There seems to be some kind of mistake in this puzzle#110
A huge number of GPU capacities have already been spent and no one could find the key.
I checked the range borders with bsgs, there is no key there.
This was the only version why we could not find the key for so long.
But she is not true. The key is not near the edge.
Someone has an idea why we can’t find the key .. even with the possibility of 400 GPU?
Topic: Pollard's kangaroo ECDLP solver - page 119. (Read 55599 times)
With the new -wsplit option:
How are you using it?
The way I read, it saves an updated file let's say every -wi 60 seconds. Got it. I've tested it.
So the only way the key can be solved on the server side, is if a collision happens in the current, being worked on, reset/smaller hash table (the one that will be saved next)?
The server doesn't compare the different saved files for a collision, so one must merge the different saved working files to check for a collision.
If you have 10, 20, 30, 100, 1000 saved files over a period of days, weeks, months, are you manually merging the saved work files every so often?
The current merge option only allows the merge of 2 files at a time...manually merging could become an hourly task (depending on how many files you have being saved each minutes/hours.)
So you have a benefit in RAM but increases the amount of work you have to do in merging files?
Grabber and merger merge files in realtime, so i do not have any split files on server. Once file exist merger merge him with masterfile.How are you using it?
The way I read, it saves an updated file let's say every -wi 60 seconds. Got it. I've tested it.
So the only way the key can be solved on the server side, is if a collision happens in the current, being worked on, reset/smaller hash table (the one that will be saved next)?
The server doesn't compare the different saved files for a collision, so one must merge the different saved working files to check for a collision.
If you have 10, 20, 30, 100, 1000 saved files over a period of days, weeks, months, are you manually merging the saved work files every so often?
The current merge option only allows the merge of 2 files at a time...manually merging could become an hourly task (depending on how many files you have being saved each minutes/hours.)
So you have a benefit in RAM but increases the amount of work you have to do in merging files?
all what you need it is set -jobtime to value (wi/2*1000). If -wi set to 60, than you need -jobtime 30000, -jobtime in ms.
@JeanLuc in next release if it possible fix please addiding date string to workfile before extention.
I mean i use file -w savework1.part but after saving got savework1.part_27May20_160341 should be savework1_27May20_160341.part I think. Thanks.
Edit:With -d 28, -wi 600, memory usage by server 10-20MB( it think -wsplit is great tools)
I mean i use file -w savework1.part but after saving got savework1.part_27May20_160341 should be savework1_27May20_160341.part I think. Thanks.
Edit:With -d 28, -wi 600, memory usage by server 10-20MB( it think -wsplit is great tools)
I recompiled a version that does similar to what you asked.
Simply changed the code in Backup.cpp
Code:
string fileName = workFile;
if(splitWorkfile)
//fileName = workFile + "_" + Timer::getTS();
fileName = Timer::getTS() + "_" + workFile;
if(splitWorkfile)
//fileName = workFile + "_" + Timer::getTS();
fileName = Timer::getTS() + "_" + workFile;
works as described.
Instead of saving as savework1.part_27May20_160341, it saves as 27May20_160341_savework1.part
This way your .part extension is at the end of the file.
With the new -wsplit option:
How are you using it?
The way I read, it saves an updated file let's say every -wi 60 seconds. Got it. I've tested it.
So the only way the key can be solved on the server side, is if a collision happens in the current, being worked on, reset/smaller hash table (the one that will be saved next)?
The server doesn't compare the different saved files for a collision, so one must merge the different saved working files to check for a collision.
If you have 10, 20, 30, 100, 1000 saved files over a period of days, weeks, months, are you manually merging the saved work files every so often?
The current merge option only allows the merge of 2 files at a time...manually merging could become an hourly task (depending on how many files you have being saved each minutes/hours.)
So you have a benefit in RAM but increases the amount of work you have to do in merging files?
How are you using it?
The way I read, it saves an updated file let's say every -wi 60 seconds. Got it. I've tested it.
So the only way the key can be solved on the server side, is if a collision happens in the current, being worked on, reset/smaller hash table (the one that will be saved next)?
The server doesn't compare the different saved files for a collision, so one must merge the different saved working files to check for a collision.
If you have 10, 20, 30, 100, 1000 saved files over a period of days, weeks, months, are you manually merging the saved work files every so often?
The current merge option only allows the merge of 2 files at a time...manually merging could become an hourly task (depending on how many files you have being saved each minutes/hours.)
