Implementing-cryptography-using-python

Chapter 8 ■ Cryptographic Applications and PKI 239
of mathematics called elliptic curves. Although the system was introduced in
the mid ’80s, it took another twenty years for the cryptosystem to gain wide
acceptance. Several factors are contributing to its increasing popularity. First, the
security of 1024-bit RSA encryption is degrading due to faster computing and
a better understanding and analysis of encryption methods. While brute force
is still unlikely to crack 1024-bit RSA keys, other approaches, including highly
intensive parallel computing in distributed computing arrays, are resulting in
more sophisticated attacks. These attacks have reduced the effectiveness of this
level of security. Even 2,048-bit encryption is estimated by the RSA Security to
be effective only until 2030. A second factor that is contributing to the adoption
of ECC is that many government entities have started to accept ECC as an
encryption method. Third, the authentication speed of ECC is faster than RSA
in terms of server authentication. Finally, certificate authorities have started
embedding ECC algorithms into their SSL certificates.
ECC was independently suggested by Neal Koblitz (University of Washington)
and Victor S. Miller (IBM) in 1985. After the introduction of Diffie-Hellman and
RSA, cryptographers started exploring other mathematics-based cryptographic
solutions looking for other algorithms that would offer easy one-way calculations
that were hard to find an inverse for; these types of functions are referred to as
trapdoors. A trapdoor function is a function that is easy to perform one way but
has a secret that is required to perform the inverse calculation efficiently. That is,
if f is a trapdoor function, then y = f(x) is easy to compute, but x = f − 1(y) is hard
to compute without some special knowledge k. Unless you have a mathematical
background, elliptic curves may be new to you; so what exactly is an elliptic
curve and how does the elliptic curve trapdoor function work?
An elliptic curve is the set of points that satisfy a specific mathematical equation.
The equation for an elliptic curve looks something like this:
2 3
y x ax b
That graphs to something that looks like Figure 8.6.
Figure 8.6: An elliptic curve