Zeta functions of algebraic cycles over finite fields

1992 ◽  
Vol 74 (1) ◽  
pp. 413-444 ◽  
Author(s):  
Daqing Wan
Keyword(s):  
Finite Fields ◽  
Zeta Functions ◽  
Algebraic Cycles

Rational points and zeta functions of some curves over finite fields

2010 ◽  
Vol 53 (11) ◽  
pp. 2855-2863
Author(s):  
Long Wang ◽  
JinQuan Luo
Keyword(s):  
Finite Fields ◽  
Zeta Functions ◽  
Rational Points ◽  
Curves Over Finite Fields

scholarly journals Universal Equivalence and Majority of Probabilistic Programs over Finite Fields

2022 ◽  
Vol 23 (1) ◽  
pp. 1-42
Author(s):  
Gilles Barthe ◽  
Charlie Jacomme ◽  
Steve Kremer
Keyword(s):  
Programming Language ◽  
Finite Fields ◽  
Differential Privacy ◽  
Zeta Functions ◽  
Equivalence Problem ◽  
Linear Recurrence ◽  
Universal Equivalence ◽  
Local Zeta Functions ◽  
Recurrence Sequences ◽  
Probabilistic Programs

We study decidability problems for equivalence of probabilistic programs for a core probabilistic programming language over finite fields of fixed characteristic. The programming language supports uniform sampling, addition, multiplication, and conditionals and thus is sufficiently expressive to encode Boolean and arithmetic circuits. We consider two variants of equivalence: The first one considers an interpretation over the finite field F q , while the second one, which we call universal equivalence, verifies equivalence over all extensions F q k of F q . The universal variant typically arises in provable cryptography when one wishes to prove equivalence for any length of bitstrings, i.e., elements of F 2 k for any k . While the first problem is obviously decidable, we establish its exact complexity, which lies in the counting hierarchy. To show decidability and a doubly exponential upper bound of the universal variant, we rely on results from algorithmic number theory and the possibility to compare local zeta functions associated to given polynomials. We then devise a general way to draw links between the universal probabilistic problems and widely studied problems on linear recurrence sequences. Finally, we study several variants of the equivalence problem, including a problem we call majority, motivated by differential privacy. We also define and provide some insights about program indistinguishability, proving that it is decidable for programs always returning 0 or 1.

scholarly journals Partitions and Compositions over Finite Fields

10.37236/2678 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Amela Muratovic-Ribic ◽  
Qiang Wang
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Zeta Functions ◽  
Exact Formula

In this paper we  find an exact formula for the number of partitions of an element  $z$ into $m$ parts over a finite field,  i.e. we find the number of nonzero solutions of the equation $x_1+x_2+\cdots +x_m=z$ over a finite field when the order of terms does not matter. This is equivalent to counting the number of $m$-multi-subsets whose sum is $z$.  When the order of the terms in a solution does matter, such a solution is called a composition of $z$. The number of compositions is useful in the study of zeta functions of toric hypersurfaces over finite fields.  We also give an application in the study of polynomials of prescribed ranges over finite fields.

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.

Classification of zeta functions of bielliptic surfaces over finite fields

2016 ◽  
Vol 99 (3-4) ◽  
pp. 397-405 ◽  
Author(s):  
S. Yu. Rybakov
Keyword(s):  
Finite Fields ◽  
Zeta Functions
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