## Computing Zeta Functions Over Finite Fields (0)

Venue: | Contemporary Mathematics |

Citations: | 15 - 3 self |

### BibTeX

@ARTICLE{Wan_computingzeta,

author = {Daqing Wan},

title = {Computing Zeta Functions Over Finite Fields},

journal = {Contemporary Mathematics},

year = {},

volume = {225},

pages = {131--141}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subject Classification: 11Y16, 11T99, 14Q15. 1. Introduction Let p be a prime number. Let F q be a finite field of q elements of characteristic p. Let X be an algebraic variety defined over F q , say an affine variety defined by the vanishing of r polynomials in n variables: f 1 (x 1 ; \Delta \Delta \Delta ; xn ) = \Delta \Delta \Delta = f r (x 1 ; \Delta \Delta \Delta ; xn ) = 0; where the polynomials f i have coefficients in F q . Let N(X) denote the number of F q -rational points on X . Problem I. Compute N(X) efficiently. In addition to its intrinsic theoretical interest, this fundamental algorithmic problem has important applications in diverse areas such as coding theory, cryptography, ...

### Citations

762 | Elliptic curve cryptosystems - Koblitz |

262 |
La conjecture de
- Deligne
- 1980
(Show Context)
Citation Context ... explicitly using p-adic methods as done by Bombieri [Bo]. The size of the coefficients in Z(X;T ) can also be bounded easily using trivial estimate. In nicer cases, one can use Deligne's deep result =-=[De]-=- to get better bounds. The general modulo idea to compute Z(X;T ) consists of two steps. The first step is to compute Z(X;T ) modulo various small primes (or prime powers). The second step is to use t... |

181 |
Elliptic curves over finite fields and the computation of square roots mod p
- Schoof
- 1985
(Show Context)
Citation Context ...arieties. In the special case that X is an elliptic curve E : y 2 = x 3 + ax + b; a; b 2 F q ; c fl1997 American Mathematical Society 1 2 DAQING WAN a polynomial time algorithm was obtained by Schoof =-=[Sc1]-=-. More practical but probabilistic versions were obtained later by Atkin, Elkies, Couveignes and a few other authors, see [Sc2] for an updated exposition. Schoof 's algorithm was generalized to abelia... |

156 | Hyperelliptic cryptosystems - Koblitz - 1989 |

91 |
On the rationality of the zeta function of an algebraic variety
- Dwork
- 1960
(Show Context)
Citation Context ... large k. Nevertheless, one can still hope to find efficient algorithms in important cases that arise from various applications. The sequence N k (X) has a nice structure. In fact, by Dwork's theorem =-=[Dw]-=-, the generating zeta function Z(X;T ) = exp( 1 X k=1 N k (X) k T k ) is a rational function. Thus, the sequence N k (X) satisfies a linear recurrence relation. Write Z(X;T ) = g(T ) h(T ) for some po... |

89 | Counting points on elliptic curves over finite fields
- Schoof
- 1995
(Show Context)
Citation Context ...ociety 1 2 DAQING WAN a polynomial time algorithm was obtained by Schoof [Sc1]. More practical but probabilistic versions were obtained later by Atkin, Elkies, Couveignes and a few other authors, see =-=[Sc2]-=- for an updated exposition. Schoof 's algorithm was generalized to abelian varieties and curves by Pila [Pi] with some improvements by Adleman-Huang [AH]. Curves and abelian varieties are the cases wh... |

54 |
Frobenius maps of Abelian varieties and finding roots of unity in finite fields
- Pila
- 1990
(Show Context)
Citation Context ...tic versions were obtained later by Atkin, Elkies, Couveignes and a few other authors, see [Sc2] for an updated exposition. Schoof 's algorithm was generalized to abelian varieties and curves by Pila =-=[Pi]-=- with some improvements by Adleman-Huang [AH]. Curves and abelian varieties are the cases which have been studied most extensively in the literature. For more general varieties, no non-trivial algorit... |

