Cryptography Reference
In-Depth Information
random key bits
K
1
,
K
2
,…,
K
n
P
1
⊕
K
1
,
P
2
⊕
K
2
,…,
P
n
⊕
K
n
⊕
ciphertext bits
plaintext bits
P
1
,
P
2
,…,
P
n
Figure 3.1.
The Vernam Cipher
Mathematics Appendix). In other words:
C
1
=
P
1
⊕
K
1
C
2
=
P
2
⊕
K
2
.
C
n
=
P
n
⊕
K
n
.
This process is depicted in Figure 3.1. Decryption is almost identical, since
the receiver computes
P
i
=
C
i
⊕
K
i
to recover each plaintext bit from each
ciphertext bit.
The Vernam Cipher is a one-time pad under the condition that every string of
key bits is used just once. Of course we cannot
guarantee
that we will not generate
the same string of bits some time in the future. So long as the bits are randomly
generated then the next time we generate a string of bits it is very unlikely that
it has been used before. We will discuss random generation in more detail in
Section 8.1.
The Vernam Cipher is closely related to all the previous one-time pads:
1. Recalling that XOR is the same as 'adding' two binary numbers (which is actually
addition modulo 2), we see that it is the same as our mathematical description
of the Caesar Cipher one-time pad, where instead of working modulo 26 we
are working modulo 2 (see Section 5.1.3 and the Mathematics Appendix for
more explanation);
2. We could write out the Vernam Cipher as an enormous Latin square, but
why bother when we already have such a simple and efficient description
of it?
When we refer to the one-time pad at any later stage in our cryptographic
discussion we could, in theory, be discussing any of these versions of this
cryptosystem. However, all of our modern cryptosystems are designed to process
strings of bits, so it is probably best to think of a one-time pad as the Vernam
Cipher.
