Number of points on certain hyperelliptic curves defined over finite fields
نویسندگان
چکیده
منابع مشابه
Rational Points on Certain Hyperelliptic Curves over Finite Fields
Let K be a field, a, b ∈ K and ab 6= 0. Let us consider the polynomials g1(x) = x n + ax + b, g2(x) = x n + ax + bx, where n is a fixed positive integer. In this paper we show that for each k ≥ 2 the hypersurface given by the equation
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We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm à la Schoof for genus 2 using Cantor’s division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods ...
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Abstract. We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of quadratic residues and non-residues in the image of such subsets over uniformly random hyperelliptic curves of given degrees. We find a constant proba...
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Preface These notes treat the problem of counting the number of rational points on a curve defined over a finite field. The notes are an extended version of an earlier set of notes Aritmetisk Algebraisk Geometri – Kurver by Johan P. Hansen [Han] on the same subject. In Chapter 1 we summarize the basic notions of algebraic geometry, especially rational points and the Riemann-Roch theorem. For th...
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Stanford University) Abstract: A curve is a one dimensional space cut out by polynomial equations, such as y2=x3+x. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables (x and y in the example) in the finite field that satisfy the given polyn...
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ژورنال
عنوان ژورنال: Finite Fields and Their Applications
سال: 2008
ISSN: 1071-5797
DOI: 10.1016/j.ffa.2006.12.007