zbMATH — the first resource for mathematics

The geometry of some parameterizations and encodings. (English) Zbl 1314.14053
The paper deals with the problem of the parameterization by radicals (of degree prime to \(q(q-1)\)) of algebraic curves of small genus (mainly genus 2) defined over a finite field \(\mathbb{F}_q\), of characteristic different from 2 and 3. A such parameterization would allow an efficient deterministic encodings into those curves.
Encoding algorithms for elliptic curves are already known, see [T. Icart, Lect. Notes Comput. Sci. 5677, 303–316 (2009; Zbl 1252.94075)], but for genus 2 curves only partial results were known, results only valid for a negligible proportion of all genus 2 curves, see Kammerer, Lercier and Renault [J.-G. Kammerer et al., Lect. Notes Comput. Sci. 6487, 278–297 (2010; Zbl 1290.94100)].
Section 2 begins remembering the basic notions of parameterizations, encoding and torsors. Then Tartaglia-Cardan formulae is formulated in the language of torsors and the proposed method of parametrization is showed (2.7 and 2.8). This general method is illustrated in Section 3 for elliptic curves.
Section 4 studies the case of genus 2 curves having two rational points whose difference has order 3 in the associated Jacobian variety. This allows to parameterize by 3-radicals a positive proportion of all genus 2 curves. An example (over \(\mathbb{F}_{83}\)) is worked in detail. Finally the paper studies other parameterizations by \(l\)-radicals, in particular for \(l=5\).

14H45 Special algebraic curves and curves of low genus
14E20 Coverings in algebraic geometry
11G20 Curves over finite and local fields
12G05 Galois cohomology
11S20 Galois theory
Full Text: DOI arXiv
[1] O. Bolza, On binary sextics with linear transformations into themselves,, Amer. J. Math., 10, 47, (1887) · JFM 19.0488.01
[2] D. Boneh, Identity-based encryption from the Weil pairing,, in Adv. Crypt. - CRYPTO’ 2001 (ed. J. Kilian), 213, (2001) · Zbl 1002.94023
[3] J. Boxall, 5-torsion points on curves of genus 2,, J. London Math. Soc., 64, 29, (2001) · Zbl 1069.14031
[4] A. Clebsch, Zur Theorie der binären Formen sechster Ordnung und zur Dreitheilung a der hyperelliptischen Funktionen,, Abh. der k. Ges. Wiss. zu Göttingen, 14, 17, (1869) · JFM 02.0066.01
[5] J.-M. Couveignes, The geometry of flex tangents to a cubic curve and its parameterizations,, J. Symb. Comput., 47, 266, (2012) · Zbl 1244.14024
[6] N. Elkies, The identification of three moduli spaces,, preprint
[7] R. R. Farashahi, Hashing into Hessian curves,, in Africa CRYPT, 278, (2011) · Zbl 1280.94050
[8] P.-A. Fouque, Deterministic encoding and hashing to odd hyperelliptic curves,, in Pairing-Based Cryptography (eds. M. Joye, 265, (2010) · Zbl 1290.94073
[9] M. Fried, Combinatorial computation of moduli dimension of Nielsen classes of covers,, in Graphs and Algorithms, 61, (1989) · Zbl 0703.14018
[10] M. Harrison, Explicit solution by radicals, gonal maps and plane models of algebraic curves of genus \(5\) or \(6\),, J. Symb. Comp., 51, 3, (2013) · Zbl 1276.14047
[11] T. Icart, How to hash into elliptic curves,, in CRYPTO, 303, (2009) · Zbl 1252.94075
[12] J.-I. Igusa, Arithmetic variety of moduli for genus two,, Ann. Math., 72, 612, (1960) · Zbl 0122.39002
[13] J.-G. Kammerer, Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time,, in Pairing, 278, (2010) · Zbl 1290.94100
[14] S. Lang, <em>Algebra</em>,, Springer, (2002) · Zbl 0984.00001
[15] R. Lercier, Fast computation of isomorphisms of hyperelliptic curves and explicit descent,, in ANTS X - Proc. 10th Algor. Number Theory Symp. (eds. E.W. Howe and K.S. Kedlaya), 463, (2013) · Zbl 1344.11049
[16] J. Neukirch, <em>Cohomology of Number Fields</em>,, Springer-Verlag, (2000) · Zbl 0948.11001
[17] G. Salmon, <em>Lessons Introductory to the Modern Higher Algebra</em>,, Chelsea Publishing Co., (1885)
[18] A. Schinzel, On equations \(y^2=x^n+k\) in a finite field,, Bull. Pol. Acad. Sci. Math., 52, 223, (2004) · Zbl 1105.11015
[19] M. Skałba, Points on elliptic curves over finite fields,, Acta Arith., 117, 293, (2005) · Zbl 1078.11044
[20] A. Shallue, Construction of rational points on elliptic curves over finite fields,, in Algorithmic Number Theory, 510, (2006) · Zbl 1143.11331
[21] H. Stichtenoth, <em>Algebraic Function Fields and Codes</em>,, Second edition, (2009) · Zbl 1155.14022
[22] M. Ulas, Rational points on certain hyperelliptic curves over finite fields,, Bull. Polish Acad. Sci. Math., 55, 97, (2007) · Zbl 1131.11039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.