[curves] Jacobi Quartic "Distortion Map"

[Brad Klee]

Hi Curves Fans,

I will shut up after this post, and get back to calculating and writing.

Whether you call it a shear transformation, a distortion map, or a
symplectomorphism, we can all agree about Jacobian
coordinate-transformation matrices, which are trivial to calculate. In
Craig Costello's preferred examples 4.5 & 4.6, the Jacobians belong to the
unitary group U(2) but not to the special unitary group SU(2), indicating
that the transformation permutes real and complex time axes [1]. Planar
shear transformations in general belong to both SU(2) and U(2). This puts
one of my preferred examples into an altogether different class [2]. But it
is like Costello's type 1, where the "distortion map" immediatly provides a
group isomorphism.

Conjecture. Super-singularity is not essential. We can look for a group of
order 2*p, with prime p. Ignore the coset by C_2. Distinct addition rules
generate disjoint order-p subgroups from appropriately chosen initial
conditions. Isomorphism between groups G1 and G2 involves a linear
intersection geometry. For g1~g2 with points in the cubic basis, a line
passes through g1^(-1), g2^(-1), and the quartic identity, usually chosen
(0,1).

~~~~~

Ex. 1. Prime 101, group C_{29}.
n*101 = -16 + 16* (x^2 + y^2) - 64 *x^4 .
n*101 = -16 + 16* (x^2 + y^2) - 64 x^2 y .
( see [3] : https://ptpb.pw/GRdL.png )
============================
{n}P: {{13, 94}, {5, 19}, {60, 3}, {27, 64}, {92, 16}, {72, 13}, {55, 35},
{98, 84}, {49, 30}, {80, 91}, {4, 42}, {32, 36}, {77, 87}, {1, 4}, {100,
4}, {24, 87}, {69, 36}, {97, 42}, {21, 91}, {52, 30}, {3, 84}, {46, 35},
{29, 13}, {9, 16}, {74, 64}, {41, 3}, {96, 19}, {88, 94}, {infty, 0}} .
[n]P: {{66, 7}, {10, 37}, {85, 88}, {43, 34}, {73, 14}, {54, 10}, {56, 65},
{17, 2}, {34, 17}, {89, 38}, {63, 66}, {71, 25}, {40, 26}, {50, 50}, {51,
50}, {61, 26}, {30, 25}, {38, 66}, {12, 38}, {67, 17}, {84, 2}, {45, 65},
{47, 10}, {28, 14}, {58, 34}, {16, 88}, {91,37}, {35, 7}, {0, 1}} .

Set P1 = {13,94} and P2 = {66,7}. The line joining inverses is: 7 + y = 8 +
55*x. Does it visit {0,1}? Yes, True.

Ex. 2. Same Curve, but with prime 223 and group C_{59}. D.I.Y. !

~~~~~

I can't see a reason to doubt that these relations will generalize to
higher prime fields, and think that a proof could probably be written. Not
sure if anything practical will follow, but interested to hear thoughts on
this question from anyone who intricately knows the encryption algorithms.
Are examples such as these of any interest to practising cryptographers?

Cheers,

Brad

[1] http://www.craigcostello.com.au/pairings/PairingsForBeginners.pdf
[2] https://www.youtube.com/watch?v=FUw4LB5v4Os
[3] https://ptpb.pw/GRdL.png
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