American Institute of Mathematical Sciences

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November  2014, 8(4): 437-458. doi: 10.3934/amc.2014.8.437

The geometry of some parameterizations and encodings

Jean-Marc Couveignes 1, and  Reynald Lercier 2,

1. 

Université de Bordeaux, IMB, UMR 5251, F-33400 Talence, France
2. 

DGA.MI, La Roche Marguerite, F-35174 Bruz, France

Received  May 2014 Revised  June 2014 Published  November 2014

We explore parameterizations by radicals of low genera algebraic curves. We prove that for $q$ a prime power that is large enough and prime to $6$, a fixed positive proportion of all genus 2 curves over the field with $q$ elements can be parameterized by $3$-radicals. This results in the existence of a deterministic encoding into these curves when $q$ is congruent to $2$ modulo $3$. We extend this construction to parameterizations by $l$-radicals for small odd integers $l$, and make it explicit for $l=5$.
Keywords: parameterizations, encodings, finite fields, torsors., deterministic algorithms, Algebraic curves, radicals.
Mathematics Subject Classification: Primary: 14H45, 14G50, 14E20; Secondary: 11G20, 12G05, 11S2.
Citation: Jean-Marc Couveignes, Reynald Lercier. The geometry of some parameterizations and encodings. Advances in Mathematics of Communications, 2014, 8 (4) : 437-458. doi: 10.3934/amc.2014.8.437
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show all references

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[1]

Amer. J. Math., 10 (1887), 47-70. doi: 10.2307/2369402.  Google Scholar

[2]

in Adv. Crypt. - CRYPTO' 2001 (ed. J. Kilian), Springer-Verlag, Berlin, 2001, 213-229. doi: 10.1007/3-540-44647-8_13.  Google Scholar

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J. London Math. Soc., 64 (2001), 29-43. doi: 10.1017/S0024610701002113.  Google Scholar

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Abh. der k. Ges. Wiss. zu Göttingen, 14 (1869), 17-75. Google Scholar

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J. Symb. Comput., 47 (2012), 266-281. doi: 10.1016/j.jsc.2011.11.003.  Google Scholar

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N. Elkies, The identification of three moduli spaces,, preprint, ().   Google Scholar

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in Africa CRYPT, 2011, 278-289. doi: 10.1007/978-3-642-21969-6_17.  Google Scholar

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in Pairing-Based Cryptography (eds. M. Joye, A. Miyaji and A. Otsuka), Springer, 2010, 265-277. doi: 10.1007/978-3-642-17455-1_17.  Google Scholar

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[17]

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[22]

Bull. Polish Acad. Sci. Math., 55 (2007), 97-104. doi: 10.4064/ba55-2-1.  Google Scholar

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