Brad Klee

2018-03-15 02:50:43 UTC

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] : Loading Image... )

============================

{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]

[3] https://ptpb.pw/GRdL.png

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] : Loading Image... )

============================

{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]

[3] https://ptpb.pw/GRdL.png