Современные информационные технологии/4. Информационная безопасность

Serdyukov Sergey D.
Novosibirsk State University, Russia
Modern Cryptography: Comparative Analysis of Asymmetric Cryptosystems Elliptical Curve Cryptography and RSA,

Their Advantages and Disadvantages

In today's world there is a pointed question about computer security due to the rapid growth of networking and the Internet. Cryptography is one of the main tools in the field of information security. However it has its own problems like any other area.

Four experts on computer security Alex Stamos, Tom Ritter, Thomas Ptacek and Javed Samuel made a disquieting statement during the summer computer security conference Black Hat USA 2013. They said that existing cryptosystems are in dangerous, because of mathematical problems’ solving progress. Therefore we have to abandon existing SSL-certificates in favor of modern cryptographic systems.

There was a lot of cyber-attack this year such as BEAST, CRIME, and Lucky13. It shows us a need in improvement of cryptographic schemes. The industry must be able to anticipate the future types of attacks before they occur, but there are many factors that impede progress. The inefficiency of crypto-ecosystem between suppliers of cryptographic tools, suppliers of certificates and browser vendors ensure the preservation of the current state of affairs. In addition, it is very difficult to move at a rapid pace of development of modern science for industry participants.

Asymmetric cryptosystems are based on two keys: first one is needed to encrypt data; second one is needed to decrypt data. It is possible because of properties of one-way functions. It is assumed that some of the mathematical operation is quite complex and can be implemented in a time growing exponentially with a linear increase in the dimension of the problem. However the existence of such functions is an unproven hypothesis. Their existence proves that complexity classes P and NP are not equal. Modern asymmetric cryptography is based on assumption that one-way functions are exist [1].

There were obtained algorithms for discrete logarithm. In fact these algorithms are of limited use. And at this moment there are not ways to use it in practical cryptography. There is a possibility that in the future there will be a polynomial solution of this problem, which would mean collapse of existing cryptography. This is indicated by the publication of the French scientist Antoine Joux, who published two articles which suggest that such a solution can be found soon. Joux applied the known techniques that have not been used for the solution of this problem. Collapse might occur soon, while alternative and more sophisticated systems are not widely spread.

The most common asymmetric algorithm – RSA – relies on the complexity of two problems: integer factorization and discrete logarithm. Modern mathematics does not have simple polynomial solutions, but quick progress in this area over the last six months causes fear after the decades of research.

The attacks on the discrete logarithm and integer factorization in RSA follow almost the same steps in terms of the polynomial selection, sifting and linear algebra. The fourth step (the calculation of square roots) is very fast, which is very dangerous for the discrete logarithm problem.

It was offered to use cryptography based on elliptic curves over finite fields (Elliptical curve cryptography). Elliptical curve cryptography is similar to other asymmetric algorithms: there is an assumption about the complexity of the mathematical problem, in this case – the discrete logarithm in the group of points on elliptic curves. In contrast to the relatively similar problems integer factorization and discrete logarithm, the positive result in one of them does not threaten the elliptical curve cryptography [1].

On the other hand there are certain difficulties in the translation to Elliptical curve cryptography. Much of the technology is patented by BlackBerry and patent issues led some manufacturers to give up their support. Protocols which use elliptical curve cryptography are not supported widely enough. Certification centers don't provide certificates of elliptical curve cryptography.

Advantages of elliptical curve cryptography:

1) Much shorter key length compared to the “classical” asymmetric cryptography.

2) The speed of the algorithm based on elliptical curve cryptography is much higher than the “classical” ones. This is explained both size of field and use binary finite field’s structure which is convenient to computer.

3) Asymmetric cryptography algorithms on elliptic curves can be used in smart cards and other devices with limited computing resources due to the small length of the key and high-speed operation.

All advantages of elliptical curve cryptography follow from one particular fact – the shorter key length.

Disadvantages of elliptical curve cryptography:

1) There are not sub-exponential algorithms for discrete logarithm on elliptical curve. However if such algorithms will appear, it will be the collapse of elliptical cryptosystems.

2) Elliptical curve cryptography is very complex and includes set of subtlety. During the mass transition to elliptic cryptography there will be a lot of bugs and vulnerability, which are could be used by malefactors.

Based on the above analysis, we can conclude that the mass transition to an elliptical cryptography is not necessary. Today it is enough to use the RSA. Progress in solving mathematical problems is, but it is not too dangerous to the existing cryptosystems as it was told by experts. However it would be great to start support elliptical cryptography now, because there is a risk and we have to be ready to make the transition without significant problems at every moment.

Bibliography:

1. Applied Cryptography. Second Edition. Bruce Schneier. John Wiley & Sons, 1996. - 784 pages.