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Papers by Ernesto Reinaldo Barreiro

Research paper thumbnail of The Emergence of Picard Jacobians in Cryptography

Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik eBooks, Feb 15, 2001

In this paper we present a new family of Jacobian Varieties defined over finite fields that provi... more In this paper we present a new family of Jacobian Varieties defined over finite fields that provides many elements whose group structure is suitable for cryptosystems based on the intractability of the discrete logarithm problem. Their security against several know attacks and the efficiency (and effectiveness) of the most important algorithms of the corresponding cryptosystems are discussed.

Research paper thumbnail of On the Jacobian Varieties of Picard curves

Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik eBooks, Oct 20, 2005

In this paper a system of coordinates for the e ective divisors on the Jacobian Variety of a Pica... more In this paper a system of coordinates for the e ective divisors on the Jacobian Variety of a Picard curve i s p r e s e n ted. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an explicit structure of algebraic variety on the Jacobian as well as an e cient algorithm for the addition of divisors. Supported partially by a D F G grant 1 n-gonal cyclic curves, also called cycloelliptic or superelliptic). These curves play a central role in some approaches to generalizations of Hilbert problems 7, 12, 21 and 22, in special di erential equations and many other researches 4], 5], 6], 9], 10], 11], 12]. The authors felt motivated to nd out, to which extent the Picard curves (or more general, the n-gonal cyclic curves) could share with the hyperelliptic curves the nice property o f m a king things to become explicit. In this paper, we s h o w our rst results in this direction. We wish to thank R.-P. Holzapfel for his valuable comments, discussions and encouragement. 2 Preliminaries Let k be an arbitrary eld. we write k for its algebraic closure. The a ne space A n k consists of all the points f(x 1 x 2 ::: x n) x i 2 kg. P oints in the projective space P n k consist of equivalence classes of points in A n+1 k n f (0 0 :: 0)g where (x 0 x 1 : : x n) a n d (y 0 y 1 :: y n) a r e equivalent if there is c 2 k (c 6 = 0), such t h a t x i = cy i for all i = 0 1 :: n. Note that A 2 k is naturally embedded in P 2 k by the map (x y) ! (x y 1). If G 2 k x y] is an homogeneous polynomial then, G de ne an algebraic subset of P 2

Research paper thumbnail of On the Jacobian Varieties of Picard curves

In this paper a system of coordinates for the e ective divisors on the Jacobian Variety of a Pica... more In this paper a system of coordinates for the e ective divisors on the Jacobian Variety of a Picard curve i s p r e s e n ted. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an explicit structure of algebraic variety on the Jacobian as well as an e cient algorithm for the addition of divisors. Supported partially by a D F G grant 1 n-gonal cyclic curves, also called cycloelliptic or superelliptic). These curves play a central role in some approaches to generalizations of Hilbert problems 7, 12, 21 and 22, in special di erential equations and many other researches 4], 5], 6], 9], 10], 11], 12]. The authors felt motivated to nd out, to which extent the Picard curves (or more general, the n-gonal cyclic curves) could share with the hyperelliptic curves the nice property o f m a king things to become explicit. In this paper, we s h o w our rst results in this direction. We wish to thank R.-P. Holzapfel for his valuable comments, discussions and encouragement. 2 Preliminaries Let k be an arbitrary eld. we write k for its algebraic closure. The a ne space A n k consists of all the points f(x 1 x 2 ::: x n) x i 2 kg. P oints in the projective space P n k consist of equivalence classes of points in A n+1 k n f (0 0 :: 0)g where (x 0 x 1 : : x n) a n d (y 0 y 1 :: y n) a r e equivalent if there is c 2 k (c 6 = 0), such t h a t x i = cy i for all i = 0 1 :: n. Note that A 2 k is naturally embedded in P 2 k by the map (x y) ! (x y 1). If G 2 k x y] is an homogeneous polynomial then, G de ne an algebraic subset of P 2

Research paper thumbnail of The Emergence of Picard Jacobians in Cryptography

In this paper we present a new family of Jacobian Varieties defined over finite fields that provi... more In this paper we present a new family of Jacobian Varieties defined over finite fields that provides many elements whose group structure is suitable for cryptosystems based on the intractability of the discrete logarithm problem. Their security against several know attacks and the efficiency (and effectiveness) of the most important algorithms of the corresponding cryptosystems are discussed.

