| 2011 |
Semiautomated editing in Wikipedia: The AutoWikiBrowser case (in English, 18 slides)
(abstract missing)
(abstract missing)
Lattices, elliptic curves over the complex numbers and isogeny graphs (in English, 24 pages)
(abstract missing)
DLP: From RSA to ECDLP and HCDLP (in German and English, 26 slides)(abstract missing)
We discuss the difficulty of the discrete logarithm problem in various finite fields. We also examine various attacks on ECDLP and focus on the isogeny attack.
| 2009 |
The discrete logarithm problem on isogenous hyperelliptic curves of genus 2 (in English, 53 slides)
In 2005, Jao, Miller, and Venkatesan proved that the DLP of elliptic curves with the same endomorhism ring is random reducible under the GRH. In this talk, we discuss a possible generalization of this result to hyperelliptic curves of genus 2 (and 3) defined over a finite field and show the difficulties involved. First, we explain the role of the endomorphism rings of the Jacobian and the polarization. Following the work of Jao, Miller and Venkatesan, we construct isogeny graphs for genus 2 curves. Specifically, we discuss the connection between isogenies and ideal classes in the Jacobian of these curves. This project is research in progress and we describe the current status of this research.
In 2005, Jao, Miller, and Venkatesan proved that the DLP of elliptic curves with the same endomorhism ring is random reducible under the GRH. In this talk, we discuss a possible generalization of this result to hyperelliptic curves of genus 2 (and 3) defined over a finite field and show the difficulties involved. First, we explain the role of the endomorphism rings of the Jacobian and the polarization. Following the work of Jao, Miller and Venkatesan, we construct isogeny graphs for genus 2 curves. Specifically, we discuss the connection between isogenies and ideal classes in the Jacobian of these curves. This project is research in progress and we describe the current status of this research.
The DLP on Curves with the same endomorphism ring: The genus 2 case (in English and partially in German, 49 slides)
We try to extend the result we presented in our last talk for higher genus curves. We give some background on the arithmetic of curves of high genus and discuss the discrete logarithm problem (DLP) in the divisor class group for curves over finite fields with Jacobian varieties having the same ring of endomorphisms. We strict ourselves to the genus 2 case with Jacobian of CM type and we present the work we have done so far. Finally, we explain which phenomena can occur for curves of genus 3.
We try to extend the result we presented in our last talk for higher genus curves. We give some background on the arithmetic of curves of high genus and discuss the discrete logarithm problem (DLP) in the divisor class group for curves over finite fields with Jacobian varieties having the same ring of endomorphisms. We strict ourselves to the genus 2 case with Jacobian of CM type and we present the work we have done so far. Finally, we explain which phenomena can occur for curves of genus 3.
The DLP on Curves with the same endomorphism ring: The genus 1 case (in English, 48 slides)
We consider elliptic curves with the same order over a finite filed and the same endomorphism ring. We ask whether the discrete logarithm problem has the same complexity. We present a result of Jao, Miller and Venkatesan who proved that the answer to our question is positive under Generalised Riemann Hypothesis. Possible generalisations on curves of higher genus will be discussed in a second talk.
We consider elliptic curves with the same order over a finite filed and the same endomorphism ring. We ask whether the discrete logarithm problem has the same complexity. We present a result of Jao, Miller and Venkatesan who proved that the answer to our question is positive under Generalised Riemann Hypothesis. Possible generalisations on curves of higher genus will be discussed in a second talk.
The DLP on Elliptic and Hyperelliptic Curves with the same endomorphism ring (in English, 39 slides)
We ask whether the discrete logarithm problem (DLP) in the divisor class group has the same complexity for all curves over finite fields with Jacobian varieties having the same ring of endomorphisms. We present a result of Jao, Miller and Venkatesan who proved that the answer to our question is positive for elliptic curves. We try to use the same methods to extend the result to the genus 2 case in the case that the Jacobian is of CM type and we present the work we have done so far. Finally, we explain which phenomena can occur for curves of genus 3.
We ask whether the discrete logarithm problem (DLP) in the divisor class group has the same complexity for all curves over finite fields with Jacobian varieties having the same ring of endomorphisms. We present a result of Jao, Miller and Venkatesan who proved that the answer to our question is positive for elliptic curves. We try to use the same methods to extend the result to the genus 2 case in the case that the Jacobian is of CM type and we present the work we have done so far. Finally, we explain which phenomena can occur for curves of genus 3.
| 2008 |
Iwasawa's theorem (in English, 15 pages)
We prove the Iwasava's Theorem, which describes the behaviour of the class number in an extension of a finite field.
[dvi] [ps] [pdf]
We prove the Iwasava's Theorem, which describes the behaviour of the class number in an extension of a finite field.
[dvi] [ps] [pdf]
Arithmetic of Quaternion Algebras: Orders and Ideals (in English, 17 pages)
The basics on the arithmetic on quaternion algebras is introduced: (maximal) orders, (principal) ideals, (reduced) norm/discriminant, ideal classes, etc.
