We give a characterization of the codomain [ℓ]E(k) of the multiplication-by-ℓ map [ℓ] in the case... more We give a characterization of the codomain [ℓ]E(k) of the multiplication-by-ℓ map [ℓ] in the case of elliptic curves over a field k of characteristic = 2, 3 with ℓ-torsion E[ℓ] = W 1 , W 2 fully defined over k, for primes ℓ different from the characteristic. We show that a point Q ∈ E(k) lies in [ℓ]E(k) if and only if h W1 (−Q) and h W2 (−Q) are ℓpowers of k, where h W1 and h W2 are functions on E with divisor div(h Wi) = ℓW i −ℓP ∞. Our characterization leads to an effective procedure to find pre-images of [ℓ] by solving an order ℓ system of linear equations and computing a polynomial gcd.
We give a method to compute the halving of a divisor in the Jacobian variety of a genus 2 curve i... more We give a method to compute the halving of a divisor in the Jacobian variety of a genus 2 curve in characteristic 2. We present explicit halving formulae for each possible divisor class for some curves (those with () =), detailing all the process to obtain them. We improve the fastest known formulae for some divisor classes in the studied type of curves.
We give an upper bound for the number of vertices in mixed abelian Cayley graphs with given degre... more We give an upper bound for the number of vertices in mixed abelian Cayley graphs with given degree and diameter.
Cayley graphs are well known objects with interesting properties, also in the context of Moore gr... more Cayley graphs are well known objects with interesting properties, also in the context of Moore graphs and digraphs. In 1978 Bosák extended Moore's property to the mixed setting (with arcs allowed along with edges), in the so called mixed Moore graphs: those having a unique trail between pairs of vertices at a distance smaller than or equal to the diameter. In this paper we adapt Cayley's construction to mixed graphs and we show certain mixed Moore graphs are Cayley while some other cannot be Cayley.
Given a genus 2 curve [Formula: see text] defined over a finite field [Formula: see text] of odd ... more Given a genus 2 curve [Formula: see text] defined over a finite field [Formula: see text] of odd characteristic such that [Formula: see text], we study the growth of the 2-adic valuation of the cardinality of the Jacobian over a tower of quadratic extensions of [Formula: see text]. In the cases of simpler regularity, we determine the exponents of the 2-Sylow subgroup of [Formula: see text].
Construction of extremal mixed graphs of diameter two
Discrete Applied Mathematics
Graphs and digraphs with maximum order allowed by its degree and diameter have been widely studie... more Graphs and digraphs with maximum order allowed by its degree and diameter have been widely studied in the context of the Degree/Diameter problem. This problem turns out to be very interesting in the mixed case, where many open problems arise, specially when the diameter is two and the order of the graph achieves the largest theoretical value given by the mixed Moore bound. These extremal graphs are known as mixed Moore graphs. In this paper we construct by voltage assignment some infinite families of mixed graphs of diameter two and order approaching the Moore bound. One of these families, in particular, yields most of the known mixed Moore graphs. We also present other families which are the result of the first known extension of the paradigmatic McKay-Miller-Širáň construction (McKay et al., 1998).
The Degree/Diameter Problem is an extremal problem in graph theory with applications in network d... more The Degree/Diameter Problem is an extremal problem in graph theory with applications in network design. One of the main research areas in the Degree/Diameter Problem consists of finding large graphs whose order approach the theoretical upper bounds as much as possible. In the case of directed graphs there exist some families that come close to the theoretical upper bound asymptotically. In the case of undirected graphs there also exist some good constructions for specific values of the parameters involved (degree and/or diameter). One such construction was given by McKay, Miller, andŠiráň in [5], which produces large graphs of diameter 2 with the aid of the voltage assignment technique. Here we show how to re-engineer the McKay-Miller-Širáň construction in order to obtain large mixed graphs of diameter 2, i.e. graphs containing both directed arcs and undirected edges.
Let E be an elliptic curve defined over a finite field Fq of odd characteristic. Let = 2 be a pri... more Let E be an elliptic curve defined over a finite field Fq of odd characteristic. Let = 2 be a prime number different from the characteristic and dividing #E(Fq). We describe how the-adic valuation of the number of points grows by taking finite extensions of the base field. We also investigate the group structure of the corresponding-Sylow subgroups.