Research paper thumbnail of Interactive Mathematical Documents on the Web

Algebra, Geometry and Software Systems, 2003

This paper deals with our work on interactive mathematical documents. These documents accomodate ... more This paper deals with our work on interactive mathematical documents. These documents accomodate various sources, users, and mathematical services. Communication of mathematics between these entities is based on the OpenMath standard and Java technology. But, for the management of the communication, more protocols and tools are needed. We describe an architecture that serves as a framework for our work on interactive documents, and we report on what we have implemented so far.

Research paper thumbnail of Efficient Reduction on the Jacobian Variety of Picard Curves

Coding Theory, Cryptography and Related Areas, 2000

In this paper, a system of coordinates for the elements on the Jacobian Variety of Picard curves ... more In this paper, a system of coordinates for the elements on the Jacobian Variety of Picard curves is presented. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an explicit structure of abelian variety for the Jacobian, as well as an e cient algorithm for the reduction and addition of divisors. Exploiting the geometry of the Picard curves, a completely e ective reduction algorithm is developed, which w orks for curves dened over any ground eld k, with char(k) = 0 o r char(k) 6 = 3 .

Research paper thumbnail of On the Jacobian Varieties of Picard Curves: Explicit Addition Law and Algebraic Structure

Mathematische Nachrichten, 2010

In this paper a system of coordinates for the effective divisors on the Jacobian Variety of a Pic... more In this paper a system of coordinates for the effective divisors on the Jacobian Variety of a Picard curve is presented. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an explicit structure of algebraic variety on the Jacobian as well as an efficient algorithm for the addition of divisors.

Research paper thumbnail of The Emergence of Picard Jacobians in Cryptography

Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik eBooks, Feb 15, 2001

In this paper we present a new family of Jacobian Varieties defined over finite fields that provi... more In this paper we present a new family of Jacobian Varieties defined over finite fields that provides many elements whose group structure is suitable for cryptosystems based on the intractability of the discrete logarithm problem. Their security against several know attacks and the efficiency (and effectiveness) of the most important algorithms of the corresponding cryptosystems are discussed.

Research paper thumbnail of On the Jacobian Varieties of Picard curves

Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik eBooks, Oct 20, 2005

In this paper a system of coordinates for the e ective divisors on the Jacobian Variety of a Pica... more In this paper a system of coordinates for the e ective divisors on the Jacobian Variety of a Picard curve i s p r e s e n ted. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an explicit structure of algebraic variety on the Jacobian as well as an e cient algorithm for the addition of divisors. Supported partially by a D F G grant 1 n-gonal cyclic curves, also called cycloelliptic or superelliptic). These curves play a central role in some approaches to generalizations of Hilbert problems 7, 12, 21 and 22, in special di erential equations and many other researches 4], 5], 6], 9], 10], 11], 12]. The authors felt motivated to nd out, to which extent the Picard curves (or more general, the n-gonal cyclic curves) could share with the hyperelliptic curves the nice property o f m a king things to become explicit. In this paper, we s h o w our rst results in this direction. We wish to thank R.-P. Holzapfel for his valuable comments, discussions and encouragement. 2 Preliminaries Let k be an arbitrary eld. we write k for its algebraic closure. The a ne space A n k consists of all the points f(x 1 x 2 ::: x n) x i 2 kg. P oints in the projective space P n k consist of equivalence classes of points in A n+1 k n f (0 0 :: 0)g where (x 0 x 1 : : x n) a n d (y 0 y 1 :: y n) a r e equivalent if there is c 2 k (c 6 = 0), such t h a t x i = cy i for all i = 0 1 :: n. Note that A 2 k is naturally embedded in P 2 k by the map (x y) ! (x y 1). If G 2 k x y] is an homogeneous polynomial then, G de ne an algebraic subset of P 2

