Equations singular points the discriminant elliptic curves f 2 elliptic curves f 3 the sum of points examples structure of ef 2 structure of ef 3 further examples notations fields of characteristics 0 1 q is the. The focus in this application of algebraic geometry to coding theory is on algebraic curves over finite fields with many rational points relative to the genus. The problem of calculating the trace of an elliptic curve over a finite field has attracted considerable interest in recent years. Let f be a finite field with q elements of characteristic p not equal to 2 or 3. The constants in the bound are related to the weierstrass ordersequence associated with the projective embedding. Masters thesis algebraic curves over finite fields. Rational points on curves and surfaces over finite fields. A vector space of polynomials over a xed nite eld f q gives rise to. Rational points on curves over finite fields by harald. Elliptic curves over f q introduction history length of ellipses why elliptic curves. For reasons to be explained later, we also toss in an. Feb 24, 2017 in algebraic geometry, curves are onedimensional varieties, and just as there is a version of the riemann hypothesis for curves over finite fields, there is also a version of the riemann hypothesis for higherdimensional varieties over finite fields, called the weil conjectures, since they were proposed by weil himself after he proved the case. Lpolynomials of curves over finite fields gary mcguire school of mathematical sciences university college dublin ireland july 2015 12th finite fields and their applications conference.
A method for encrypting messages using elliptic curves over finite field is proposed in 7, where each character in the message is encoded to a point on the curve by using a code table which is agreed upon by communicating parties and each message point is encrypted to a pair of cipher points. The main result is that the odddimensional ktheory of the extension of x to the algebraic closure is the sum of two copies of the ktheory of the field. Maximal curves are the curves attaining the upper bound of. A method for encrypting messages using elliptic curves over finite field is proposed in 7, where each character in the message is encoded to a point on the curve by using a code table which is.
On curves over finite fields by arnaldo garcia emis. Weierstrass points and curves over finite fields stohr. Details concerning their construction, arithmetic and basic properties may be found in chapter elliptic curves. Rational point counts for del pezzo surfaces over finite fields and coding theory abstract the goal of this thesis is to apply an approach due to elkies to study the distribution of rational point counts for certain families of curves and surfaces over nite elds. You are advised to consult the publishers version publishers pdf if you. Encryption of data using elliptic curve over finite fields.
The present work extends the research already carried out by the authors for genus 2 and presented in the th international conference on finite fields and applications, held in gaeta, in june 2017. Ii 953 here we would like to record our thanks to the referee, who among other things pointed out the isogeny between ex 4. Rational points on curves and surfaces over finite fields and. The quadratic residue weight enumerator for quadrics 53 4. Elliptic and modular curves over finite fields and. We study the isogeny graphs of supersingular elliptic curves over nite elds, with an emphasis on the vertices corresponding to elliptic curves of jinvariant 0 and 1728.
Curves over finite fields attaining the hasseweil upper bound. Elliptic curves over finite fields university of colorado. Introduction jacobi was the rst person to suggest in 1835 using the group law on a cubic curve e. This yields a method for computing the order of the jacobian. Let k f q be the finite field with q elements and e an elliptic curve defined over k. The most important arithmetic quantity associated with such a curve is its number of rational points. Thenweusethe connectionbetween the theory of linear codes and algebraic curves over finite fields goppacodes10. Pdf encryption of data using elliptic curve over finite fields. Here, the authors relate an important application of such curves, namely, to the construction of lowdiscrepancy sequences, needed for numerical methods in diverse areas. Computing zeta functions of artinschreier curves over finite. While the precise number of rational points of an elliptic curve e over k is in general rather difficult to compute, hasses theorem on elliptic curves gives us, including the point at infinity, the following estimate.
Rational torsion of prime order in elliptic curves over number fields asterisque no. Differential operators and hyperelliptic curves over finite. It is the functions z that are designed to multiply, to get global zeta functions. This chapter describes the specialised facilities for elliptic curves defined over finite fields. Rational curves on cubic hypersurfaces over finite fields 3 method for the function. In 2005, this was embedded in a larger project of determining n. Pdf encryption of data using elliptic curve over finite. The constants in the bound are related to the weierstrass order. University of groningen algebraic curves over finite fields. Algebraic curves over a finite field princeton university press. Elliptic curves over finite fields indian statistical institute. This can be done over any eld over which there is a rational point. Pdf legendre elliptic curves over finite fields jaap top.
Pdf the genus of curves over finite fields with many rational points. The bounds obtained lead to a proof of the riemann hypothesis for curves over finite fields and yield several improvements on it. Here we explain the main tools for finding rational points on a curve over a finite field and provide the necessary background on ring and field theory. The riemann hypothesis for curves over finite fields. In practice it makes z a rational function of t, something that is interesting even in the case of v an elliptic curve over finite field. Macwilliams theorem for the quadratic residue weight enumerator 50 3. If the genus of the curve is large with respect to the cardinality of the finite field, ihara noticed. The chordtangent method does give rise to a group law if a point is. Counting points on elliptic curves over finite fields numdam.
Recently, the authors discovered another important application of such curves, namely to the construction of lowdiscrepancy sequences. Decomposing jacobians of curves over finite fields in the absence of algebraic structure, j number theory, nov 2015. A method for encrypting messages using elliptic curves over finite field is proposed in 7, where each. The chordtangent method does give rise to a group law if a point is xed as the zero element. This method has important applications in cryptography, where groups of prime order are used as the basis. Quadratic residue weight enumerators and elliptic curves over finite fields 47 1. Lpolynomials of curves over finite fields skidmore college. Formal finiteness and the torsion conjecture on elliptic curves. On isogeny graphs of supersingular elliptic curves over finite fields gora adj, omran ahmadi, and alfred menezes abstract. Let c be a superspecial hyperelliptic curve of genus g over k. Elliptic curves over finite fields and the computation of. Elliptic curves are studied over finite and infinite fields. Pdf twists of genus three curves over finite fields. N we present a new method for constructing genus 2 curves over a nite eld fn with a given number of points on its jacobian.
Curves over finite fields attaining the hasseweil upper bound 3 in other terms, the second largest genus g of a maximal curve over f q2 is given by g q 124. This book provides an accessible and selfcontained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, errorcorrecting codes, and cryptology. Those involve different finite fields for example the whole family of fields zpz as p runs over all prime numbers. Pdf twists of genus three curves over finite fields jaap.
In this chapter we study elliptic curves defined over a finite field. Rational points on algebraic curves over finite fields is a key topic for algebraic geometers and coding theorists. Elliptic curves over finite fields and the computation of square roots mod p by rene schoof abstract. University of groningen algebraic curves over finite.
Let x be a smooth projective curve over a finite field. Algebraic complexities and algebraic curves over finite fields. Ii 953 here we would like to record our thanks to the referee, who among other things pointed out the isogeny between ex 4 and es given above and also suggested the first and third. In this paper we present a deterministic algorithm to compute the number of fpoints of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation. We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in legendre form, with the sole exception of a minimal respectively maximal elliptic. Stange stanford university boise reu, june 14th, 2011. Lauder and daqing wan abstract the authors present a practical polynomialtime algorithm for computing the zeta function of certain artinschreier curves over. Elliptic curves over finite fields lsu mathematics. Pdf the group structure of bachet elliptic curves over.
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