50 |
Formule de Lefschetz et rationalité des fonctions L , Sém
- Grothendieck
- 1964
(Show Context)
Citation Context ...spective about the potential of these methods in computing zeta functions. No results and/or theorems are given in this section. `-adic methods. One can try to use Grothendieck's `-adic trace formula =-=[Gr]-=- modulo ` m : Z(X;T ) j 2dim(X) Y i=0 det(I \Gamma FT jH i c (X\Omega FqsF q ; Z=` m Z) (\Gamma1) i\Gamma1 ( mod ` m ); 4 DAQING WAN where H i c denotes the `-adic 'etale cohomology with compact suppo... |

48 |
Factoring polynomials over finite fields
- Berlekamp
(Show Context)
Citation Context ...urns out that the subspace R 1 (F ) is also useful for the harder problem of factoring f(x) into irreducible factors over F q . COMPUTING ZETA FUNCTIONS OVER FINITE FIELDS 7 Theorem 3.1.4. (Berlekamp =-=[Be]-=-). The eigenspace R 1 (F ) can be used to factor f(x) over F q . The idea is as follows. By the Chinese remainder theorem, the eigenspace R 1 (F ) has a basis of the form fy 1 (x) f(x) f 1 (x) a1 ; \D... |

47 |
On exponential sums in finite fields
- Bombieri
- 1966
(Show Context)
Citation Context ...odular approach for computing zeta functions Let X be an algebraic variety defined over a finite field F q . The degree of Z(X;T ) can be estimated explicitly using p-adic methods as done by Bombieri =-=[Bo]-=-. The size of the coefficients in Z(X;T ) can also be bounded easily using trivial estimate. In nicer cases, one can use Deligne's deep result [De] to get better bounds. The general modulo idea to com... |

36 |
Formal cohomology
- Monsky, Washnitzer
- 1968
(Show Context)
Citation Context ...s approach. p-adic methods. There are various p-adic formulas for Z(X;T ). All of them can be made to be explicit in some sense. The earliest one is due to Dwork, with generalizations by Reich-Monsky =-=[Mo]-=-. There are also p-adic cohomological formulas in terms of formal cohomology, crystalline cohomology and rigid cohomology [Ber]. We believe that some of these formulas and their variants can be very u... |

35 |
Géométrie rigide et cohomologie des variétés algebriques de caractéristique p, in Introductions aux cohomologies p-adiques (Luminy
- Berthelot
- 1984
(Show Context)
Citation Context ...e. The earliest one is due to Dwork, with generalizations by Reich-Monsky [Mo]. There are also p-adic cohomological formulas in terms of formal cohomology, crystalline cohomology and rigid cohomology =-=[Ber]-=-. We believe that some of these formulas and their variants can be very useful in computing the zeta function in the general case if the characteristic p is not too large. In particular, we expect it ... |

23 |
Travaux de Dwork, Seminaire Bourbaki
- Katz
- 1971
(Show Context)
Citation Context ...is not suitable for computing the whole zeta function. This is because the p-adic 'etale formula gives only the p-adic unit root part of the zeta function, not the whole thing, as conjectured by Katz =-=[Ka1]-=- and proved in the zeta case by Etesse-Le Stum [ES]. Our general feeling is as follow. If p is small (q can be large), p-adic methods should be much more efficient. If p is large and the degree of X i... |

22 |
Counting rational points on curves and Abelian varieties over finite
- Adleman, Huang
- 1996
(Show Context)
Citation Context ...kies, Couveignes and a few other authors, see [Sc2] for an updated exposition. Schoof 's algorithm was generalized to abelian varieties and curves by Pila [Pi] with some improvements by Adleman-Huang =-=[AH]-=-. Curves and abelian varieties are the cases which have been studied most extensively in the literature. For more general varieties, no non-trivial algorithm is known unless the variety is defined ove... |

19 |
B.: Fonctions L associées aux F-isocristaux surconvergents
- Etesse, Stum
- 1993
(Show Context)
Citation Context ...n. This is because the p-adic 'etale formula gives only the p-adic unit root part of the zeta function, not the whole thing, as conjectured by Katz [Ka1] and proved in the zeta case by Etesse-Le Stum =-=[ES]-=-. Our general feeling is as follow. If p is small (q can be large), p-adic methods should be much more efficient. If p is large and the degree of X is small, then `-adic methods should be better assum... |