[dvi] [ps] [pdf]
The basics on the arithmetic on quaternion algebras is introduced: (maximal) orders, (principal) ideals, (reduced) norm/discriminant, ideal classes, etc.
[dvi] [ps] [pdf]
The DLP on Elliptic Curves with the same order (in English, 20 pages)
We ask whether the discrete logarithm problem (DLP) has the same difficulty for all curves with the same order over a finite field. We present the result of Jao, Miller and Venkatesan who proved that the answer to our question is positive if you limit ourselves to curves with the same endomorphism ring.
[ps] [pdf]
We ask whether the discrete logarithm problem (DLP) has the same difficulty for all curves with the same order over a finite field. We present the result of Jao, Miller and Venkatesan who proved that the answer to our question is positive if you limit ourselves to curves with the same endomorphism ring.
[ps] [pdf]
| 2007 |
The Tensor Product Theorem (in English, 11 pages)
The Tensor Product Theorem from Flath asserts that if A is the adele ring of a global field F and G is a reductive algebraic group over F, then G(A) decomposes into a "restricted tensor product" of representations of the groups G(Fυ). We give a proof of the theorem.
[dvi] [ps] [pdf]
The Tensor Product Theorem from Flath asserts that if A is the adele ring of a global field F and G is a reductive algebraic group over F, then G(A) decomposes into a "restricted tensor product" of representations of the groups G(Fυ). We give a proof of the theorem.
[dvi] [ps] [pdf]
| 2006 |
Modular forms of weight 1 (in English, 22 pages)
We study modular forms and Galois representations over finite and fields and over the complex numbers. We give the proof of an important theorem from Serre and Deligne that in every modular form of weight 1 we can attach a linear representation. This representation is unique up to isomorphism.
[dvi] [ps] [pdf]
We study modular forms and Galois representations over finite and fields and over the complex numbers. We give the proof of an important theorem from Serre and Deligne that in every modular form of weight 1 we can attach a linear representation. This representation is unique up to isomorphism.
[dvi] [ps] [pdf]
| 2004 |
Primes of the form x2 + ny2 (in English, 24 pages)
We study ring class fields of orders in imaginary quadratic fields to determine which primes are of the form x2 + ny2, where x, y integers, for arbitrary n. We give certain examples how our result works in practice.
[dvi] [ps] [pdf]
We study ring class fields of orders in imaginary quadratic fields to determine which primes are of the form x2 + ny2, where x, y integers, for arbitrary n. We give certain examples how our result works in practice.
[dvi] [ps] [pdf]
Optimal linear codes over GF(4) (in Greek, 18 pages)
A central problem in coding theory is that of finding the smallest length for which there exists a linear code of dimension k and minimum distance d, over a filed of q elements. We consider here the problem for quaternary codes (q = 4), solving the problem for k < 5 for all values of d.
[doc] [ps] [pdf]
A central problem in coding theory is that of finding the smallest length for which there exists a linear code of dimension k and minimum distance d, over a filed of q elements. We consider here the problem for quaternary codes (q = 4), solving the problem for k < 5 for all values of d.
[doc] [ps] [pdf]
| 2003 |
Primality test (Algorithms and Complexity) (in Greek, 26 pages)
We consider the primality problem, to decide whether a number is prime or composite. In this survey we show that PRIMES is in coNP and in NP. Then we try a probabilistic approach and we show that PRIMES is in coRP and in ZPP. Finally we present one of the most significant results of the last years: that PRIMES is in P.
Last update: Aug 31, 2005
[doc] [mdi] [pdf]
We consider the primality problem, to decide whether a number is prime or composite. In this survey we show that PRIMES is in coNP and in NP. Then we try a probabilistic approach and we show that PRIMES is in coRP and in ZPP. Finally we present one of the most significant results of the last years: that PRIMES is in P.
Last update: Aug 31, 2005
[doc] [mdi] [pdf]
Smooth numbers and the quadratic sieve (in Greek, 12 pages)
With the help of Analytic Number Theory we consider the problem of optimizing the bound used in the quadratic sieve to factorise numbers.
[doc] [ps] [pdf]
With the help of Analytic Number Theory we consider the problem of optimizing the bound used in the quadratic sieve to factorise numbers.
[doc] [ps] [pdf]
The main linear coding theory problem (in Greek, 27 pages)
Central problem in coding theory is that of constructing optimal codes for a variable (length, dimension, minimum distance), over a field of q elements, while keeping the other two constant. Here we present one version of the problem, with the help of Finite Geometries, and all the known results until now.
Last update: Jan 14, 2004
[doc] [ps] [pdf]
Central problem in coding theory is that of constructing optimal codes for a variable (length, dimension, minimum distance), over a field of q elements, while keeping the other two constant. Here we present one version of the problem, with the help of Finite Geometries, and all the known results until now.
Last update: Jan 14, 2004
[doc] [ps] [pdf]
| 2002 |
