I recently ran across elliptic curve crypto-systems:


An Introduction to the Theory of Elliptic Curves (Brown University)
Elliptic Curve Cryptography (Wikipedia)
Performance analysis of identity management in the Session Initiation Protocol (SIP) (IEEE)
Overview of Elliptic Curve Cryptosystems. (RSA.com)


It seemed to me to be great alternative to RSA as the de-facto cryptosystems to be used in banking and financial systems and in the public key infrastructure for certificates, but is not used! If someone can explain why this is not done, it would be very helpful. A comparison between traditional RSA and an elliptic curve cryptology would be helpful.

To begin with:

Advantage of RSA:


Well established.


Advantages of elliptic curve:


Shorter keys are as strong as long key for RSA (see the IEEE paper)
Low on CPU consumption.
Low on memory usage.


RSA was there first. That's actually enough for explaining its preeminence. RSA was first published in 1978 and the PKCS#1 standard (which explains exactly how RSA should be used, with unambiguous specification of which byte goes where) has been publicly and freely available since 1993. The idea of using elliptic curves for cryptography came to be in 1985, and relevant standards have existed since the late 1990s. Also, both RSA and elliptic curves have been covered by patents, but the RSA patents have expired in 2000, while some elliptic curve patents are still alive.

One perceived, historical advantage of RSA is that RSA is two algorithms, one for encryption and one for signatures, that could both use the same key and the same core implementation. But this is not a real advantage because it is usually a bad idea to use the same key for both encryption and signatures. Also, you can mathematically use the same private key for ECDH (key exchange) and for ECDSA (signatures), so that's really not an "advantage" of RSA over EC at all.

Another advantage of RSA is that its mathematics are somewhat simpler than those involved for elliptic curves, so many engineers feel that they "understand" RSA more than elliptic curves; again, a fallacious argument, since implementation of cryptographic algorithms is fraught with subtle details and best left to professionals -- and there is no need to understand the internal mathematics of a library to simply use it (we could make this argument semi-valid by pointing out that RSA relies on the hardness of factorization, which has been studied for 2500 years, whereas discrete logarithm on elliptic curves can only sport about 25 years of research).

The only scientifically established advantaged of RSA over elliptic curves cryptography is that public key operations (e.g. signature verification, as opposed to signature generation) are faster with RSA. But public-key operations are rarely a bottleneck, and we are talking about 8000 ECDSA verifications per second, vs 20000 RSA verifications per second.

An additional interoperability issue is that elliptic-curve operations can be made over curves of distinct types, and can be widely optimized if you stick to a specific curve known when the code was written. There is no security issue in using the same curve for many distinct people with distinct key pairs. But it means that most implementations will only support two or three specific curves. NIST has defined 15 standard curves. However, in practice, many implementations only support two of them, P-256 and P-384, because that's what is recommended by NSA (under the name "suite B")(a notorious example is NSS, the cryptography library used by the Firefox Web browser for SSL).

There are two ANSI standards for elliptic curves, X9.62 for signatures (partially redundant with FIPS 186-3, but much more detailed), and X9.63 for asymmetric encryption.

So there is a lot of political push for the adoption of elliptic curves in cryptography, by both academic researchers and institutional organizations. But inertia of the firmly entrenched RSA will take time to defeat. Also, the perceived mathematical complexity, and the potential legal risks related to patents, still hinder wide acceptance of elliptic curves.

(To your list, you can add "key generation time": generating a new key pair for ECDH or ECDSA is widely faster than generating a new RSA key pair.)