Resumen. Los grafos construidos a partir de grupos (grafos de Cayley) han sido ampliamente estudi... more Resumen. Los grafos construidos a partir de grupos (grafos de Cayley) han sido ampliamente estudiados y gozan de propiedades que los hacen mu utiles en el con-texto de los grafos y digrafos de Moore. En el año 1978, Bosák generalizó los grafos y digrafos de Moore a grafos mezclados de Moore, es decir, grafos donde coexisten aristas y arcos simultaneamente y para los que existe u unico recorrido entre dos pa-res de vértices cualesquiera a distancia más pequeña o igual que el diámetro. En este trabajo extendemos el concepto de grafo de Cayley para los grafos mezclados, mos-tramos como algunos grafos mezclados de Moore son de tipo Cayley y descartamos la existencia de este tipo de grafos en otros casos.
By reversing reduction in divisor class arithmetic we provide efficient trisection algorithms for... more By reversing reduction in divisor class arithmetic we provide efficient trisection algorithms for supersingular Jacobians of genus 2 curves over finite fields of characteristic 2. With our technique we obtain new results for these Jacobians: we show how to find their 3-torsion subgroup, we prove there is none with 3-torsion subgroup of rank 3 and we prove that the maximal 3power order subgroup is isomorphic to either Z/3 v Z or (Z/3 v 2 Z) 2 or (Z/3 v 4 Z) 4 , where v is the 3-adic valuation v 3 (#Jac(C)(F 2 m)). Ours are the first trisection formulae available in literature.
Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, ... more Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated since the full torsion subgroup has rank 2g. In this paper we prove that distortion maps always exist for supersingular curves of genus g>1 and we construct distortion maps in genus 2 (for
Proceedings of the Japan Academy Series a-Mathematical Sciences, 2009
We show how to invert the multiplication-by-2 map in Jacobians of genus 2 curves C over finite fi... more We show how to invert the multiplication-by-2 map in Jacobians of genus 2 curves C over finite fields F q of odd characteristic. For any divisor D 2 JacðCÞðF q Þ we provide a method to construct the coordinates of all divisors D 0 2 JacðCÞðF q Þ such that 2D 0 ¼ D.
Properties of mixed Moore graphs of directed degree one
Discrete Mathematics, 2015
ABSTRACT Mixed graphs of order such that for any pair of vertices there is a unique trail of leng... more ABSTRACT Mixed graphs of order such that for any pair of vertices there is a unique trail of length at most between them are known as mixed Moore graphs. These extremal graphs may only exist for diameter and certain (infinitely many) values of . In this paper we give some properties of mixed Moore graphs of directed degree one and reduce their existence to the existence of some (undirected) strongly regular graphs.
We give a characterization of the codomain [ℓ]E(k) of the multiplication-by-ℓ map [ℓ] in the case... more We give a characterization of the codomain [ℓ]E(k) of the multiplication-by-ℓ map [ℓ] in the case of elliptic curves over a field k of characteristic = 2, 3 with ℓ-torsion E[ℓ] = W 1 , W 2 fully defined over k, for primes ℓ different from the characteristic. We show that a point Q ∈ E(k) lies in [ℓ]E(k) if and only if h W1 (−Q) and h W2 (−Q) are ℓpowers of k, where h W1 and h W2 are functions on E with divisor div(h Wi) = ℓW i −ℓP ∞. Our characterization leads to an effective procedure to find pre-images of [ℓ] by solving an order ℓ system of linear equations and computing a polynomial gcd.
We give a method to compute the halving of a divisor in the Jacobian variety of a genus 2 curve i... more We give a method to compute the halving of a divisor in the Jacobian variety of a genus 2 curve in characteristic 2. We present explicit halving formulae for each possible divisor class for some curves (those with () =), detailing all the process to obtain them. We improve the fastest known formulae for some divisor classes in the studied type of curves.
We give an upper bound for the number of vertices in mixed abelian Cayley graphs with given degre... more We give an upper bound for the number of vertices in mixed abelian Cayley graphs with given degree and diameter.
Cayley graphs are well known objects with interesting properties, also in the context of Moore gr... more Cayley graphs are well known objects with interesting properties, also in the context of Moore graphs and digraphs. In 1978 Bosák extended Moore's property to the mixed setting (with arcs allowed along with edges), in the so called mixed Moore graphs: those having a unique trail between pairs of vertices at a distance smaller than or equal to the diameter. In this paper we adapt Cayley's construction to mixed graphs and we show certain mixed Moore graphs are Cayley while some other cannot be Cayley.
Given a genus 2 curve [Formula: see text] defined over a finite field [Formula: see text] of odd ... more Given a genus 2 curve [Formula: see text] defined over a finite field [Formula: see text] of odd characteristic such that [Formula: see text], we study the growth of the 2-adic valuation of the cardinality of the Jacobian over a tower of quadratic extensions of [Formula: see text]. In the cases of simpler regularity, we determine the exponents of the 2-Sylow subgroup of [Formula: see text].
