scholarly journals Moments of zeta functions associated to hyperelliptic curves over finite fields

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140307 ◽  
Author(s):  
Michael O. Rubinstein ◽  
Kaiyu Wu
Keyword(s):  
Asymptotic Formula ◽  
Finite Fields ◽  
Remainder Term ◽  
Prime Power ◽  
Zeta Functions ◽  
Number Fields ◽  
Hyperelliptic Curves ◽  
Central Point ◽  
Characteristic Polynomials ◽  
Curves Over Finite Fields

Let q be an odd prime power, and denote the set of square-free monic polynomials D ( x )∈ F q [ x ] of degree d . Katz and Sarnak showed that the moments, over , of the zeta functions associated to the curves y 2 = D ( x ), evaluated at the central point, tend, as , to the moments of characteristic polynomials, evaluated at the central point, of matrices in USp (2⌊( d −1)/2⌋). Using techniques that were originally developed for studying moments of L -functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments for q fixed and . We provide theoretical and numerical evidence in favour of their conjecture. In some cases, we are able to work out exact formulae for the moments and use these to precisely determine the size of the remainder term in the predicted moments.

Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields

2007 ◽  
Vol 50 (3) ◽  
pp. 409-417 ◽  
Author(s):  
Florian Luca ◽  
Igor E. Shparlinski
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Complex Multiplication ◽  
Number Fields ◽  
Quadratic Number ◽  
Endomorphism Rings ◽  
Quadratic Number Field ◽  
Curves Over Finite Fields

AbstractWe show that, for most of the elliptic curves E over a prime finite field p of p elements, the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over p is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over p.

A Note on Relations Between the Zeta-Functions of Galois Coverings of Curves Over Finite Fields

1990 ◽  
Vol 33 (3) ◽  
pp. 282-285 ◽  
Author(s):  
Amilcar Pacheco
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Group Ring ◽  
Algebraic Curve ◽  
Zeta Functions ◽  
Finite Subgroup ◽  
Rational Group ◽  
Galois Coverings ◽  
Curves Over Finite Fields ◽  
Special Case

AbstractLet C be a complete irreducible nonsingular algebraic curve defined over a finite field k. Let G be a finite subgroup of the group of automorphisms Aut(C) of C. We prove that certain idempotent relations in the rational group ring Q[G] imply other relations between the zeta-functions of the quotient curves C/H, where H is a subgroup of G. In particular we generalize some results of Kani in the special case of curves over finite fields.

scholarly journals Computing the Characteristic Polynomials of a Class of Hyperelliptic Curves for Cryptographic Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Lin You ◽  
Guangguo Han ◽  
Jiwen Zeng ◽  
Yongxuan Sang
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Hyperelliptic Curve ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Common Method ◽  
Characteristic Polynomials ◽  
Computational Data ◽  
Prime Factors

Hyperelliptic curves have been widely studied for cryptographic applications, and some special hyperelliptic curves are often considered to be used in practical cryptosystems. Computing Jacobian group orders is an important operation in constructing hyperelliptic curve cryptosystems, and the most common method used for the computation of Jacobian group orders is by computing the zeta functions or the characteristic polynomials of the related hyperelliptic curves. For the hyperelliptic curveCq:v2=up+au+bover the fieldFqwithqbeing a power of an odd primep, Duursma and Sakurai obtained its characteristic polynomial forq=p,a=−1,andb∈Fp. In this paper, we determine the characteristic polynomials ofCqover the finite fieldFpnforn=1, 2 anda,b∈Fpn. We also give some computational data which show that many of those curves have large prime factors in their Jacobian group orders, which are both practical and vital for the constructions of efficient and secure hyperelliptic curve cryptosystems.

Sign in / Sign up

Export Citation Format

Share Document