So you have a benefit in RAM but increases the amount of work you have to do in merging files?
You fell from heaven with this tool, man! Just tell me how to set -ext knowing that the latest release adds the ending without .part?
-ext you can set what ever you want, just it was a word in the name of small files.Example. you have -w mysmallfile on server, when it will be save with -wsplit it will look like mysmallfile_27May20_232415
So you need set -ext like smallfile or small.. if only the grabbber could distinguish this file from others
Jean_Luc, I want to look at the table which is saved to workfile. But as I understood, only 128bit are saved for X-coordinate and 126bit for distance (together with 1 bit for sign and 1 bit for kangaroo type).
Anyway, what is the easiest way to receive the whole table in txt format. I easily could read from binary file the head, dp, start/stop ranges, x/y coordinates for key. After that the hash table is saved with a lot of 0 bytes....
Can you just briefly describe the hash table structure which is saved to binary file?
Anyway, what is the easiest way to receive the whole table in txt format. I easily could read from binary file the head, dp, start/stop ranges, x/y coordinates for key. After that the hash table is saved with a lot of 0 bytes....
Can you just briefly describe the hash table structure which is saved to binary file?
if you still haven't managed to export the workfile, here are the changes I made to the code to export the DPs to txt.
The X-coord is 128+18 bits and distance 126bits. The X-coordinate could be re-generated from the distance if necessary.
https://github.com/PatatasFritas/FriedKangaroo/commit/1669df5f91ef2cc8a7619b21f66883fa164ab602
If someone wants to run a solver with small DPs, but the server’s resources don’t allow it, then you can use the -wsplit option,
which appears in version 1.7 of the solver.
But in any case, you must have a PC that can merge files. I just had such a problem.
Now I can safely merge files on my home PC. In order not to do it all manually, you need a grabber and a merger.
File grabber is launched on the server, merger is launched on the home PC.
Merger communicates with the grabber and requests files from him. The graber sends, if any, and then removes them from the server.
The merger, in turn, after receiving the file from the grabber, starts the merge process, during which it is possible to find the key, after merge temp file deleted.
Grabber gives files only to an authorized merger.
If it helps someone, archive with source codes, compiled programs and example .bat files: https://drive.google.com/file/d/1YB3tz_tkJWTilXMCm_3GIfKsODGzDTqi
As before, the sources under Purebasic.
mergeServer(grabber):
which appears in version 1.7 of the solver.
But in any case, you must have a PC that can merge files. I just had such a problem.
Now I can safely merge files on my home PC. In order not to do it all manually, you need a grabber and a merger.
File grabber is launched on the server, merger is launched on the home PC.
Merger communicates with the grabber and requests files from him. The graber sends, if any, and then removes them from the server.
The merger, in turn, after receiving the file from the grabber, starts the merge process, during which it is possible to find the key, after merge temp file deleted.
Grabber gives files only to an authorized merger.
If it helps someone, archive with source codes, compiled programs and example .bat files: https://drive.google.com/file/d/1YB3tz_tkJWTilXMCm_3GIfKsODGzDTqi
As before, the sources under Purebasic.
mergeServer(grabber):
Code:
-pass >password for merger authorization
-port >listening port, where merger will be connect
-ext >by this extension, the grabber will search for files in the folder,
for ex. using -ext part, than you should start server with -w xxxx.part
mergeClient(merger):-port >listening port, where merger will be connect
-ext >by this extension, the grabber will search for files in the folder,
for ex. using -ext part, than you should start server with -w xxxx.part
Code:
-jobtime 60000>request a file from the grabber every 60s
-name >it is name of your merger, useless, just for stats
-pass >password for authorization(should be the same as in grabber)
-server >host:port grabber
-workfile >name of your masterfile
-merger >Kangaroo.exe by default
-name >it is name of your merger, useless, just for stats
-pass >password for authorization(should be the same as in grabber)
-server >host:port grabber
-workfile >name of your masterfile
-merger >Kangaroo.exe by default
You fell from heaven with this tool, man! Just tell me how to set -ext knowing that the latest release adds the ending without .part?
If someone wants to run a solver with small DPs, but the server’s resources don’t allow it, then you can use the -wsplit option,
which appears in version 1.7 of the solver.
But in any case, you must have a PC that can merge files. I just had such a problem.
Now I can safely merge files on my home PC. In order not to do it all manually, you need a grabber and a merger.
File grabber is launched on the server, merger is launched on the home PC.
Merger communicates with the grabber and requests files from him. The graber sends, if any, and then removes them from the server.