18 | Meromorphic continuation of L-functions of p-adic representations
- Wan
- 1996
(Show Context)
Citation Context ...s proves that / q (f q\Gamma1 h) 2 R(d). The following simple congruence formula for zeta functions can be easily proved using the reduction of the Dwork trace formula. It is implicit in section 7 of =-=[Wa2]-=-. Theorem 4.2. We have the congruence formula Z(X;T ) (\Gamma1) n j det(I \Gamma (/ q ffi f q\Gamma1 )T jR(d)) ( mod p): In the projective case, a similar congruence formula but with a much harder pro... |

14 | Computational aspects of curves of genus at least 2
- Poonen
- 1996
(Show Context)
Citation Context ...r a small finite field F q (q = 2 or 3 for instance), computing N 1 (X) is easy since q is small. Even in this special case, computing Z(X;T ) already seems to be difficult as Katz-Sarnak pointed out =-=[Po]-=-. This case was motivated by their investigation of the distribution of the zeros and poles of zeta functions. The results of Adleman-Huang give a doubly exponential algorithm for a general plane curv... |

12 |
Zeta functions, one-way functions and pseudorandom number generators
- Anshel, Goldfeld
- 1997
(Show Context)
Citation Context ...gree is large. Computing the zeta function of a hyperellitptic curve is very useful in the hyperelliptic curve cryptosystem proposed by Koblitz [Ko1-2]. Additional applications may be found in [AH2], =-=[AG]-=- and [Ts]. Computing zeta functions of algebraic varieties should also be the first key step in computing zeta functions of more general Hilbert sets and definable sets, see [Wa1] and [FHJ]. A very sp... |

11 | Counting curves and their projections
- GATHEN, KARPINSKI, et al.
- 1996
(Show Context)
Citation Context ...s which have been studied most extensively in the literature. For more general varieties, no non-trivial algorithm is known unless the variety is defined over a small subfield of F q . It is shown in =-=[GKS]-=- that this problem is #P-hard (at least NP-hard) for sparse plane curves if one uses sparse input size. Thus, in order to get efficient algorithms to compute N(X), one might need to put some restricti... |

10 |
Entireness of L-functions of ϕ-sheaves on affine complete intersections
- Taguchi, Wan
- 1997
(Show Context)
Citation Context ... projective case, a similar congruence formula but with a much harder proof is given by Katz [Ka2]. Very general congruence formulas but acting on infinite dimensional space can be found in [TW1] and =-=[TW2]-=-. Using Theorem 4.2, we get Corollary 4.3. The zeta function Z(X, T) modulo p can be computed in time that is a polynomial in ( ) d n log q (the input and output size) if p is small and n is fixed. Th... |

6 |
Effective counting of the points of definable sets over finite fields
- Fried, Haran, et al.
(Show Context)
Citation Context ...d in [AH2], [AG] and [Ts]. Computing zeta functions of algebraic varieties should also be the first key step in computing zeta functions of more general Hilbert sets and definable sets, see [Wa1] and =-=[FHJ]-=-. A very special case is already considered in [GKS]. Our purpose here is to give a brief general discussion of the modular approach for computing zeta functions, indicating some theoretical tools tha... |

5 |
Factoring polynomials over finite fields using differential equations and normal bases
- Niederreiter
- 1994
(Show Context)
Citation Context ...as the admissible basis ff 0 1 (x) f(x) f 1 (x) ; \Delta \Delta \Delta ; f 0 k (x) f(x) f k (x) g; where f 0 i (x) denotes the derivative of f(x). It follows that we have Theorem 3.2.1. (Niederreiter =-=[Ni]-=-). The eigenspace R 1 (D) can be used to factor f(x) over F q in a way similar to Berlekamp's algorithm. Using the Chinese remainder theorem, normal basis and basic properties of the operators / q and... |

