this is something that has not been tried yet, and i think it relates really well to what we are trying to solve. i linked BENALOHPAILLIER, alice and bob, how deep the rabbit hole goes, and time reveals all. someone has access to running this in a script hopefully, i did all the leg work, just need someone to solve. divided up into a few parts. He said he would give us a REALLY HARD ONE, that can be solved, i think this would qualify.
its based on
A common use of elliptic curve cryptography is to generate a shared secret key for use in secret key systems. The Diffie-Hellman protocol can be applied to elliptic curves as follows:
A’s secret number = 2101
Scalar Multiply: 2101*(2,9)
Calculating 1*(2,9) = 0*(2,9) + 1*(2,9) = (Zero) + (2,9)
Adding (Zero) and (2,9)
Sum is (2,9)
Working result: 1*(2,9) = (2,9)
Doubling 1*(2,9) = (2,9)
x1^2 = 4
3x1^2 = 12
3x1^2+a = 16
2y1 = 18
lambda = (3x1^2+a)/(2y1) = 64
lambda^2 = 49
x3 = lambda^2-x1-x1 = 45
x1-x3 = 28
lambda(x1-x3) = 17
y3 = lambda*(x-x3)-x1 = 8
Double value: 2*(2,9) = (45,8)
Doubling 2*(2,9) = (45,8)
x1^2 = 37
3x1^2 = 40
3x1^2+a = 44
2y1 = 16
lambda = (3x1^2+a)/(2y1) = 56
lambda^2 = 12
x3 = lambda^2-x1-x1 = 64
x1-x3 = 52
lambda(x1-x3) = 1
y3 = lambda*(x-x3)-x1 = 64
Double value: 4*(2,9) = (64,64)
Calculating 5*(2,9) = 1*(2,9) + 4*(2,9) = (2,9) + (64,64)
Adding (2,9) and (64,64)
y2-y1 = 55
x2-x1 = 62
lambda = (y2-y1)/(x2-x1) = 57
lambda^2 = 54
x3 = lambda - x1 - x2 = 59
x1-x3 = 14
lambda*(x1-x3) = 17
y3 = lambda*(x1-x3)-y1 = 8
Sum is (59,8)
Working result: 5*(2,9) = (59,8)
Doubling 4*(2,9) = (64,64)
x1^2 = 49
3x1^2 = 5
3x1^2+a = 9
2y1 = 57
lambda = (3x1^2+a)/(2y1) = 45
lambda^2 = 37
x3 = lambda^2-x1-x1 = 51
x1-x3 = 13
lambda(x1-x3) = 17
y3 = lambda*(x-x3)-x1 = 24
Double value: 8*(2,9) = (51,24)
Doubling 8*(2,9) = (51,24)
x1^2 = 45
3x1^2 = 64
3x1^2+a = 68
2y1 = 48
lambda = (3x1^2+a)/(2y1) = 31
lambda^2 = 38
x3 = lambda^2-x1-x1 = 7
x1-x3 = 44
lambda(x1-x3) = 15
y3 = lambda*(x-x3)-x1 = 62
Double value: 16*(2,9) = (7,62)
Calculating 21*(2,9) = 5*(2,9) + 16*(2,9) = (59,8) + (7,62)
Adding (59,8) and (7,62)
y2-y1 = 54
x2-x1 = 19
lambda = (y2-y1)/(x2-x1) = 29
lambda^2 = 60
