24 research outputs found
On special divisors and the two variable zeta function of algebraic curves over finite fields
The gonality sequence of a plane curve is computed. A two variable zeta function for curves over a finite field is defined and the rationality and a functional equation are proved
Error-correcting pairs for a public-key cryptosystem
Code-based Cryptography (CBC) is a powerful and promising alternative for quantum resistant cryptography. Indeed, together with lattice-based cryptography, multivariate cryptography and hash-based cryptography are the principal available techniques for post quantum cryptography. CBC was first introduced by McEliece where he designed one of the most efficient Public-Key encryption schemes with exceptionally strong security guarantees and other desirable properties that still resist to attacks based on Quantum Fourier Transform and Amplitude Amplification
Trinomial curves with many rational points
The authors discuss a method to get plane curves with many rational points and a construction to get asymptotically good codes and curves, that is closely related to finding bivariate polynomials representing designs. The number of rational points and the genus is computed for plane curves that have a defining equation with three monomials
Error-correcting pairs for a public-key cryptosystem
Code-based cryptography is an interesting alternative to classic number-theory PKC since it is conjectured to be secure against quantum computer attacks. Many families of codes have been proposed for these cryptosystems, one of the main requirements is having high performance t-bounded decoding algorithms which in the case of having high an error-correcting pair is achieved. In this article the class of codes with a t-ECP is proposed for the McEliece cryptosystem. The hardness of retrieving the t-ECP for a given code is considered. As a first step distinguishers of several subclasses are given
Quantum BCH and reed-solomon entanglement-assisted codes
Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some resources can be used to improve the parameters of these codes, e.g., entanglement. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this talk, a general method to construct QUENTA codes via cyclic codes is shown. Afterwards, the method is applied to BCH and Reed-Solomon codes
Defining the q-analogue of a matroid
\u3cp\u3eThis paper defines the q-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a q-matroid, and why they are (not) cryptomorphic. Also, we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid.\u3c/p\u3
On defining generalized rank weights
\u3cp\u3eThis paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over L, where L is a finite extension of a field K. This is a generalization of the case where K = Fq and L = Fqm of Gabidulin codes to arbitrary characteristic. We show equivalence to previous definitions, in particular the ones by Kurihara-Matsumoto-Uyematsu [12, 13], Oggier-Sboui [16] and Ducoat [6]. As an application of the notion of generalized rank weights, we discuss codes that are degenerate with respect to the rank metric.\u3c/p\u3
The extended and generalized rank weight enumerator of a code
This paper investigates the rank weight enumerator of a code over L, where L is a finite extension of a field K. This is a generalization of the case where K = F_q and L = F_{q^m} of Gabidulin codes to arbitrary characteristic. We use the notion of counting polynomials, to define the (extended) rank weight enumerator, since in this generality the set of codewords of a given rank weight is no longer finite. Also the extended and generalized rank weight enumerator are studied in analogy with previous work on codes with respect to the Hamming metric