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On l-th roots and division by l

Josep M. Miret, Jordi Pujolàs, Nicolas Thériault

Abstract

We give a characterization of the codomain $[\ell]E(k)$ of the multiplication-by-$\ell$ map $[\ell]$ in the case of elliptic curves over a field $k$ of characteristic $\ne 2,3$ with $\ell$-torsion $E[\ell]=\langle W_1,W_2 \rangle$ fully defined over $k$, for primes $\ell$ different from the characteristic. We show that a point $Q\in E(k)$ lies in $[\ell]E(k)$ if and only if $h_{W_1}(-Q)$ and $h_{W_2}(-Q)$ are $\ell$-powers of $k$, where $h_{W_1}$ and $h_{W_2}$ are functions on $E$ with divisor ${\rm div}(h_{W_i})=\ell W_i- \ell P_{\infty}$. Our characterization leads to an effective procedure to find pre-images of $[\ell]$ by solving an order $\ell$ system of linear equations and computing a polynomial gcd.

On l-th roots and division by l

Abstract

We give a characterization of the codomain of the multiplication-by- map in the case of elliptic curves over a field of characteristic with -torsion fully defined over , for primes different from the characteristic. We show that a point lies in if and only if and are -powers of , where and are functions on with divisor . Our characterization leads to an effective procedure to find pre-images of by solving an order system of linear equations and computing a polynomial gcd.
Paper Structure (9 sections, 15 theorems, 97 equations)

This paper contains 9 sections, 15 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Miller functions $h_{W_i}$
  3. The function $g_P$
  4. Division by $\ell$ and $\ell$-powers
  5. A characterization of $[\ell]E(k)$
  6. Computation of pre-images
  7. Application to elliptic curve point counting
  8. Conclusion
  9. Funding statement

Key Result

Proposition 2.1

If $W_i=(\gamma_i,\delta_i), W_j=(\gamma_j,\delta_j), W_k=(\gamma_k,\delta_k)\in E[\ell]$ with $W_k = W_i + W_j$, then where $L_{ij}=0$ is the line joining $W_i$ and $W_j$.

Theorems & Definitions (38)

  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • ...and 28 more