Research paper thumbnail of On the Jacobian Varieties of Picard curves

In this paper a system of coordinates for the e ective divisors on the Jacobian Variety of a Pica... more In this paper a system of coordinates for the e ective divisors on the Jacobian Variety of a Picard curve i s p r e s e n ted. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an explicit structure of algebraic variety on the Jacobian as well as an e cient algorithm for the addition of divisors. Supported partially by a D F G grant 1 n-gonal cyclic curves, also called cycloelliptic or superelliptic). These curves play a central role in some approaches to generalizations of Hilbert problems 7, 12, 21 and 22, in special di erential equations and many other researches 4], 5], 6], 9], 10], 11], 12]. The authors felt motivated to nd out, to which extent the Picard curves (or more general, the n-gonal cyclic curves) could share with the hyperelliptic curves the nice property o f m a king things to become explicit. In this paper, we s h o w our rst results in this direction. We wish to thank R.-P. Holzapfel for his valuable comments, discussions and encouragement. 2 Preliminaries Let k be an arbitrary eld. we write k for its algebraic closure. The a ne space A n k consists of all the points f(x 1 x 2 ::: x n) x i 2 kg. P oints in the projective space P n k consist of equivalence classes of points in A n+1 k n f (0 0 :: 0)g where (x 0 x 1 : : x n) a n d (y 0 y 1 :: y n) a r e equivalent if there is c 2 k (c 6 = 0), such t h a t x i = cy i for all i = 0 1 :: n. Note that A 2 k is naturally embedded in P 2 k by the map (x y) ! (x y 1). If G 2 k x y] is an homogeneous polynomial then, G de ne an algebraic subset of P 2

Research paper thumbnail of The Emergence of Picard Jacobians in Cryptography

In this paper we present a new family of Jacobian Varieties defined over finite fields that provi... more In this paper we present a new family of Jacobian Varieties defined over finite fields that provides many elements whose group structure is suitable for cryptosystems based on the intractability of the discrete logarithm problem. Their security against several know attacks and the efficiency (and effectiveness) of the most important algorithms of the corresponding cryptosystems are discussed.

Research paper thumbnail of Interactive Mathematical Documents on the Web

Algebra, Geometry and Software Systems, 2003

This paper deals with our work on interactive mathematical documents. These documents accomodate ... more This paper deals with our work on interactive mathematical documents. These documents accomodate various sources, users, and mathematical services. Communication of mathematics between these entities is based on the OpenMath standard and Java technology. But, for the management of the communication, more protocols and tools are needed. We describe an architecture that serves as a framework for our work on interactive documents, and we report on what we have implemented so far.

Research paper thumbnail of Efficient Reduction on the Jacobian Variety of Picard Curves

Coding Theory, Cryptography and Related Areas, 2000

In this paper, a system of coordinates for the elements on the Jacobian Variety of Picard curves ... more In this paper, a system of coordinates for the elements on the Jacobian Variety of Picard curves is presented. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an explicit structure of abelian variety for the Jacobian, as well as an e cient algorithm for the reduction and addition of divisors. Exploiting the geometry of the Picard curves, a completely e ective reduction algorithm is developed, which w orks for curves dened over any ground eld k, with char(k) = 0 o r char(k) 6 = 3 .

Research paper thumbnail of On the Jacobian Varieties of Picard Curves: Explicit Addition Law and Algebraic Structure

Mathematische Nachrichten, 2010

In this paper a system of coordinates for the effective divisors on the Jacobian Variety of a Pic... more In this paper a system of coordinates for the effective divisors on the Jacobian Variety of a Picard curve is presented. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an explicit structure of algebraic variety on the Jacobian as well as an efficient algorithm for the addition of divisors.