x3 = lambda - x1 - x2 = 65
x1-x3 = 65
lambda*(x1-x3) = 39
y3 = lambda*(x1-x3)-y1 = 31
Sum is (65,31)
Working result: 21*(2,9) = (65,31)
Doubling 16*(2,9) = (7,62)
x1^2 = 49
3x1^2 = 5
3x1^2+a = 9
2y1 = 53
lambda = (3x1^2+a)/(2y1) = 35
lambda^2 = 18
x3 = lambda^2-x1-x1 = 4
x1-x3 = 3
lambda(x1-x3) = 34
y3 = lambda*(x-x3)-x1 = 43
Double value: 32*(2,9) = (4,43)
Calculating 53*(2,9) = 21*(2,9) + 32*(2,9) = (65,31) + (4,43)
Adding (65,31) and (4,43)
y2-y1 = 12
x2-x1 = 10
lambda = (y2-y1)/(x2-x1) = 58
lambda^2 = 27
x3 = lambda - x1 - x2 = 29
x1-x3 = 36
lambda*(x1-x3) = 29
y3 = lambda*(x1-x3)-y1 = 69
Sum is (29,69)
Working result: 53*(2,9) = (29,69)
Doubling 32*(2,9) = (4,43)
x1^2 = 16
3x1^2 = 48
3x1^2+a = 52
2y1 = 15
lambda = (3x1^2+a)/(2y1) = 65
lambda^2 = 36
x3 = lambda^2-x1-x1 = 28
x1-x3 = 47
lambda(x1-x3) = 2
y3 = lambda*(x-x3)-x1 = 30
Double value: 64*(2,9) = (28,30)
Doubling 64*(2,9) = (28,30)
x1^2 = 3
3x1^2 = 9
3x1^2+a = 13
2y1 = 60
lambda = (3x1^2+a)/(2y1) = 44
lambda^2 = 19
x3 = lambda^2-x1-x1 = 34
x1-x3 = 65
lambda(x1-x3) = 20
y3 = lambda*(x-x3)-x1 = 61
Double value: 128*(2,9) = (34,61)
Doubling 128*(2,9) = (34,61)
x1^2 = 20
3x1^2 = 60
3x1^2+a = 64
2y1 = 51
lambda = (3x1^2+a)/(2y1) = 11
lambda^2 = 50
x3 = lambda^2-x1-x1 = 53
x1-x3 = 52
lambda(x1-x3) = 4
y3 = lambda*(x-x3)-x1 = 14
Double value: 256*(2,9) = (53,14)
Doubling 256*(2,9) = (53,14)
x1^2 = 40
3x1^2 = 49
3x1^2+a = 53
2y1 = 28
lambda = (3x1^2+a)/(2y1) = 45
lambda^2 = 37
x3 = lambda^2-x1-x1 = 2
x1-x3 = 51
lambda(x1-x3) = 23
y3 = lambda*(x-x3)-x1 = 9
Double value: 512*(2,9) = (2,9)
Doubling 512*(2,9) = (2,9)
x1^2 = 4
3x1^2 = 12
3x1^2+a = 16
2y1 = 18
lambda = (3x1^2+a)/(2y1) = 64
lambda^2 = 49
x3 = lambda^2-x1-x1 = 45
x1-x3 = 28
lambda(x1-x3) = 17
y3 = lambda*(x-x3)-x1 = 8
Double value: 1024*(2,9) = (45,8)
Doubling 1024*(2,9) = (45,8)
x1^2 = 37
3x1^2 = 40
3x1^2+a = 44
2y1 = 16
lambda = (3x1^2+a)/(2y1) = 56
lambda^2 = 12
x3 = lambda^2-x1-x1 = 64
x1-x3 = 52
lambda(x1-x3) = 1
y3 = lambda*(x-x3)-x1 = 64
Double value: 2048*(2,9) = (64,64)
Calculating 2101*(2,9) = 53*(2,9) + 2048*(2,9) = (29,69) + (64,64)
Adding (29,69) and (64,64)
y2-y1 = 66
x2-x1 = 35
lambda = (y2-y1)/(x2-x1) = 10
