Non-hyperelliptic curves of genus three over finite fields of characteristic two

Let k be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non-singular quartic plane curves defined over k. We find explicit rational models and closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non-hyperelliptic locus of the moduli space M₃ of curves of genus 3. By adding to these computations the results of Oort [Moduli of abelian varieties and Newton polygons, C.R. Acad. Sci. Paris 312 (1991) 385-389] and Nart and Sadornil [Hyperelliptic curves of genus three over finite fields of characteristic two, Finite Fields Appl. 10 (2004) 198-200] on the hyperelliptic locus we obtain a complete picture of these strata for M₃.     

Twists of genus three curves over finite fields

In this article we recall how to describe the twists of a curve over a finite field and we show how to compute the number of rational points on such a twist by methods of linear algebra. We illustrate this in the case of plane quartic curves with at least 16 automorphisms. In particular we treat the twists of the Dyck–Fermat and Klein quartics. Our methods show how in special cases non-Abelian cohomology can be explicitly computed. They also show how questions which appear difficult from a function field perspective can be resolved by using the theory of the Jacobian variety.     

Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic

This paper is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus 4 case and the finite fields are of even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.     

Binomial permutations over finite fields with even characteristic

Designs, Codes and Cryptography - In this paper, we study binomials having the form $$x^r(a+x^{3(q-1)})$$ over the finite field $$\mathbb {F}_{q^2}$$ with $$q=2^m$$ , and determine all the...     

Optimal curves of low genus over finite fields

We investigate maximal and minimal curves of genus 4 and 5 over finite fields with discriminant −11 and −19. As a result the Hasse–Weil–Serre bound is improved.     

Symmetric bilinear forms over finite fields of even characteristic

Let Sm be the set of symmetric bilinear forms on an m-dimensional vector space over GF(q), where q is a power of two. A subset Y of Sm is called an (m,d)-set if the difference of every two distinct elements in Y has rank at least d. Such objects are closely related to certain families of codes over Galois rings of characteristic four. An upper bound on the size of (m,d)-sets is derived, and in certain cases, the rank distance distribution of an (m,d)-set is explicitly given. Constructions of (m,d)-sets are provided for all possible values of m and d.     

Isomorphism classes of hyperelliptic curves of genus 2 over finite fields with characteristic 2

In this paper we study the computation of the number of isomorphism classes of hyperelliptic curves of genus 2 over finite fields Fq with q even. We show the formula of the number of isomorphism classes, that is, for q = 2m, if 4 m, then the formula is 2q3 q2 - q; if 4 | m, then the formula is 2q3 q2 - q 8. These results can be used in the classification problems and the hyperelliptic curve cryptosystems.     

Zeta-functions of curves of genus 3 over finite fields

In this paper, we determine zeta-functions of some curves of genus 3 over finite fields by gluing three elliptic curves based on Xing＇s research, and the examples show that there exists a maximal curve of genus 3 over F49.     

On non-monomial APcN permutations over finite fields of even characteristic

Recently, a new concept called the c-differential uniformity was proposed by Ellingsen et al. (2020), which generalizes the notion of differential uniformity measuring the resistance against differential cryptanalysis. Since then, finding functions having low c-differential uniformity has attracted the attention of many researchers. However it seems that, at this moment, there are not many non-monomial permutations having low c-differential uniformity. In this paper, we present new classes of (almost) perfect c-nonlinear non-monomial permutations over a binary field.     

Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with odd characteristic

This paper is devoted to computing the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus-4 case and the finite fields are of odd characteristic. The number of isomorphism classes is computed. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The results have applications in the classification problems and in the hyperelliptic curve cryptosystems.     

Several classes of complete permutation polynomials over finite fields of even characteristic

In this paper, we find three classes of complete permutation polynomials over finite fields of even characteristic. The first class of quadrinomials is complete in the sense of addition. The second and third classes of binomials and trinomials are complete in multiplication. Moreover, a result related to the complete property in multiplication of a special class of polynomials is also given.     

Efficient algorithms for Koblitz curves over fields of characteristic three

The nonadjacent form method of Koblitz [Advances in Cryptology (CRYPTO'98), in: Lecture Notes in Comput. Sci., vol. 1462, 1998, pp. 327–337] is an efficient algorithm for point multiplication on a family of supersingular curves over a finite field of characteristic 3. In this paper, a further discussion of the method is given. A window nonadjacent form method is proposed and its validity is proved. Efficient reduction and pre-computations are given. Analysis shows that more than 30% of saving can be achieved.     

The genus of curves over finite fields with many rational points

We prove the following result which was conjectured by Stichtenoth and Xing: letg be the genus of a projective, irreducible non-singular algebraic curve over the finite field and whose number of -rational points attains the Hasse-Weil bound; then either 4g≤(q−1)² or 2g=(q−1)q. Supported by a grant from the International Atomic Energy and UNESCOCorrespondence to: F. Torres This article was processed by the author using theLatex style file from Springer-Verlag.