The merger, in turn, after receiving the file from the grabber, starts the merge process, during which it is possible to find the key, after merge temp file deleted.
Grabber gives files only to an authorized merger.
If it helps someone, archive with source codes, compiled programs and example .bat files: https://drive.google.com/file/d/1wQWLCRsYY2s4DH2OZHmyTMMxIPn8kdsz
Edit: fixed little memory leak at grabber side.
As before, the sources under Purebasic.
mergeServer(grabber):
which appears in version 1.7 of the solver.
But in any case, you must have a PC that can merge files. I just had such a problem.
Now I can safely merge files on my home PC. In order not to do it all manually, you need a grabber and a merger.
File grabber is launched on the server, merger is launched on the home PC.
Merger communicates with the grabber and requests files from him. The graber sends, if any, and then removes them from the server.
The merger, in turn, after receiving the file from the grabber, starts the merge process, during which it is possible to find the key, after merge temp file deleted.
Grabber gives files only to an authorized merger.
If it helps someone, archive with source codes, compiled programs and example .bat files: https://drive.google.com/file/d/1wQWLCRsYY2s4DH2OZHmyTMMxIPn8kdsz
Edit: fixed little memory leak at grabber side.
As before, the sources under Purebasic.
mergeServer(grabber):
Code:
-pass >password for merger authorization
-port >listening port, where merger will be connect
-ext >by this extension, the grabber will search for files in the folder,
for ex. using -ext part, than you should start server with -w xxxx.part
mergeClient(merger):-port >listening port, where merger will be connect
-ext >by this extension, the grabber will search for files in the folder,
for ex. using -ext part, than you should start server with -w xxxx.part
Code:
-jobtime 60000>request a file from the grabber every 60s
-name >it is name of your merger, useless, just for stats
-pass >password for authorization(should be the same as in grabber)
-server >host:port grabber
-workfile >name of your masterfile
-merger >Kangaroo.exe by default
-name >it is name of your merger, useless, just for stats
-pass >password for authorization(should be the same as in grabber)
-server >host:port grabber
-workfile >name of your masterfile
-merger >Kangaroo.exe by default
I tested -wsplit in the 89bit range. Works perfet! Thanks JeanLuc.
Grabber take files from server and send to local PC, where merger merged this files with masterfile.
Key solved during merge after 32minutes on 2080ti. I did not wait until the key is solved on the server, enough that it was found through merge.
Grabber take files from server and send to local PC, where merger merged this files with masterfile.
Key solved during merge after 32minutes on 2080ti. I did not wait until the key is solved on the server, enough that it was found through merge.
Many thanks for the test
I'm currently working on the merger trying to speed it up and decrease memory consumption.
I tested -wsplit in the 89bit range. Works perfet! Thanks JeanLuc.
Grabber take files from server and send to local PC, where merger merged this files with masterfile.
Key solved during merge after 32minutes on 2080ti. I did not wait until the key is solved on the server, enough that it was found through merge.
Grabber take files from server and send to local PC, where merger merged this files with masterfile.
Key solved during merge after 32minutes on 2080ti. I did not wait until the key is solved on the server, enough that it was found through merge.
-snip-
Different private keys, different positions, different points, different public keys.
Different private keys, different positions, different points, different public keys.
Ok, got it. It is the group axiom.
-snip-
1) for each private key there is only 1 point (n private keys, n points, n is the order of the curve)
-snip-
1) for each private key there is only 1 point (n private keys, n points, n is the order of the curve)
-snip-
I know that EC collision is not proved as there is no example.
Yes, for each private key there is only 1 point, however not necessary that if we have n private key we will have n different points.
Simple example for the group of 10 elements:
Code:
Key Point
1 7
2 6
3 5
4 7
5 3
6 2
7 7
8 0
9 9
0 7
1 7
2 6
3 5
4 7
5 3
6 2
7 7
8 0
9 9
0 7
You can see that for each key (0..9) there is only one Point, i.e. we do not receive 2 or 3 or more points for each key. We have only one point. However key 1,4,7 and 0 lead to Point 7 (collision). For group of 10 elements it is easy to check.
However how could you be sure that for group with almost 2^256 elements for every private key we have the unique Point?
'Group' has a specific meaning in math, it is not a simple set of elements, moreover a group with a prime number of elements has the property to be cyclic, i. e. you can 'generate' each element using only one element of the group (but not zero) and using the operation of the group: adding it to itself: G, G+G=2*G, G+G+G=3*G, ..... it is like to assign a number, a scalar, a order to each element from the point of view of G.
