Elliptic and modular curves over finite fields and. Masters thesis algebraic curves over finite fields. Pdf the genus of curves over finite fields with many rational points. Pdf encryption of data using elliptic curve over finite. Encryption of data using elliptic curve over finite fields. This chapter describes the specialised facilities for elliptic curves defined over finite fields. 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.
You are advised to consult the publishers version publishers pdf if you. The chordtangent method does give rise to a group law if a point is xed as the zero element. Computing zeta functions of artinschreier curves over finite. 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. 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. In this chapter we study elliptic curves defined over a finite field. This can be done over any eld over which there is a rational point. 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. Macwilliams theorem for the quadratic residue weight enumerator 50 3. Stange stanford university boise reu, june 14th, 2011. Rational torsion of prime order in elliptic curves over number fields asterisque no. Decomposing jacobians of curves over finite fields in the absence of algebraic structure, j number theory, nov 2015.
Elliptic curves over finite fields and the computation of square roots mod p by rene schoof abstract. Lauder and daqing wan abstract the authors present a practical polynomialtime algorithm for computing the zeta function of certain artinschreier curves over. 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. Elliptic curves over finite fields indian statistical institute. Formal finiteness and the torsion conjecture on elliptic curves. The bounds obtained lead to a proof of the riemann hypothesis for curves over finite fields and yield several improvements on it.
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. The chordtangent method does give rise to a group law if a point is. 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. 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. This yields a method for computing the order of the jacobian.
Let c be a superspecial hyperelliptic curve of genus g over k. Pdf the group structure of bachet elliptic curves over. On curves over finite fields by arnaldo garcia emis. 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. 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. Lpolynomials of curves over finite fields skidmore college.
Rational points on curves and surfaces over finite fields. 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. 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. Weierstrass points and curves over finite fields stohr. In 2005, this was embedded in a larger project of determining n. The quadratic residue weight enumerator for quadrics 53 4. Elliptic curves are studied over finite and infinite fields. If the genus of the curve is large with respect to the cardinality of the finite field, ihara noticed. Quadratic residue weight enumerators and elliptic curves over finite fields 47 1. Pdf we prove the following result which was conjectured by stichtenoth and xing. A method for encrypting messages using elliptic curves over finite field is proposed in 7, where each. This method has important applications in cryptography, where groups of prime order are used as the basis.
Recently, the authors discovered another important application of such curves, namely to the construction of lowdiscrepancy sequences. The constants in the bound are related to the weierstrass order. 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 constants in the bound are related to the weierstrass ordersequence associated with the projective embedding. Pdf twists of genus three curves over finite fields jaap.
For reasons to be explained later, we also toss in an. The riemann hypothesis for curves over finite fields. It is the functions z that are designed to multiply, to get global zeta functions. Thenweusethe connectionbetween the theory of linear codes and algebraic curves over finite fields goppacodes10. Algebraic complexities and algebraic curves over finite fields. A vector space of polynomials over a xed nite eld f q gives rise to. On isogeny graphs of supersingular elliptic curves over finite fields gora adj, omran ahmadi, and alfred menezes abstract. Let f be a finite field with q elements of characteristic p not equal to 2 or 3. 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.
Algebraic curves over a finite field princeton university press. Let x be a smooth projective curve over a finite field. Introduction jacobi was the rst person to suggest in 1835 using the group law on a cubic curve e. Those involve different finite fields for example the whole family of fields zpz as p runs over all prime numbers. University of groningen algebraic curves over finite. Differential operators and hyperelliptic curves over finite. University of groningen algebraic curves over finite fields. 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. Elliptic curves over finite fields lsu mathematics.
Curves over finite fields attaining the hasseweil upper bound. Pdf encryption of data using elliptic curve over finite fields. Maximal curves are the curves attaining the upper bound of. 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. Rational points on algebraic curves over finite fields is a key topic for algebraic geometers and coding theorists. 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. Here, the authors relate an important application of such curves, namely, to the construction of lowdiscrepancy sequences, needed for numerical methods in diverse areas. Elliptic curves over finite fields university of colorado. Pdf twists of genus three curves over finite fields. 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. Rational points on curves and surfaces over finite fields and.
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. 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. The problem of calculating the trace of an elliptic curve over a finite field has attracted considerable interest in recent years. Elliptic curves over f q introduction history length of ellipses why elliptic curves. Most of the machinery has been constructed with elliptic curve cryptography in mind. Rational curves on cubic hypersurfaces over finite fields 3 method for the function.757 430 251 1215 1548 728 1434 512 579 1363 1377 174 372 712 47 710 364 1496 591 970 931 709 379 35 53 716 1446 524 1211 203 566 1258