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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 \(l=5\).

MSC:
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
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References:
[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
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