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tweedle

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Generator and supporting evidence for security of the Tweedledum/Tweedledee pair of elliptic curves suitable for Halo

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amicable.sage
animation-p.webm
animation-q.webm
animation.sh
checksumsets.py
clean.sh
injectivitylemma.py
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Tweedledum/Tweedledee supporting evidence
-----------------------------------------

This repository contains supporting evidence that the amicable pair of
prime-order curves:

* Ep : y^2 = x^3 + 5 over GF(p) of order q, called Tweedledum;
* Eq : y^2 = x^3 + 5 over GF(q) of order p, called Tweedledee;

with

* p = 2^254 + 4707489545178046908921067385359695873
* q = 2^254 + 4707489544292117082687961190295928833

satisfy *some* of the [SafeCurves criteria](https://safecurves.cr.yp.to/index.html).

The criteria that are *not* satisfied are, in summary:

* large-magnitude CM discriminant (both curves have CM discriminant of absolute value 3,
  as a consequence of how they were constructed);
* completeness (complete formulae are possible, but not according to the Safe curves
  criterion);
* ladder support (not possible for prime-order curves);
* Elligator 2 support (indistinguishability is possible using
  [Elligator Squared](https://ifca.ai/pub/fc14/paper_25.pdf), but not using Elligator 2).

Tweedledum/Tweedledee is the first cycle output by
``sage amicable.sage --sequential --nearpowerof2 255 32``.

(The `--sequential` option makes the output completely deterministic and so resolves
ambiguity about which result is "first". For exploratory searches it is faster not to
use `--sequential`.)

**The cycle we call Tweedledum/Tweedledee has changed from the initial (September 2019) draft of the Halo paper.**

Note that although there is no known security problem with the Tweedle cycle, there are efficiency
and interoperability reasons to prefer the [Pasta cycle](https://github.com/zcash/pasta), as
explained in [this blog post](https://electriccoin.co/blog/the-pasta-curves-for-halo-2-and-beyond/).

Prerequisites:

* ``apt-get install sagemath``

Run ``sage verify.sage Ep`` and ``sage verify.sage Eq``; or ``./run.sh`` to run both
and also print out the results.

``amicable.sage`` also outputs isogenies (of degree up to ``ISOGENY_DEGREE_MAX``) suitable
for use with the "simplified SWU" method for hashing to an elliptic curve. This is based
on code from Appendix A of [Wahby and Boneh 2019](https://eprint.iacr.org/2019/403.pdf).
Note that simplified SWU is not necessarily the preferred method to hash to a given curve.
In particular it probably is not for the Tweedle curves; they only have suitable isogenies
of degree 23, which is rather large.

To check the correctness of the endomorphism optimization described in the Halo paper, run
``python3 injectivitylemma.py`` and ``python3 checksumsets.py``. To also generate animations
showing the minimum distances between multiples of ζ used in the proof, run ``./animation.sh``.

``animation.sh`` has the following prerequisites:

* ``apt-get install ffmpeg ffcvt``
* ``pip3 install bintrees Pillow``

``checksumsets.py`` on its own only requires the ``bintrees`` Python package.