Index calculus in the trace zero variety
Elisa Gorla Maike Massierer
Advances in Mathematics of Communications 2015, 9(4): 515-539 doi: 10.3934/amc.2015.9.515
We discuss how to apply Gaudry's index calculus algorithm for abelian varieties to solve the discrete logarithm problem in the trace zero variety of an elliptic curve. We treat in particular the practically relevant cases of field extensions of degree 3 or 5. Our theoretical analysis is compared to other algorithms present in the literature, and is complemented by results from a prototype implementation.
keywords: trace zero variety. index calculus discrete logarithm problem Elliptic curve cryptography
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
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

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