lambda^2 = 29
x3 = lambda - x1 - x2 = 7
x1-x3 = 22
lambda*(x1-x3) = 7
y3 = lambda*(x1-x3)-y1 = 9
Sum is (7,9)
Working result: 2101*(2,9) = (7,9)
Product is (7,9)
A sends (7,9)
B’s secret number = 2103
Scalar Multiply: 2103*(2,9)
Calculating 1*(2,9) = 0*(2,9) + 1*(2,9) = (Zero) + (2,9)
Adding (Zero) and (2,9)
Sum is (2,9)
Working result: 1*(2,9) = (2,9)
Doubling 1*(2,9) = (2,9)
x1^2 = 4
3x1^2 = 12
3x1^2+a = 16
2y1 = 18
lambda = (3x1^2+a)/(2y1) = 64
lambda^2 = 49
x3 = lambda^2-x1-x1 = 45
x1-x3 = 28
lambda(x1-x3) = 17
y3 = lambda*(x-x3)-x1 = 8
Double value: 2*(2,9) = (45,8)
Calculating 3*(2,9) = 1*(2,9) + 2*(2,9) = (2,9) + (45,8)
Adding (2,9) and (45,8)
y2-y1 = 70
x2-x1 = 43
lambda = (y2-y1)/(x2-x1) = 33
lambda^2 = 24
x3 = lambda - x1 - x2 = 48
x1-x3 = 25
lambda*(x1-x3) = 44
y3 = lambda*(x1-x3)-y1 = 35
Sum is (48,35)
Working result: 3*(2,9) = (48,35)
Doubling 2*(2,9) = (45,8)
x1^2 = 37
3x1^2 = 40
3x1^2+a = 44
2y1 = 16
lambda = (3x1^2+a)/(2y1) = 56
lambda^2 = 12
x3 = lambda^2-x1-x1 = 64
x1-x3 = 52
lambda(x1-x3) = 1
y3 = lambda*(x-x3)-x1 = 64
Double value: 4*(2,9) = (64,64)
Calculating 7*(2,9) = 3*(2,9) + 4*(2,9) = (48,35) + (64,64)
Adding (48,35) and (64,64)
y2-y1 = 29
x2-x1 = 16
lambda = (y2-y1)/(x2-x1) = 24
lambda^2 = 8
x3 = lambda - x1 - x2 = 38
x1-x3 = 10
lambda*(x1-x3) = 27
y3 = lambda*(x1-x3)-y1 = 63
Sum is (38,63)
Working result: 7*(2,9) = (38,63)
Doubling 4*(2,9) = (64,64)
x1^2 = 49
3x1^2 = 5
3x1^2+a = 9
2y1 = 57
lambda = (3x1^2+a)/(2y1) = 45
lambda^2 = 37
x3 = lambda^2-x1-x1 = 51
x1-x3 = 13
lambda(x1-x3) = 17
y3 = lambda*(x-x3)-x1 = 24
Double value: 8*(2,9) = (51,24)
Doubling 8*(2,9) = (51,24)
x1^2 = 45
3x1^2 = 64
3x1^2+a = 68
2y1 = 48
lambda = (3x1^2+a)/(2y1) = 31
lambda^2 = 38
x3 = lambda^2-x1-x1 = 7
x1-x3 = 44
lambda(x1-x3) = 15
y3 = lambda*(x-x3)-x1 = 62
Double value: 16*(2,9) = (7,62)
Calculating 23*(2,9) = 7*(2,9) + 16*(2,9) = (38,63) + (7,62)
Adding (38,63) and (7,62)
y2-y1 = 70
x2-x1 = 40
lambda = (y2-y1)/(x2-x1) = 55
lambda^2 = 43
x3 = lambda - x1 - x2 = 69
x1-x3 = 40
lambda*(x1-x3) = 70
y3 = lambda*(x1-x3)-y1 = 7
Sum is (69,7)
Working result: 23*(2,9) = (69,7)
Doubling 16*(2,9) = (7,62)
x1^2 = 49
3x1^2 = 5
3x1^2+a = 9
