MOR Cryptosystem and classical Chevalley groups in odd characteristic

In this paper we study the MOR cryptosystem using finite classical Chevalley groups over a finite field of odd characteristic. In the process we develop an algorithm for these Chevalley groups in the same spirit as the row-column operation for special linear group. We focus our study on orthogonal and symplectic groups. We find the hardness of the proposed MOR cryptosystem for these groups

Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U

Role of causality in ensuring unconditional security of relativistic quantum cryptography

The problem of unconditional security of quantum cryptography (i.e. the security which is guaranteed by the fundamental laws of nature rather than by technical limitations) is one of the central points in quantum information theory. We propose a relativistic quantum cryptosystem and prove its unconditional security against any eavesdropping attempts. Relativistic causality arguments allow to demonstrate the security of the system in a simple way. Since the proposed protocol does not employ collective measurements and quantum codes, the cryptosystem can be experimentally realized with the present state-of-art in fiber optics technologies. The proposed cryptosystem employs only the individual measurements and classical codes and, in addition, the key distribution problem allows to postpone the choice of the state encoding scheme until after the states are already received instead of choosing it before sending the states into the communication channel (i.e. to employ a sort of ``antedate'' coding).Comment: 9 page

The decoding failure probability of MDPC codes

Moderate Density Parity Check (MDPC) codes are defined here as codes which have a parity-check matrix whose row weight is $O(\sqrt{n})$ where $n$ is the length $n$ of the code. They can be decoded like LDPC codes but they decode much less errors than LDPC codes: the number of errors they can decode in this case is of order $\Theta(\sqrt{n})$. Despite this fact they have been proved very useful in cryptography for devising key exchange mechanisms. They have also been proposed in McEliece type cryptosystems. However in this case, the parameters that have been proposed in \cite{MTSB13} were broken in \cite{GJS16}. This attack exploits the fact that the decoding failure probability is non-negligible. We show here that this attack can be thwarted by choosing the parameters in a more conservative way. We first show that such codes can decode with a simple bit-flipping decoder any pattern of $O\left(\frac{\sqrt{n} \log \log n}{\log n}\right)$ errors. This avoids the previous attack at the cost of significantly increasing the key size of the scheme. We then show that under a very reasonable assumption the decoding failure probability decays almost exponentially with the codelength with just two iterations of bit-flipping. With an additional assumption it has even been proved that it decays exponentially with an unbounded number of iterations and we show that in this case the increase of the key size which is required for resisting to the attack of \cite{GJS16} is only moderate