In the secp256k1 the group is a group of points, a cyclic group, and each private key is the 'position' of each point from the point of view of G (a chosen point). Because it is cyclic, when you perform G, G+G=2*G, G+G+G=3*G, ...., you generate all the elements of the group before to return to G.
Different private keys, different positions, different points, different public keys.
If you apply the Fermat's little theorem in a additive group, a*(n+1) = a, a+a+a+a+a....+a=a and
(n+1)*G = G.
-snip-
1) for each private key there is only 1 point (n private keys, n points, n is the order of the curve)
-snip-
1) for each private key there is only 1 point (n private keys, n points, n is the order of the curve)
-snip-
I know that EC collision is not proved as there is no example.
Yes, for each private key there is only 1 point, however not necessary that if we have n private key we will have n different points.
Simple example for the group of 10 elements:
Code:
Key Point
1 7
2 6
3 5
4 7
5 3
6 2
7 7
8 0
9 9
0 7
1 7
2 6
3 5
4 7
5 3
6 2
7 7
8 0
9 9
0 7
You can see that for each key (0..9) there is only one Point, i.e. we do not receive 2 or 3 or more points for each key. We have only one point. However key 1,4,7 and 0 lead to Point 7 (collision). For group of 10 elements it is easy to check.
However how could you be sure that for group with almost 2^256 elements for every private key we have the unique Point?
-snip-
There are about 2^256 points, then 2^255 different X-coordinates (k*G and -k*G have the same x-coordinate).
-snip-
There are about 2^256 points, then 2^255 different X-coordinates (k*G and -k*G have the same x-coordinate).
-snip-
Is this proved fact?
I mean why every private key leads to the unique X-coordinate? (except for symmetry point).
In other words, if k and (order-k) leads to x-coordinate Xk, how could we be sure that there are no other key m leads to Xk as well? (m differs from k and (order-k) )
Because it is proved that:
1) for each private key there is only 1 point (n private keys, n points, n is the order of the curve)
2) for each x-coordinate, the y-coordinate must fulfil this relation: y^2 = x^3 + 7 mod n;
each equation of the form y^2 = a has or 2 opposite solutions (+y and -y -> k*G / -k*G), or has no solution.
Therefor there can't be 3 different points with the same x-coordinate and with 3 different y-coordinate, because there are no 3 different y solutions for the equation y^2 = a mod n.
We know that there are exactly (n-1)/2 different x-coordinates (and (n-1)/3 different y-coordinates)
That's how the math works. The points on the curve form a group (https://en.wikipedia.org/wiki/Group_%28mathematics%29) and the point G is a generator of that group.
-snip-
There are about 2^256 points, then 2^255 different X-coordinates (k*G and -k*G have the same x-coordinate).
-snip-
There are about 2^256 points, then 2^255 different X-coordinates (k*G and -k*G have the same x-coordinate).
-snip-
Is this proved fact?
I mean why every private key leads to the unique X-coordinate? (except for symmetry point).
In other words, if k and (order-k) leads to x-coordinate Xk, how could we be sure that there are no other key m leads to Xk as well? (m differs from k and (order-k) )
Because it is proved that:
1) for each private key there is only 1 point (n private keys, n points, n is the order of the curve)
2) for each x-coordinate, the y-coordinate must fulfil this relation: y^2 = x^3 + 7 mod p;
each equation of the form y^2 = a mod p has or 2*** opposite solutions (+y and -y -> k*G / -k*G), or has no solution.
Therefor there can't be 3 different points with the same x-coordinate and with 3 different y-coordinates, because there are no 3 different y solutions for the equation y^2 = a mod p.
We know that there are exactly (n-1)/2 different x-coordinates (and (n-1)/3 different y-coordinates)
***EDIT:
according to the Fermat's "little theorem"
https://brilliant.org/wiki/fermats-little-theorem/
a^p = a, then a^(p+1) = a^2 mod p then a^(p+1)/4 = sqrt(a) mod p
(note: in the secp256k1 p = 3 mod 4 then p+1 = 4 mod 4)
if y = a^(p+1)/4 is the sqrt(a) then y = -(a^(p+1)/4) is the sqrt(a) too, i.e. to the power of 2 they both produce the same result, a, that in our case is X^3 + 7.
-snip-
There are about 2^256 points, then 2^255 different X-coordinates (k*G and -k*G have the same x-coordinate).
