Cryptography Reference
In-Depth Information
2
7
= 128 (i.e., it uses only 7 bits of each byte). There is also an extended ASCII
character set with 2
8
= 256 characters or symbols (i.e., it uses all 8 bits of each
byte). For the purpose of this topic, however, we don't distinguish between the two
ASCII character sets.
Ta b l e 6 . 1
ASCII Character Set with Hexadecimal Values
0x00
0x10
0x20
0x30
0x40
0x50
0x60
0x70
+0 NUL
DLE
0
@
P
'
p
+1 SOH
DC1
!
1
A
Q
a
q
+2 STX
DC2
"
2
B
R
b
r
+3 ETX
DC3
#
3
C
S
c
s
+4 EOT
DC4
$
4
D
T
d
t
+5 ENQ
NAK
%
5
E
U
e
u
+6 ACK
SYN
&
6
F
V
f
v
+7 BEL
ETB
'
7
G
W
g
w
+8 BS
CAN
(
8
H
X
h
x
+9 HT
EM
)
9
I
Y
i
y
+A LF
SUB
:
J
Z
j
z
*
+B VT
ESC
+
;
K
[
k
{
+C FF
FS
,
<
L
\
l
|
+D CR
GS
-
=
M
]
m
}
+E SO
RS
.
>
N
ˆ
n
˜
?
+F SI
US
/
O
_
o
DEL
Instead of directly using letters or ASCII characters, computer systems nor-
mally operate on
binary digits
(or
bits
). Consequently, the alphabet most frequently
used in computer science is Σ=
=2.
If an alphabet Σ is finite (which is almost always the case), then its length is
less than infinity (i.e.,
{
0
,
1
}
and its length is
|{
0
,
1
}|
). In this case, the
n
elements of Σ can also
be associated with the
n
elements (residue classes) of
|
Σ
|
=
n<
∞
Z
n
=
{
0
,
1
,...,n
−
1
}
.
Consequently, it is possible to work in
Z
n
instead of any character set with
n
elements. This simplifies things considerably. In particular, it allows us to work with
mathematical structures we know and with which we are familiar.
Let Σ be an alphabet. The term
word
(or
string
) over Σ refers to a finite
sequence of characters or symbols from Σ, including, for example, the empty word
ε
. The length of a word
w
over Σ, denoted as
|
w
|
, corresponds to the number of
characters. The empty word has length zero (i.e.,
=0). The set of all words over
Σ (again, including the empty word) is referred to as Σ
∗
.Forevery
n
|
ε
|
, Σ
n
∈
N
n
refers to the set of all words of length
n
over Σ. For example,
{
0
,
1
}
denotes the
}
∗
denotes the set of all binary words. This can
set of all
n
-bit sequences, and
{
0
,
1
be formally expressed as follows:
