## Research Statement

### BibTeX

@MISC{Uzunkol_researchstatement,

author = {Osmanbey Uzunkol},

title = {Research Statement},

year = {}

}

### OpenURL

### Abstract

My research is mainly in algorithmic number theory and arithmetic geometry with particular interest in cryptography, coding theory, complex multiplication theory and explicit class field theory. This outline is intended as a brief description of the research projects I have undertaken. It projects forward to research projects I am currently working on and others I am planning to undertake after my Phd. 1 Construction of elliptic and hyperelliptic curves over finite fields with CM method Since Kronecker, number theorists have kept exploiting the idea of generating abelian extensions of number fields k by means of special values of appropriately chosen analytic functions. In the simplest case, i. e. k = Q, the Kronecker-Weber theorem says that the abelian extensions of Q are completely classified by using the special values of the transcendental function z ↦ → e 2πiz at points of finite order on the circle R/Z, see [Gr]. Hence, the question of extending this theorem to any base number field k, i. e. the famous Hilbert’s 12th problem, can be formulated whether abelian extensions of k can be generated by adjoining torsion points of suitable abelian groups.

### Citations

494 |
Introduction to the arithmetic theory of automorphic functions
- Shimura
- 1971
(Show Context)
Citation Context ... Ô ∗ t = ∏ p (Ot ⊗Z Zp) ∗ of ̂ k ∗ is the inverse image under A of G(k ab /Ωt), hence one needs to check whether the action of Ô∗ t on f(τ) induced by A is trivial. Now Shimura’s reciprocity law, see =-=[Sh]-=-, p. 160, says that the image of x ∈ Ô∗ t can be obtained as the value in τ of a modular function that is conjugate to f over Q(j). We have precisely a map ξ from Ô∗ t to GL2( ̂ Z) with and a fundamen... |

231 |
Advanced Topics in the Arithmetic of Elliptic Curves
- Silverman
- 1994
(Show Context)
Citation Context ...ulated whether abelian extensions of k can be generated by adjoining torsion points of suitable abelian groups. Besides Q, this problem is completely solved only for imaginary quadratic fields k, see =-=[Sil2]-=-, by using the theory of complex multiplication. The maximal abelian extension of k = Q(τ), where τ is an imaginary quadratic number on the upper half plane H, can be obtained by adjoining the special... |

111 | Elliptic Functions - Lang - 1973 |

100 | Algorithmic algebraic number theory - Pohst, Zassenhaus - 1989 |

91 |
Abelian Varieties With Complex Multiplication And Modular Functions
- Shimura
- 1998
(Show Context)
Citation Context ...modular functions. These modular functions play the role of exponential function in the case of imaginary quadratic fields. The theory of complex multiplication can be generalized to CM fields k, see =-=[Sh1]-=-. However, in general there are no known modular functions, which give the maximal abelian extension of a given CM field k, if [k : Q] > 2. The first main theorem of complex multiplication implies tha... |

78 | A taxonomy of pairing-friendly elliptic curves. Preprint 2006, Available at http://eprint.iacr
- Freeman, Scott, et al.
- 1994
(Show Context)
Citation Context ...ptic curves with a known number of rational points with the theory of complex multiplication, see [AtMr] and [Mor] (see also for applications in group and pairing based cryptography [BSS1],[BSS2] and =-=[FST]-=-). An elliptic curve E over C is analytically isomorphic to a complex torus C/Λ with the 1endomorphism ring {α ∈ C|αΛ ⊆ Λ} ∼ = Z or O, where O is an order in Q(τ) and {τ, 1}, τ ∈ H, forms a basis of ... |

54 |
Recognizing primes in random polynomial time
- Adleman, Huang
- 1987
(Show Context)
Citation Context ...method of genus 2, which would enable to propose hyperelliptic curves for use in pairing based cryptosystems. 3.3 Primality Proving Adleman and Huang give a probabilistic polynomial time algorithm in =-=[Ahu]-=-, which can prove or disprove that a given number N is a prime. Using the CM method I want to give a first practical version of this algorithm, like the practical version of primality proving using el... |