Construction of extremal mixed graphs of diameter two
Discrete Applied Mathematics
Graphs and digraphs with maximum order allowed by its degree and diameter have been widely studie... more Graphs and digraphs with maximum order allowed by its degree and diameter have been widely studied in the context of the Degree/Diameter problem. This problem turns out to be very interesting in the mixed case, where many open problems arise, specially when the diameter is two and the order of the graph achieves the largest theoretical value given by the mixed Moore bound. These extremal graphs are known as mixed Moore graphs. In this paper we construct by voltage assignment some infinite families of mixed graphs of diameter two and order approaching the Moore bound. One of these families, in particular, yields most of the known mixed Moore graphs. We also present other families which are the result of the first known extension of the paradigmatic McKay-Miller-Širáň construction (McKay et al., 1998).
The Degree/Diameter Problem is an extremal problem in graph theory with applications in network d... more The Degree/Diameter Problem is an extremal problem in graph theory with applications in network design. One of the main research areas in the Degree/Diameter Problem consists of finding large graphs whose order approach the theoretical upper bounds as much as possible. In the case of directed graphs there exist some families that come close to the theoretical upper bound asymptotically. In the case of undirected graphs there also exist some good constructions for specific values of the parameters involved (degree and/or diameter). One such construction was given by McKay, Miller, andŠiráň in [5], which produces large graphs of diameter 2 with the aid of the voltage assignment technique. Here we show how to re-engineer the McKay-Miller-Širáň construction in order to obtain large mixed graphs of diameter 2, i.e. graphs containing both directed arcs and undirected edges.
Let E be an elliptic curve defined over a finite field Fq of odd characteristic. Let = 2 be a pri... more Let E be an elliptic curve defined over a finite field Fq of odd characteristic. Let = 2 be a prime number different from the characteristic and dividing #E(Fq). We describe how the-adic valuation of the number of points grows by taking finite extensions of the base field. We also investigate the group structure of the corresponding-Sylow subgroups.
Resumen. Los grafos construidos a partir de grupos (grafos de Cayley) han sido ampliamente estudi... more Resumen. Los grafos construidos a partir de grupos (grafos de Cayley) han sido ampliamente estudiados y gozan de propiedades que los hacen mu utiles en el con-texto de los grafos y digrafos de Moore. En el año 1978, Bosák generalizó los grafos y digrafos de Moore a grafos mezclados de Moore, es decir, grafos donde coexisten aristas y arcos simultaneamente y para los que existe u unico recorrido entre dos pa-res de vértices cualesquiera a distancia más pequeña o igual que el diámetro. En este trabajo extendemos el concepto de grafo de Cayley para los grafos mezclados, mos-tramos como algunos grafos mezclados de Moore son de tipo Cayley y descartamos la existencia de este tipo de grafos en otros casos.
By reversing reduction in divisor class arithmetic we provide efficient trisection algorithms for... more By reversing reduction in divisor class arithmetic we provide efficient trisection algorithms for supersingular Jacobians of genus 2 curves over finite fields of characteristic 2. With our technique we obtain new results for these Jacobians: we show how to find their 3-torsion subgroup, we prove there is none with 3-torsion subgroup of rank 3 and we prove that the maximal 3power order subgroup is isomorphic to either Z/3 v Z or (Z/3 v 2 Z) 2 or (Z/3 v 4 Z) 4 , where v is the 3-adic valuation v 3 (#Jac(C)(F 2 m)). Ours are the first trisection formulae available in literature.
Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, ... more Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated since the full torsion subgroup has rank 2g. In this paper we prove that distortion maps always exist for supersingular curves of genus g>1 and we construct distortion maps in genus 2 (for
Proceedings of the Japan Academy Series a-Mathematical Sciences, 2009
We show how to invert the multiplication-by-2 map in Jacobians of genus 2 curves C over finite fi... more We show how to invert the multiplication-by-2 map in Jacobians of genus 2 curves C over finite fields F q of odd characteristic. For any divisor D 2 JacðCÞðF q Þ we provide a method to construct the coordinates of all divisors D 0 2 JacðCÞðF q Þ such that 2D 0 ¼ D.
Properties of mixed Moore graphs of directed degree one
Discrete Mathematics, 2015
ABSTRACT Mixed graphs of order such that for any pair of vertices there is a unique trail of leng... more ABSTRACT Mixed graphs of order such that for any pair of vertices there is a unique trail of length at most between them are known as mixed Moore graphs. These extremal graphs may only exist for diameter and certain (infinitely many) values of . In this paper we give some properties of mixed Moore graphs of directed degree one and reduce their existence to the existence of some (undirected) strongly regular graphs.
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Papers by Jordi Pujolàs