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

with

• p = 2^254 + 11429413694214642624661040171709366273
• q = 2^254 + 11429413694209135470422256387130130433

satisfy some of the SafeCurves criteria.

The criteria that are not satisfied are, in summary:

• large CM discriminant (both curves have CM discriminant 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, but not using Elligator 2);
• twist security.

(Provisional) Tweedledum/Tweedledee is the first cycle output by sage amicable.sage --nearpowerof2 255 30.

Which cycle we call Tweedledum/Tweedledee is subject to change as we make further optimizations to Halo.

Prerequisites:

• apt-get install sagemath
• pip install sortedcontainers

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