33 | The complexity of class polynomial computation via floating point approximations
- Enge
(Show Context)
Citation Context ...rder N. The most time consuming step when constructing elliptic curves with CM-method is the construction of the polynomial HD(x). There are 3 methods to construct HD(x), namely complex analytic, see =-=[Enge1]-=-, using p−adic lifting algorithm, see [Brk], and a CRT-method, see [Ag]. 1.2 Hyperelliptic curve construction The efficient generation of genus 2 hyperelliptic curves with a Jacobian of nearly prime o... |

27 | Implementing the asymptotically fast version of the elliptic curve primality proving algorithm
- MORAIN
(Show Context)
Citation Context ...field and computing the number of rational points, it is possible to construct special elliptic curves with a known number of rational points with the theory of complex multiplication, see [AtMr] and =-=[Mor]-=- (see also for applications in group and pairing based cryptography [BSS1],[BSS2] and [FST]). An elliptic curve E over C is analytically isomorphic to a complex torus C/Λ with the 1endomorphism ring ... |

22 |
Die Klassenkörper der komplexen Multiplikation
- Deuring
- 1958
(Show Context)
Citation Context ...imal polynomial of g(τ) is W−204(x) = x 6 − 16 · x 5 − 12 · x 4 + 48 · x 3 + 144 · x 2 + 64 · x + 64, which has clearly much smaller coefficients compared to H−204(x). Using a theorem of Deuring, see =-=[Deu]-=-, p. 43, I proved the following theorems, which prove that most of these class invariants are units in the corresponding ring class fields, Theorem 4. Let g(τ) be one of the class invariants as in the... |

22 | Constructing elliptic curves over finite fields using double eta-quotients, Journal de Théorie des Nombres de Bordeaux 16 - Enge, Schertz - 2004 |

19 | A CRT algorithm for constructing genus 2 curves over finite fields
- Eisenträger, Lauter
(Show Context)
Citation Context ...ariants, i = 1, 2, 3. Like in the case of elliptic curve construction, there are 3 methods to compute these polynomails, namely analytic, see [Weng], p-adic, see [Gau] and [CaKoL], and CRT based, see =-=[EiLa]-=-. After solving the roots of the polynomials over a chosen prime field and using Mestre’s algorithm, it is possible to find a model of the hyperelliptic curve for which the order of its Jacobian has b... |

18 | Constructing elliptic curves with known number of points over a prime field, High primes and misdemeanours: lectures in honour of the 60th birthday
- Agashe, Lauter, et al.
- 2004
(Show Context)
Citation Context ...h CM-method is the construction of the polynomial HD(x). There are 3 methods to construct HD(x), namely complex analytic, see [Enge1], using p−adic lifting algorithm, see [Brk], and a CRT-method, see =-=[Ag]-=-. 1.2 Hyperelliptic curve construction The efficient generation of genus 2 hyperelliptic curves with a Jacobian of nearly prime order for use in cryptography is an important problem. For curves over f... |

13 |
Konstruktion kryptographisch geeigneter Kurven mit komplexer Multiplikation
- Weng
- 2001
(Show Context)
Citation Context ...ction of polynomials Hi(x) of the absolute Igusa class invariants, i = 1, 2, 3. Like in the case of elliptic curve construction, there are 3 methods to compute these polynomails, namely analytic, see =-=[Weng]-=-, p-adic, see [Gau] and [CaKoL], and CRT based, see [EiLa]. After solving the roots of the polynomials over a chosen prime field and using Mestre’s algorithm, it is possible to find a model of the hyp... |

