Elementary Abstract Algebra- Examples and Applications, 2019a

552 CHAPTER 13 FURTHER TOPICS IN CRYPTOGRAPHY
slowing down communication. Elliptic curve cryptography (ECC) is one
approach to the public key sharing dilemma that offers greater security with
smaller keys. The table in Figure 13.2.1 shows the relationship between key
length and security for three different cryptosystems: RSA, Diffie-Hellman,
and ECC. In the table, ‘key length’ refers to the number of binary bits in the
key: for comparison, a 160 bit key has 49 decimal digits, while a 1024 bit key
has 309 decimal digits. The ’security level’ is also measured in bits (80, 128,
192, 256) where these bits refer to the size of the number of computational
steps necessary to break the code. For example, a security level of 80 means
that 2 80 computations are required to break the code (from reference(5)).
To give you an idea of what this means practically, in 2002 it took 10, 000
computers (mainly PCs) running 24 hours a day for 549 days to break an
ECC system with a 109 bit key length. A 160-bit ECC would be 2 25 times
more secure: which means that (with 2002 technology) it would take one
billion computers over 500 years to crack it.
Figure 13.2.1. Key bit lengths of cryptosystems for different security levels
recommended by the National Institute of Standards and Technology, (from
reference (8))
Referencing the table in Figure 13.2.1, we can see that an ECC cryptosystem
with a 160 bit key has a similar security level to RSA and Diffie-
Hellman with 1024-bit keys. This means the same security with 6 times less
information–a significant difference!
Exercise 13.2.1. How long would it take to crack a cryptosystem with a
128 bit security, using a billion modern computers with 2 GHz processors?
(Note: A 2 GHz processor is able to perform 2·10 9 computations per second.)
♦
Now that we’ve described the benefits of ECC, let’s see what elliptic
curves are all about. Our discussion will be quite wide-ranging, and touch on