| The study of elliptic curves by algebraists, algebraic geometers and number theorists
dates back to the middle of the nineteenth century. There now exists an extensive literature
that describes the beautiful and elegant properties of these marvelous objects. In
1984, Hendrik Lenstra described an ingenious algorithm for factoring integers that relies
on properties of elliptic curves. This discovery prompted researchers to investigate
other applications of elliptic curves in cryptography and computational number theory.
Public-key cryptography was conceived in 1976 by Whitfield Diffie andMartin Hellman.
The first practical realization followed in 1977 when Ron Rivest, Adi Shamir and
Len Adleman proposed their now well-known RSA cryptosystem, in which security is
based on the intractability of the integer factorization problem. Elliptic curve cryptography
(ECC) was discovered in 1985 by Neal Koblitz and Victor Miller. Elliptic curve
cryptographic schemes are public-key mechanisms that provide the same functionality
as RSA schemes. However, their security is based on the hardness of a different
problem, namely the elliptic curve discrete logarithm problem (ECDLP). Currently
the best algorithms known to solve the ECDLP have fully exponential running time,
in contrast to the subexponential-time algorithms known for the integer factorization
problem. This means that a desired security level can be attained with significantly
smaller keys in elliptic curve systems than is possible with their RSA counterparts.
For example, it is generally accepted that a 160-bit elliptic curve key provides the same
level of security as a 1024-bit RSA key. The advantages that can be gained from smaller
key sizes include speed and efficient use of power, bandwidth, and storage. |
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