-snip-
There are about 2^256 points, then 2^255 different X-coordinates (k*G and -k*G have the same x-coordinate).
-snip-
Is this proved fact?
I mean why every private key leads to the unique X-coordinate? (except for symmetry point).
In other words, if k and (order-k) leads to x-coordinate Xk, how could we be sure that there are no other key m leads to Xk as well? (m differs from k and (order-k) )
@JeanLuc in next release if it possible fix please addiding date string to workfile before extention.
I mean i use file -w savework1.part but after saving got savework1.part_27May20_160341 should be savework1_27May20_160341.part I think. Thanks.
Edit:With -d 28, -wi 600, memory usage by server 10-20MB( it think -wsplit is great tools)
I mean i use file -w savework1.part but after saving got savework1.part_27May20_160341 should be savework1_27May20_160341.part I think. Thanks.
Edit:With -d 28, -wi 600, memory usage by server 10-20MB( it think -wsplit is great tools)
Does anybody know how to estimate/calculate the total possible DPs within the range? Moreover, it is also possible that for some DP value there will not be any DPs.
For example, the minimum X-coordinate value I know is for private key 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 and for 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1
Their X-coordinate has 90 leading zeros, so it is DP=90
Probably there are only 2 X-coordinates with 90 leading zeros, and no more.
For example, the minimum X-coordinate value I know is for private key 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 and for 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1
Their X-coordinate has 90 leading zeros, so it is DP=90
Probably there are only 2 X-coordinates with 90 leading zeros, and no more.
There are about 2^256 points, then 2^255 different X-coordinates (k*G and -k*G have the same x-coordinate).
Probably there are:
2^255 / 2 X-coordinates with 1 zero leading bit
2^255 / 4 X-coordinates with 2 zeros leading bits
….
2^255 / 2^90 X-coordinates with 90 zeros leading bits
…
If you sum 2^254 + 2^253 + ….+ 2^165 + …. = 2^254 (1/2 + 1/4 + 1/8 + ...) = 2^254 X-coordinates with at least 1 zero leading bit.
With DP very very large (like DP=250) you can manually determine how many x-coordinates there are (if there are) such that Y^2 = X^3 + 7 mod p
On average 1 number each 2 is a valid X-coordinate (for a estimate).
Here there are 150 X-coordinates with (at least) 40 zeros leading bits:
Format:
X-coordinate private key
* stays for: private key * lambda
** stays for: private key * lambda * lambda
Code:
0000000000e1387e34763704d1b149a7cdc086395346907a49ed7963ae5e4837 3400000000000000000000000000000000000000000000000000000035d80cda *
00000000000782001b949535174e508842e6945495da73f59adb47064176d233 010000000000000000000000000000000000000000000000000000007a44c2f0
000000000078584f51938406db35438ed5c4189f2fd2b1e6ca8a6a2179bbd808 400000000000000000000000000000000000000000000000000000007121b07e **
0000000000eaf2052284bdad2bf8cd17d99b2ffe046244a0044b261f8df674a8 4300000000000000000000000000000000000000000000000000000127924a01 **
0000000000746daafebc7e7d83963f657a9546fff669bbab8ec785946a7059ca 4e0000000000000000000000000000000000000000000000000000039345bfcd **
00000000000e258648b5edddf578a965f74363983fee1fa2fc9af00720c3c79d 2000000000000000000000000000000000000000000000000000000564b8a5ec **
000000000038641aed1bb3ea0b9eba5f75da61de20da2f9b5c2ba9e7f8ce47e5 04000000000000000000000000000000000000000000000000000006c2049732
00000000003161df5d55df74ea73f01281bd8da8926bd9fab14e69c0162f9821 4d000000000000000000000000000000000000000000000000000008bf8f3af4 **
00000000000ba89625cdfc9561b2ba99b8726d904115cbeae6823bde71aa52b0 03000000000000000000000000000000000000000000000000000009339b1d3a
000000000048d030ecd0e435656bb8645cc65914af61b30834b4793062e9c306 07000000000000000000000000000000000000000000000000000009b9cd0716 **
000000000037527338a68f66379f010f6e001ec69175d8dde8606687068b64b6 2e00000000000000000000000000000000000000000000000000000a88164703 **
0000000000114ab80b0e1804be9a7720ebd23000298d45f1e126c4329026b387 3600000000000000000000000000000000000000000000000000000bc53beb3d
