Cryptography Reference
In-Depth Information
As an example with
a
=
2,
b
=
3 and
c
=
4:
(2
3
)
4
8
4
4096
=
2
12
2
3
×
4
=
=
=
.
We will also use the fact that if we raise
a
to the power
b
, and then raise the result
to the power
c
, then this is the same as raising
a
to the power
c
and then raising
the result to the power
b
. In other words:
(
a
b
)
c
(
a
c
)
b
=
.
Using our previous example:
(2
3
)
4
(2
4
)
3
=
4096
=
.
CONCATENATION
At various points in our coverage of cryptographic mechanisms we will need a
mathematical notation to mean the very simple process of writing two pieces
of data (or numbers) 'next to one another'. We say that the data (or number)
x
is
concatenated
to the data
y
, and write this as
x
y
consists of
x
(written on the left) next to
y
(written on the right). For example, the
concatenation of
x
=
1101 and
y
=
11100011 is:
||
y
. In other words,
x
||
x
||
y
=
110111100011
.
1.6.2
Key lengths and keyspaces
Before proceeding further, it is important to understand various concepts relating
to the number of possible different decryption keys in a cryptosystem, which we
refer to as the
size
of the keyspace. This is important because one strategy for an
attacker of a cryptosystem is to try to determine the decryption key, hence the
size of the keyspace is certainly something that the attacker will be interested in.
The majority of cryptosystems have a fixed size of keyspace. However, it is worth
noting that:
• Some cryptosystems can provide a choice of size of keyspace. For example,
the encryption algorithm AES can be used in three different 'settings', each of
which has a different size of keyspace (see Section 4.5). While a cryptosystem
using AES may select just one of these 'settings', it is also possible that it could
support more than one.
• For some cryptosystems the size of the keyspace is highly flexible. For
example, both the Vigenère Cipher (see Section 2.2.4) and one-time pad (see
Section 3.1.3) have keyspaces whose sizes can (at least in theory) be made
arbitrarily large.
Since the size of the keyspace in modern cryptosystems can be enormous, we
tend to focus attention on the
length
of a cryptographic key (often also referred
to as the
size
or
strength
of the key), which is the number of bits that it takes to