12 |
Fast evaluation of modular functions using Newton iterations and the AGM
- Dupont
- 2006
(Show Context)
Citation Context ...ere gi := g(τi), The asymptotically fastest algorithm for the numerical computation of the n most significant bits of one of the Thetanullwerte and Dedekind η− function evaluated at τ ∈ H is given by =-=[Dup]-=-. We showed that we can evaluate the class invariants as quotients of Thetanullwerte more efficiently than the invariants as quotients of values of the ηfunction, see [Uz] or [Uz1]. As an example we c... |

11 |
On zeta-functions of algebraic number fields
- Brauer
- 1947
(Show Context)
Citation Context ...ts of the class polynomials HD(x) are huge compared to the field discriminant. We have approximately √ |D| coefficients to compute, as Brauer-Siegel theorem says that ht grows like |D| 1/2+o(1) , see =-=[Br]-=-. Even worse, the coefficients grow exponentiallly in the size of |D|. As an example, we have the following class polynomial for D = −204 H−204(x) = x 6 − 30703802307926880672 · x 5 + 9586484163799611... |

8 | Die singulären Werte der Weberschen Funktionen f, f1, f2, γ2, γ3, Journal für die reine und angewandte Mathematik 286/287 - Schertz - 1976 |

7 |
Weber’s class invariants revisited. Journal de Théorie des Nombres de
- Schertz
- 2002
(Show Context)
Citation Context ...eorem of complex multiplication. Hence, its minimal polynomial HD(x) has coefficients in Z. We know G(Ωt/k) ∼ = Clt, where Clt is the ring class group modulo t with ht := |Clt| and Ωt := k(j(τ)), see =-=[Sch]-=-. Accordingly, j(τi), i = 1, 2, · · · , ht form a complete system of conjugates over k and also over Q, which means ht ∏ HD(x) = (x − j(τi)) ∈ Z[x]. i=1 In order to construct an elliptic curve over Fp... |

7 |
Silverman: The Arithmetic of Elliptic Curves
- H
- 1992
(Show Context)
Citation Context ...alytically isomorphic to a complex torus C/Λ with the 1endomorphism ring {α ∈ C|αΛ ⊆ Λ} ∼ = Z or O, where O is an order in Q(τ) and {τ, 1}, τ ∈ H, forms a basis of the lattice Λ up to homothety, see =-=[Sil1]-=-, p. 164. If the endomorphism ring is larger than Z, we say that E has a complex multiplication. Let E be an ordinary elliptic curve over Fp, then by Deuring lifting theorem, we know that there exists... |

3 |
An elementary proof of the Kronecker-Weber theorem
- Greenberg
- 1974
(Show Context)
Citation Context ...r-Weber theorem says that the abelian extensions of Q are completely classified by using the special values of the transcendental function z ↦→ e 2πiz at points of finite order on the circle R/Z, see =-=[Gr]-=-. Hence, the question of extending this theorem to any base number field k, i. e. the famous Hilbert’s 12th problem, can be formulated whether abelian extensions of k can be generated by adjoining tor... |

2 |
Courbe Algébriques et Cryptologie
- Enge
(Show Context)
Citation Context ..., 3, 5, 7, 11, 13 and 17 and τ ∈ H, the generalized Weber functions are defined as wl := η( τ l ) η(τ) . (Note that wl = f1 for l = 2.) Using these functions, one can obtain new class invariants (see =-=[Enge]-=-, p. 15 and 16 or [GeSt], p. 450). In [Uz] and [Uz2], the possibility of representing these class invariants as quotients of Thetanullwerte is discussed. Moreover, it is shown that most of these invar... |

2 |
Generating Class Fields Using
- Gee, Stevenhagen
- 1998
(Show Context)
Citation Context ... and τ ∈ H, the generalized Weber functions are defined as wl := η( τ l ) η(τ) . (Note that wl = f1 for l = 2.) Using these functions, one can obtain new class invariants (see [Enge], p. 15 and 16 or =-=[GeSt]-=-, p. 450). In [Uz] and [Uz2], the possibility of representing these class invariants as quotients of Thetanullwerte is discussed. Moreover, it is shown that most of these invariants are units as in th... |

