66 research outputs found
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
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
A simple generalization of El-Gamal cryptosystem to non-abelian groups
In this paper we study the MOR cryptosystem. We use the group of
unitriangular matrices over a finite field as the non-abelian group in the MOR
cryptosystem. We show that a cryptosystem similar to the El-Gamal cryptosystem
over finite fields can be built using the proposed groups and a set of
automorphisms of these groups. We also show that the security of this proposed
MOR cryptosystem is equivalent to the El-Gamal cryptosystem over finite fields
Gaussian elimination in split unitary groups with an application to public-key cryptography
Gaussian elimination is used in special linear groups to solve the word problem. In this paper, we extend Gaussian elimination to split unitary groups. These algorithms have an application in building a public-key cryptosystem, we demonstrate that
The MOR Cryptosystem in Classical Groups with a Gaussian Elimination Algorithm for Symplectic and Orthogonal Groups
In this chapter, we study the MOR cryptosystem with symplectic and orthogonal groups over finite fields of odd characteristics. There are four infinite families of finite classical Chevalley groups. These are special linear groups SL(d, q), orthogonal groups O(d, q), and symplectic groups Sp(d, q). The family O(d, q) splits into two different families of Chevalley groups depending on the parity of d. The MOR cryptosystem over SL(d, q) was studied by the second author. In that case, the hardness of the MOR cryptosystem was found to be equivalent to the discrete logarithm problem in
F
q
d
. In this chapter, we show that the MOR cryptosystem over Sp(d, q) has the security of the discrete logarithm problem in
F
q
d
. However, it seems likely that the security of the MOR cryptosystem for the family of orthogonal groups is
F
q
d
2
. We also develop an analog of row-column operations in symplectic and orthogonal groups which is of independent interest as an appendix
The discrete logarithm problem in the group of non-singular circulant matrices
The discrete logarithm problem is one of the backbones in public key
cryptography. In this paper we study the discrete logarithm problem in the
group of circulant matrices over a finite field. This gives rise to secure and
fast public key cryptosystems
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