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Efficient $(3,3)$-isogenies on fast Kummer surfaces

Maria Corte-Real Santos, Craig Costello, Benjamin Smith

Abstract

We give an alternative derivation of $(N,N)$-isogenies between fast Kummer surfaces which complements existing works based on the theory oftheta functions. We use this framework to produce explicit formulae for the case of $N = 3$, and show that the resulting algorithms are more efficient than all prior $(3, 3)$-isogeny algorithms.

Efficient $(3,3)$-isogenies on fast Kummer surfaces

Abstract

We give an alternative derivation of -isogenies between fast Kummer surfaces which complements existing works based on the theory oftheta functions. We use this framework to produce explicit formulae for the case of , and show that the resulting algorithms are more efficient than all prior -isogeny algorithms.
Paper Structure (26 sections, 2 theorems, 56 equations, 1 table, 6 algorithms)

This paper contains 26 sections, 2 theorems, 56 equations, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Software.
  4. Fast Kummer surfaces and their arithmetic
  5. Genus-2 curves and their Jacobians
  6. Isogenies and Kummer surfaces
  7. Fast Kummer surfaces
  8. Nodes of the Kummer surface
  9. Operations on the Kummer surface
  10. (N,N)-isogenies on fast Kummer surfaces
  11. A warm-up with N = 2
  12. The general case: odd N
  13. Explicit (3,3)-isogenies on fast Kummers
  14. Explicit formulae for (3,3)-isogenies.
  15. Evaluating points under the (3,3)-isogeny.
  16. ...and 11 more sections

Key Result

Lemma 3.4

Fix Kummer surface $\mathcal{K}\xspace$ with coordinates $X_1, X_2, X_3, X_4$ and associated biquadratic forms $B_{i,j}$ for $1 \leq i,j \leq 4$. Let $N$ be an odd prime number, and fix a point $R \in \mathcal{K}\xspace[N]$ of order $N$. We denote by $I \in \{1,2,3,4\}^N$ a list of indices $I = (i_1 Then, for each $I \in \{1,2,3,4\}^N$ we define where $C_N$ is the cyclic group of order $N$. Then,

Theorems & Definitions (11)

  • Definition 2.1
  • Example 3.1
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • Remark 3.5
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Remark 5.1
  • ...and 1 more