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.

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, 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.

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.

When amicable.sage is used with the --isogenies option, the output includes isogenies 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. 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.