2y1 = 53
lambda = (3x1^2+a)/(2y1) = 35
lambda^2 = 18
x3 = lambda^2-x1-x1 = 4
x1-x3 = 3
lambda(x1-x3) = 34
y3 = lambda*(x-x3)-x1 = 43
Double value: 32*(2,9) = (4,43)
Calculating 55*(2,9) = 23*(2,9) + 32*(2,9) = (69,7) + (4,43)
Adding (69,7) and (4,43)
y2-y1 = 36
x2-x1 = 6
lambda = (y2-y1)/(x2-x1) = 6
lambda^2 = 36
x3 = lambda - x1 - x2 = 34
x1-x3 = 35
lambda*(x1-x3) = 68
y3 = lambda*(x1-x3)-y1 = 61
Sum is (34,61)
Working result: 55*(2,9) = (34,61)
Doubling 32*(2,9) = (4,43)
x1^2 = 16
3x1^2 = 48
3x1^2+a = 52
2y1 = 15
lambda = (3x1^2+a)/(2y1) = 65
lambda^2 = 36
x3 = lambda^2-x1-x1 = 28
x1-x3 = 47
lambda(x1-x3) = 2
y3 = lambda*(x-x3)-x1 = 30
Double value: 64*(2,9) = (28,30)
Doubling 64*(2,9) = (28,30)
x1^2 = 3
3x1^2 = 9
3x1^2+a = 13
2y1 = 60
lambda = (3x1^2+a)/(2y1) = 44
lambda^2 = 19
x3 = lambda^2-x1-x1 = 34
x1-x3 = 65
lambda(x1-x3) = 20
y3 = lambda*(x-x3)-x1 = 61
Double value: 128*(2,9) = (34,61)
Doubling 128*(2,9) = (34,61)
x1^2 = 20
3x1^2 = 60
3x1^2+a = 64
2y1 = 51
lambda = (3x1^2+a)/(2y1) = 11
lambda^2 = 50
x3 = lambda^2-x1-x1 = 53
x1-x3 = 52
lambda(x1-x3) = 4
y3 = lambda*(x-x3)-x1 = 14
Double value: 256*(2,9) = (53,14)
Doubling 256*(2,9) = (53,14)
x1^2 = 40
3x1^2 = 49
3x1^2+a = 53
2y1 = 28
lambda = (3x1^2+a)/(2y1) = 45
lambda^2 = 37
x3 = lambda^2-x1-x1 = 2
x1-x3 = 51
lambda(x1-x3) = 23
y3 = lambda*(x-x3)-x1 = 9
Double value: 512*(2,9) = (2,9)
Doubling 512*(2,9) = (2,9)
x1^2 = 4
3x1^2 = 12
3x1^2+a = 16
2y1 = 18
lambda = (3x1^2+a)/(2y1) = 64
lambda^2 = 49
x3 = lambda^2-x1-x1 = 45
x1-x3 = 28
lambda(x1-x3) = 17
y3 = lambda*(x-x3)-x1 = 8
Double value: 1024*(2,9) = (45,8)
Doubling 1024*(2,9) = (45,8)
x1^2 = 37
3x1^2 = 40
3x1^2+a = 44
2y1 = 16
lambda = (3x1^2+a)/(2y1) = 56
lambda^2 = 12
x3 = lambda^2-x1-x1 = 64
x1-x3 = 52
lambda(x1-x3) = 1
y3 = lambda*(x-x3)-x1 = 64
Double value: 2048*(2,9) = (64,64)
Calculating 2103*(2,9) = 55*(2,9) + 2048*(2,9) = (34,61) + (64,64)
Adding (34,61) and (64,64)
y2-y1 = 3
x2-x1 = 30
lambda = (y2-y1)/(x2-x1) = 64
lambda^2 = 49
x3 = lambda - x1 - x2 = 22
x1-x3 = 12
lambda*(x1-x3) = 58
y3 = lambda*(x1-x3)-y1 = 68
Sum is (22,68)
Working result: 2103*(2,9) = (22,68)
Product is (22,68)
B sends (22,68)