0000000000ca3291e849f0b2923ab4217c419a1a665c7aba06a5c422df7387a2 2f00000000000000000000000000000000000000000000000000000d9dafab78
00000000006abf9e3898b80935072e5e54f75d78139698321a079765fbb87077 4400000000000000000000000000000000000000000000000000000f7cc1c6e0 *
000000000082d7e503357813e06a1d496304099dd11156d6059837880a8a3852 400000000000000000000000000000000000000000000000000000112b4092c2 *
0000000000be7f258426d9fa89cbc7c04ffbdef4279bd9b797dc41951cec9ea8 47000000000000000000000000000000000000000000000000000013433071df *
0000000000be1e4abfe878e51be079b3da54f95ef87043ef131a38555d90b288 210000000000000000000000000000000000000000000000000000144097c2c6 **
000000000062eb05bda779f9d06aabd0710c96f9b9835e92af344df20e16b280 4a000000000000000000000000000000000000000000000000000014892c0ecc *
0000000000e14b6206085b683ca9cb4d18ba897fc140b72d5d4a98e85e848393 2e000000000000000000000000000000000000000000000000000012e841e014
000000000099d3ee7e7e15306ec2ed021fee5a4bf7446e3ed801a6b9ad43bc7e 3200000000000000000000000000000000000000000000000000001586220c38
0000000000290459244c0db66eb013fd5ed2fbc4b065e93c4bbe7789431840ef 4900000000000000000000000000000000000000000000000000001795d5ee70 **
00000000009370afb5c0787686cf10b47aa16de5b57bfcee6085c863cbdde7be 0f00000000000000000000000000000000000000000000000000001c4b397168 **
00000000002970866989ac7adc6e39f092b750b157d1fd0ff88b7cab6c15069d 2c00000000000000000000000000000000000000000000000000001aee157f73 **
000000000056a12f7211ec469bda752225200948a9df628ec29fdd71d7b1f255 4e00000000000000000000000000000000000000000000000000001b25cfb7fa
0000000000ffe1af9405c6cb2cd0301d205e438c3acfcb89f6706c0991747dff 4700000000000000000000000000000000000000000000000000001eb6d634ed
00000000005b407be3fd7fce7e511f36b98c0d6c6d2b41e36be06b1fa55b9d18 010000000000000000000000000000000000000000000000000000206053b7e1 **
00000000004d0c8af1e64a02cbe2515dbed58305db33d0fe607ff8d65e59fb4d 2600000000000000000000000000000000000000000000000000002059016305
0000000000d607ab339cd53605e3daed98ab12b56454a9c390be43b776e2bbd8 4700000000000000000000000000000000000000000000000000002192d3ae07 **
0000000000ba690b40254396533f88548f47e370aba84161d0a6acd3ae7d98d5 0f000000000000000000000000000000000000000000000000000022802481ec *
00000000005e014bd8b52e557973bad624de8338f64fdb4966c90c2c6c30f072 4400000000000000000000000000000000000000000000000000002305df5f3b *
00000000006dd652e5f2dbf67105fa9f4848a2802650923b8dd8fec131d2c1e5 2100000000000000000000000000000000000000000000000000002483eca1c2
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The lowest X-coordinate I have found is:
000000000000aeea7a7a5c04504f6e4e45452940431c9a264011686f3b746f92
and its private key is:
2000000000000000000000000000000000000000000000000000013a80306e86
If you want a pubkey with a X having more than 90 leading zeros you will need to perform ~2^90 group operation to find it.
The 2 points you mentioned are the privilege of the gui who have chosen G.
The 2 points you mentioned are the privilege of the gui who have chosen G.
Does anybody know how to estimate/calculate the total possible DPs within the range? Moreover, it is also possible that for some DP value there will not be any DPs.
For example, the minimum X-coordinate value I know is for private key 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 and for 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1
Their X-coordinate has 90 leading zeros, so it is DP=90
Probably there are only 2 X-coordinates with 90 leading zeros, and no more.
Does anybody know some examples with more than 90 leading zeros in x-coordinate with known private keys?
PS. Example above is for (order-1)/2 and for (order-1)/2+1
For example, the minimum X-coordinate value I know is for private key 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 and for 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1
Their X-coordinate has 90 leading zeros, so it is DP=90
Probably there are only 2 X-coordinates with 90 leading zeros, and no more.
Does anybody know some examples with more than 90 leading zeros in x-coordinate with known private keys?
PS. Example above is for (order-1)/2 and for (order-1)/2+1
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