5 |
Algebraic geometry lattices and codes
- Tsfasman
- 1996
(Show Context)
Citation Context ...arge. Computing the zeta function of a hyperellitptic curve is very useful in the hyperelliptic curve cryptosystem proposed by Koblitz [Ko1-2]. Additional applications may be found in [AH2], [AG] and =-=[Ts]-=-. Computing zeta functions of algebraic varieties should also be the first key step in computing zeta functions of more general Hilbert sets and definable sets, see [Wa1] and [FHJ]. A very special cas... |

3 |
Une formule de congruence pour la fonction ζ, Exposé XXII, in Groupes de Monodromie en Géometrie Algébrique
- Katz
- 1973
(Show Context)
Citation Context ...ve the congruence formula Z(X;T ) (\Gamma1) n j det(I \Gamma (/ q ffi f q\Gamma1 )T jR(d)) ( mod p): In the projective case, a similar congruence formula but with a much harder proof is given by Katz =-=[Ka2]-=-. Very general congruence formulas but acting on infinite dimensional space can be found in [TW1] and [TW2]. Using Theorem 4.2, we get Corollary 4.3. The zeta function Z(X;T ) modulo p can be computed... |

3 |
Subspaces and polynomial factorization over finite fields
- Lee, Vanstone
- 1995
(Show Context)
Citation Context ...the next two subsections, we describe two more such admissible subspaces and hence two more such algorithms for factoring polynomials. Admissible subspaces are studied in some details in Lee-Vanstone =-=[LV]-=-. In particular, the congruence formula in section 3.3 gives a new admissible subspace and hence answers a question in [LV]. 3.2. Zeta function and Niederreiter's algorithm Let / q be the F q -linear ... |

3 |
Entireness of L-functions of '-sheaves on affine complete intersections
- Taguchi, Wan
- 1997
(Show Context)
Citation Context ... projective case, a similar congruence formula but with a much harder proof is given by Katz [Ka2]. Very general congruence formulas but acting on infinite dimensional space can be found in [TW1] and =-=[TW2]-=-. Using Theorem 4.2, we get Corollary 4.3. The zeta function Z(X;T ) modulo p can be computed in time that is a polynomial in \Gamma d n \Delta log q (the input and output size) if p is small and n is... |

3 |
Hilbert sets and zeta function over finite fields
- Wan
(Show Context)
Citation Context ...ay be found in [AH2], [AG] and [Ts]. Computing zeta functions of algebraic varieties should also be the first key step in computing zeta functions of more general Hilbert sets and definable sets, see =-=[Wa1]-=- and [FHJ]. A very special case is already considered in [GKS]. Our purpose here is to give a brief general discussion of the modular approach for computing zeta functions, indicating some theoretical... |

2 |
On polynomial factorization over finite fields
- Gunji, Arnon
- 1981
(Show Context)
Citation Context ...be the d \Theta d matrix whose ij entry is (i; j), where 1si; jsd. As indicated by Schwarz, the matrix A is invertible. In fact, an explicit formula for the inverse matrix of A can also be found, see =-=[GA]-=-. The map F acting on R can be computed efficiently by repeated squaring. The above theorems show that all the k(i) and hence all the s i (1sisd) can be computed in polynomial time. Thus, the zeta fun... |

2 |
L-functions of Drinfeld modules and '-sheaves
- Taguchi, Wan
(Show Context)
Citation Context ...p): In the projective case, a similar congruence formula but with a much harder proof is given by Katz [Ka2]. Very general congruence formulas but acting on infinite dimensional space can be found in =-=[TW1]-=- and [TW2]. Using Theorem 4.2, we get Corollary 4.3. The zeta function Z(X;T ) modulo p can be computed in time that is a polynomial in \Gamma d n \Delta log q (the input and output size) if p is smal... |

1 |
L-functions of Drinfeld modules and ϕ-sheaves
- Taguchi, Wan
(Show Context)
Citation Context ...p). In the projective case, a similar congruence formula but with a much harder proof is given by Katz [Ka2]. Very general congruence formulas but acting on infinite dimensional space can be found in =-=[TW1]-=- and [TW2]. Using Theorem 4.2, we get Corollary 4.3. The zeta function Z(X, T) modulo p can be computed in time that is a polynomial in ( ) d n log q (the input and output size) if p is small and n is... |