Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties

We consider the problem of uniform sampling of points on an algebraic variety. Specifically, we develop a randomized algorithm that, given a small set of multivariate polynomials over a sufficiently large finite field, produces a common zero of the polynomials almost uniformly at random. The statistical distance between the output distribution of the algorithm and the uniform distribution on the set of common zeros is polynomially small in the field size, and the running time of the algorithm is polynomial in the description of the polynomials and their degrees provided that the number of the polynomials is a constant.

[1]  A. Weil.  Numbers of solutions of equations in finite fields , 1949 .

[2]  Ming-Deh A. Huang,et al.  Counting rational points on curves over finite fields , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[3]  Michael O. Rabin,et al.  Probabilistic Algorithms in Finite Fields , 1980, SIAM J. Comput..

[4]  R. Schoof.  Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p , 1985 .

[5]  Zeev Dvir,et al.  Deterministic Extractors for Algebraic Sources , 2008, Electron. Colloquium Comput. Complex..

[6]  Leonard M. Adleman,et al.  Counting Rational Points on Curves and Abelian Varieties over Finite Fields , 1996, ANTS.

[7]  Ming-Deh A. Huang,et al.  Primality Testing and Abelian Varieties over Finite Fields , 1992 .

[8]  F. S. Macaulay.  Some Formulæ in Elimination , 1902 .

[9]  Luca Trevisan,et al.  Extracting randomness from samplable distributions , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[10]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[11]  Leslie G. Valiant,et al.  Random Generation of Combinatorial Structures from a Uniform Distribution , 1986, Theor. Comput. Sci..

[12]  J. Pila.  Frobenius maps of Abelian varieties and finding roots of unity in finite fields , 1990 .

[13]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[14]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[15]  Avi Wigderson,et al.  Extractors And Rank Extractors For Polynomial Sources , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[16]  Mihir Bellare,et al.  Uniform Generation of NP-Witnesses Using an NP-Oracle , 2000, Inf. Comput..

[17]  S. Lang,et al.  NUMBER OF POINTS OF VARIETIES IN FINITE FIELDS. , 1954 .

[18]  Victor Shoup,et al.  New algorithms for finding irreducible polynomials over finite fields , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.