2 | On the computation of class polynomials with “Thetanullwerte” and its applications to the unit group computation
- Leprévost, Pohst, et al.
(Show Context)
Citation Context ...)θ10(τ) , F2(τ) := 2θ10(τ)2 θ00(τ)θ01(τ) . I proved the following theorem, which shows that we have an identity between modified Schläfli functions and sixth powers of Schläfli functions, see [Uz] or =-=[Uz1]-=-, Theorem 2. For τ ∈ H: • F(τ) = f(τ) 6 , • F1(τ) = f1(τ) 6 , • F2(τ) = f2(τ) 6 . 2.2 Class invariants with Thetanullwerte Using Theorem 2, and the class invariants constructed as values of small powe... |

2 |
Über die Konstruktion algebraischer Kurven mittels komplexer Multiplikation
- Uzunkol
- 2010
(Show Context)
Citation Context ...)2 θ00(τ)θ10(τ) , F2(τ) := 2θ10(τ)2 θ00(τ)θ01(τ) . I proved the following theorem, which shows that we have an identity between modified Schläfli functions and sixth powers of Schläfli functions, see =-=[Uz]-=- or [Uz1], Theorem 2. For τ ∈ H: • F(τ) = f(τ) 6 , • F1(τ) = f1(τ) 6 , • F2(τ) = f2(τ) 6 . 2.2 Class invariants with Thetanullwerte Using Theorem 2, and the class invariants constructed as values of s... |

1 |
Constructing Elliptic Curves of Prescribed Order
- Bröke
- 2006
(Show Context)
Citation Context ...ism ring Ot. We compute a root j of H (p+1−N) 2 −4p(x) modulo p. Then, for j ̸= 0, 1, the elliptic curve y 2 = x 3 + ax − a, a = 27j/(4(1728 − j)) or its quadratic twist is a curve with N points, see =-=[Brk]-=-, p. 38. Similarly, for j = 0 (j = 1728), the elliptic curve y 2 = x 3 + 1 (resp. y 2 = x 3 + x) or one of its twists is a curve of order N. The most time consuming step when constructing elliptic cur... |

1 |
Higher Dimensional 3−adic
- Carls, Kohel, et al.
(Show Context)
Citation Context ...the absolute Igusa class invariants, i = 1, 2, 3. Like in the case of elliptic curve construction, there are 3 methods to compute these polynomails, namely analytic, see [Weng], p-adic, see [Gau] and =-=[CaKoL]-=-, and CRT based, see [EiLa]. After solving the roots of the polynomials over a chosen prime field and using Mestre’s algorithm, it is possible to find a model of the hyperelliptic curve for which the ... |

1 | Pohst A Lower Regulator Bound for Number Fields - Fieker, M |

1 |
The 2−adic CM Method for Genus 2 Curves with Apllication to Cryptography
- Gaudry, Houtmann, et al.
- 2006
(Show Context)
Citation Context ... Hi(x) of the absolute Igusa class invariants, i = 1, 2, 3. Like in the case of elliptic curve construction, there are 3 methods to compute these polynomails, namely analytic, see [Weng], p-adic, see =-=[Gau]-=- and [CaKoL], and CRT based, see [EiLa]. After solving the roots of the polynomials over a chosen prime field and using Mestre’s algorithm, it is possible to find a model of the hyperelliptic curve fo... |

1 | On the Generalized Class Invariants with ”Thetanullwerte
- Uzunkol
(Show Context)
Citation Context ...eber functions are defined as wl := η( τ l ) η(τ) . (Note that wl = f1 for l = 2.) Using these functions, one can obtain new class invariants (see [Enge], p. 15 and 16 or [GeSt], p. 450). In [Uz] and =-=[Uz2]-=-, the possibility of representing these class invariants as quotients of Thetanullwerte is discussed. Moreover, it is shown that most of these invariants are units as in theorem 4, and that better cla... |