An algebraic approach for decoding spread codes
Elisa Gorla Felice Manganiello Joachim Rosenthal
Advances in Mathematics of Communications 2012, 6(4): 443-466 doi: 10.3934/amc.2012.6.443
In this paper we study spread codes: a family of constant-dimension codes for random linear network coding. In other words, the codewords are full-rank matrices of size $k\times n$ with entries in a finite field $\mathbb F_q$. Spread codes are a family of optimal codes with maximal minimum distance. We give a minimum-distance decoding algorithm which requires $\mathcal{O}((n-k)k^3)$ operations over an extension field $\mathbb F_{q^k}$. Our algorithm is more efficient than the previous ones in the literature, when the dimension $k$ of the codewords is small with respect to $n$. The decoding algorithm takes advantage of the algebraic structure of the code, and it uses original results on minors of a matrix and on the factorization of polynomials over finite fields.
keywords: spread codes decoding algorithm. Random linear network coding
Zig-zag and replacement product graphs and LDPC codes
Christine A. Kelley Deepak Sridhara Joachim Rosenthal
Advances in Mathematics of Communications 2008, 2(4): 347-372 doi: 10.3934/amc.2008.2.347
It is known that the expansion property of a graph influences the performance of the corresponding code when decoded using iterative algorithms. Certain graph products may be used to obtain larger expander graphs from smaller ones. In particular, the zig-zag product and replacement product may be used to construct infinite families of constant degree expander graphs. This paper investigates the use of zig-zag and replacement product graphs for the construction of codes on graphs. A modification of the zig-zag product is also introduced, which can operate on two unbalanced biregular bipartite graphs, and a proof of the expansion property of this modified zig-zag product is presented.
keywords: expander graphs Codes on graphs replacement product of a graph LDPC codes zig-zag product
Cryptanalysis of the CFVZ cryptosystem
Joan-Josep Climent Elisa Gorla Joachim Rosenthal
Advances in Mathematics of Communications 2007, 1(1): 1-11 doi: 10.3934/amc.2007.1.1
The paper analyzes CFVZ, a new public key cryptosystem whose security is based on a matrix version of the discrete logarithm problem over an elliptic curve. It is shown that the complexity of solving the underlying problem for the proposed system is dominated by the complexity of solving a fixed number of discrete logarithm problems in the group of an elliptic curve. Using an adapted Pollard rho algorithm it is shown that this problem is essentially as hard as solving one discrete logarithm problem in the group of an elliptic curve. Hence, the CFVZ cryptosystem has no advantages over traditional elliptic curve cryptography and should not be used in practice.
keywords: generalized birthday problem. elliptic curve cryptography Diffie-Hellman protocol Public key cryptography
Public key cryptography based on semigroup actions
Gérard Maze Chris Monico Joachim Rosenthal
Advances in Mathematics of Communications 2007, 1(4): 489-507 doi: 10.3934/amc.2007.1.489
A generalization of the original Diffie-Hellman key exchange in $(\mathbb Z$∕$p\mathbb Z)$* found a new depth when Miller [27] and Koblitz [16] suggested that such a protocol could be used with the group over an elliptic curve. In this paper, we propose a further vast generalization where abelian semigroups act on finite sets. We define a Diffie-Hellman key exchange in this setting and we illustrate how to build interesting semigroup actions using finite (simple) semirings. The practicality of the proposed extensions rely on the orbit sizes of the semigroup actions and at this point it is an open question how to compute the sizes of these orbits in general and also if there exists a square root attack in general.
   In Section 5 a concrete practical semigroup action built from simple semirings is presented. It will require further research to analyse this system.
keywords: one-way trapdoor functions Public key cryptography semigroup actions Diffie-Hellman protocol simple semirings.

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