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linear distance

asked 2021-07-15 23:14:35 +0200

Miroslaw gravatar image

updated 2022-04-14 10:58:29 +0200

FrédéricC gravatar image

Hello

I want to take linear distance beetween two points on Eliptic curve:

curve : y2=x3+7 secp256k1

p_string = 'FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F'
p = ZZ( '0x' + p_string.replace(' ', ''))
p
E = EllipticCurve( GF(p), [0, 7] )  
E
n_string = 'FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141'
n = ZZ( '0x' + n_string.replace(' ', '') )



xP = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
yP = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
P = E.point( (xP, yP) ) # Generator point

We have two point on curve y2=x3+7 secp256k1 with above parameter.

Point no 1:

p_1_x=0x28f3c7982219bf067d9723a54bef8e0ae7d970b4285a148685ce98819bd59c99
p_1_y=0x3da6945a85ba67249e96534f3cb93af144301e3ef03f848ebf17fedd55addd14
P1 = E.point( (p_1_x, p_1_y) )

Point no 2:

p_2_x=0xc69b120facc137ad7ecf7407b97f3e1bcecec27ed6617ad1e3bb4f5a265c2230
p_2_y=0xdab461d8de9d194b9aae05cf3fc361e31b0cd172985720a9963a1fd9cb4dab13
P2 = E.point( (p_2_x, p_2_y) )

and we want to know what is P1-P2.

P1 it is p1 * P ( integer * Generator)
P2 it is p2 * P ( integer * Generator)

so : (p1P) - (p2P), but we don't know p1 and p2 , we know only P1 and P2 as point on curve.

but we know when the 2 points P1 and P2 are equal .

the question is there any way to verify and take value P1-P2 as integer ? for example : if p1=5 and p2 =9 , then p1-p2 or P1-P2 = -4

calculation : perform add or mult for equal :

a= 82096204097375626523300693245739453783745399685178560726932375997561515334942 
P1_mult = P1 * a

b= 29894233847412895230616850443375888414306408440111129860511310128693381347651 
P2_add = P2 + b* P

so P1_mult == P2_add

P1 == P2 as P1_mult and P2_add , after calculation so from this expression can we calculate what is P1-P2? so what is linear distance in integer beetween P1 and P2 if we know above information?

at the moment no idea, any tips, solution, How to do?

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Comments

Please define "linear distance".

Max Alekseyev gravatar imageMax Alekseyev ( 2021-08-19 01:35:54 +0200 )edit

linear distance: How far are points from each other in integer values not in x,y coordinate

Miroslaw gravatar imageMiroslaw ( 2021-08-19 20:32:29 +0200 )edit

This is still unclear. Please give a formal definition.

Max Alekseyev gravatar imageMax Alekseyev ( 2021-08-19 23:35:17 +0200 )edit

edited question

Miroslaw gravatar imageMiroslaw ( 2021-08-20 09:20:23 +0200 )edit

Doesn't running P1 - P2 in Sage answer your question? It gives

(88142788360802579680349986501633147205599719003358746502313153312017233825713 : 3050891168603837968756581014575221549926190068273143424690722519788244245701 : 1)
Max Alekseyev gravatar imageMax Alekseyev ( 2021-08-20 23:04:52 +0200 )edit

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answered 2021-08-28 08:53:02 +0200

Miroslaw gravatar image

solved task

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Asked: 2021-07-15 23:14:35 +0200

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Last updated: Aug 28 '21