2019-09-08 08:23:57 -07:00
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Tweedledum/Tweedledee supporting evidence
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This repository contains supporting evidence that the amicable pair of
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prime-order curves:
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* Ep : y^2 = x^3 + 5 over GF(p) of order q, called Tweedledum;
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* Eq : y^2 = x^3 + 5 over GF(q) of order p, called Tweedledee;
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with
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* p = 57896044618658097711785492504343953925989756877607163761872965584918954377217
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* q = 57896044618658097711785492504343953925989756877607147991657089165100807356417
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satisfy *some* of the [SafeCurves criteria](https://safecurves.cr.yp.to/index.html).
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The criteria that are *not* satisfied are, in summary:
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* large CM discriminant (both curves have CM discriminant 3, as a consequence of how
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they were constructed);
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* completeness (complete formulae are possible, but not according to the Safe curves
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criterion);
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* ladder support (not possible for prime-order curves);
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* Elligator 2 support (indistinguishability is possible using
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[Elligator Squared](https://ifca.ai/pub/fc14/paper_25.pdf), but not using Elligator 2);
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* twist security;
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* rigidity.
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2017-11-01 22:52:36 -07:00
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2019-09-10 08:24:08 -07:00
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Tweedledum/Tweedledee is one of the cycles output by ``sage amicable.sage --nearpowerof2 255 32``
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(the first one with constant 5 for both curves and gcd(p-1, 5) = 1, gcd(q-1, 5) = 1).
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2017-11-01 22:52:36 -07:00
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Prerequisites:
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* apt-get install sagemath
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* pip install sortedcontainers
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2019-09-08 08:23:57 -07:00
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Run ``sage verify.sage Ep`` and ``sage verify.sage Eq``; or ``./run.sh`` to run both
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and also print out the results.
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