Cryptography Reference
In-Depth Information
6.5 Cryptographic Application of Exponentiation
We have worked hard in this chapter in our calculation of powers, and it is
reasonable to ask at this point what modular exponentiation might have to offer
to cryptographic applications. The first example to come to mind is, of course,
the RSA procedure, which requires a modular exponentiation for encryption and
decryption—assuming suitable keys. However, the author would like to ask his
readers for a bit (or perhaps even a byte) of patience, since for the RSA procedure
we still must collect a few more items, which we do in the next chapter. We shall
return to this extensively in Chapter 17.
For those incapable of waiting, we offer as examples of the application of
exponentiation two important algorithms, namely, the procedure suggested
in 1976 by Martin E. Hellman and Whitfield Diffie [Diff] for the exchange of
cryptographic keys and the encryption procedure of Taher ElGamal as an
extension of the Diffie-Hellman procedure.
The Diffie-Hellman procedure represents a cryptographic breakthrough,
namely, the first public key, or asymmetric, cryptosystem (see Chapter 17).
Two years after its publication, Rivest, Shamir, and Adleman published the RSA
procedure (see [Rive]). Variants of the Diffie-Hellman procedure are used today
for key distribution in the Internet communications and security protocols
IPSec, IPv6, and SSL, which were developed to provide security in the transfer of
data packets in the IP protocol layer and the transfer of data at the application
level, for example from the realms of electronic commerce. This principle of
key distribution thus has a practical significance that would be difficult to
overestimate.
3
With the aid of the Diffie-Hellman protocol two communicators, Ms.
A
and
Mr.
B
, say, can negotiate in a simple way a secret key that then can be used for the
encryption of communications between the two. After
A
and
B
have agreed on
a large prime number
p
and a primitive root
a
modulo
p
(we shall return to this
below), the Diffie-Hellman protocol runs as follows.
Protocol for key exchange à la Diffie-Hellman
1
and sends
y
A
:=
a
x
A
mod
p
as her
1.
A
chooses an arbitrary value
x
A
≤
p
−
public key to
B
.
2.
B
chooses an arbitrary value
x
B
≤ p −
1
and sends
y
B
:=
a
x
B
mod
p
as his
public key to
A
.
3
IP Security (IPSec), developed by the Internet Engineering Task Force (IETF), is, as an extensive
security protocol, a part of the future Internet protocol IPv6. It was created so that it could also
be used in the framework of the then current Internet protocol (IPv4). Secure Socket Layer
(SSL) is a security protocol developed by Netscape that lies above the TCP protocol, which
offers end-to-end security for applications such as HTTP, FTP, and SMTP (for all of this see
[Stal], Chapters 13 and 14).
