0
1
2
3
4
0
1
2
3
4
⋅
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
1
2
3
4
0
2
3
4
0
1
3
4
0
1
2
4
0
1
2
3
0
0
0
0
0
0
1
2
3
4
0
2
4
1
3
0
3
1
4
2
0
4
3
2
1
Discrete Mathematics – Modular Arithmetic II
335
Applications:
Criptography
One of the oldest cryptosystems is the Caesar cipher.
He made messages secret by shifting each letter
three letters forward.
Thus
B
becomes
E,
and
X
is sent to
A
To express this process mathematically we first
replace letters by integers from
0
to
25. For example,
A
is replaced by
0,
K
by 10.
Next, to encrypt a message we add 3 modulo 25 to every letter.
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 Spring '10
 AndreiBulatov
 Number Theory, Division, Prime number, Modular Arithmetic II

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