Sofar, i figured out that you can divide any pub keys with infinite numbers, and still get infinite valid results which goes back to the original pub key which results in infinite amount of collisions which results in an a big blackhole loop, that will never let you escape from it.
That's why I love Math. I saw some guy describing this as POINT_AT_INFINITY which i really agree with. Very well and nice describing
Topic: Public Key Division - Number Theore (Read 850 times)
Here is what I found, I'm not a mathematician but this should work and it can bend elliptic curve or rather straighten it.
56900346571233247÷5 = 11380069314246649.4 + n/5 + n/4 = 11380069314246649, I just couldn't find any meaningful result before, but I have tried the above method and it works.
It seems we can just divide it after all, but this won't help much if private key is unknown, these things are like fire for a child when they see it for the first time.
I wanted to say a monkey when you perform a magic trick in front of him, have you seen how excited they get? Lol.
n works like zero in real numbers, n/x = n, n+x = x and so on. 56900346571233247÷5 = 11380069314246649.4 + n/5 + n/4 = 11380069314246649, I just couldn't find any meaningful result before, but I have tried the above method and it works.
It seems we can just divide it after all, but this won't help much if private key is unknown, these things are like fire for a child when they see it for the first time.
I wanted to say a monkey when you perform a magic trick in front of him, have you seen how excited they get? Lol.
Quote
56900346571233247÷5 = 11380069314246649.4 + n/5 + n/4 = 11380069314246649, I just couldn't find any meaningful result before, but I have tried the above method and it works.
O, f,it is really work ??
Quote
Here is a problem I'm facing, it seems really easy to figure out, but I can't.
Because there is no solution. You can always divide by 5. In 20% cases, you will reach smaller key, in 80% some bigger one, just because every fifth number is divisible by 5. You can always divide by any number in range from 1 to n-1. Always.You can clearly see that when trying to reach 1/2. In real numbers, it is some value between zero and one. But in ECDSA world, it could be for example 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1. You won't get any error, like "this division is impossible". It will never be rounded down to zero, or rounded up to one.
Division, exactly in the same way as multiplication, is always possible for all numbers from 1 to n-1. If you explore it on some smaller values, for example n=67 from my avatar, then you can clearly see that. You will rotate like in a clock, always getting "some" result, and never getting to the point of "no answer". And this is the beauty of the finite field, this is what makes it strong, and when you understand it, you will see, why you cannot "just divide it".
56900346571233247÷5 = 11380069314246649.4 + n/5 + n/4 = 11380069314246649, I just couldn't find any meaningful result before, but I have tried the above method and it works.
It seems we can just divide it after all, but this won't help much if private key is unknown, these things are like fire for a child when they see it for the first time.
I wanted to say a monkey when you perform a magic trick in front of him, have you seen how excited they get? Lol.
Theoretically, if we could find a big number that by dividing it to the public key resulting in an exact number, we could solve the private key from public key.
But that's way too hard i guess, it's like guessing a whole 254-255 bits..
But that's way too hard i guess, it's like guessing a whole 254-255 bits..
Quote
Here is a problem I'm facing, it seems really easy to figure out, but I can't.
Because there is no solution. You can always divide by 5. In 20% cases, you will reach smaller key, in 80% some bigger one, just because every fifth number is divisible by 5. You can always divide by any number in range from 1 to n-1. Always.You can clearly see that when trying to reach 1/2. In real numbers, it is some value between zero and one. But in ECDSA world, it could be for example 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1. You won't get any error, like "this division is impossible". It will never be rounded down to zero, or rounded up to one.
Division, exactly in the same way as multiplication, is always possible for all numbers from 1 to n-1. If you explore it on some smaller values, for example n=67 from my avatar, then you can clearly see that. You will rotate like in a clock, always getting "some" result, and never getting to the point of "no answer". And this is the beauty of the finite field, this is what makes it strong, and when you understand it, you will see, why you cannot "just divide it".
obviously, if some private key is not divisible by 5, then you won't reach smaller key, you could get some bigger one as well. However, all ECDSA equations will be preserved in a given modulo.
I have this key :
Code:
02228c3cbe2d11af48c3b98bc2955cd2c293375f748e9aea9040ffba42c3f6be1e
When I subtract it from this key :Code:
0000000000000000000000000000000000000400000000000000000000000000
I get this result : *first resultCode:
0352a1087d5f64b39c33d54e02f678c50955770b736e2c7c646d0ee04e28fc83e7
Now here is another key :
Code:
038840d5318a6369971b6610cbef8a7cb998d8e106534881da28aabfefeec2f41b
If we subtract the above from this one :
Code:
0000000000000000000000000000000000000080000000000000000000000000
We will reach this key : which is 1/8 of *first result.Code:
037f32d1fbba6ec4ed91109538ea93afc9bb4f13e3052d319263c7bb05fcb109a7
Do you think there is a way to find any of the public keys above?
Quote
I tried inversion and it doesn't work
It has to work. Let's test it on some key, where none of us knows the private key:Code:
03 678AFDB0FE5548271967F1A67130B7105CD6A828E03909A67962E0EA1F61DEB6 Satoshi
02 2959350037EACA04CF26538AECA4402432BC2DEC271496446A326B22B7E08742 Satoshi/5=first
02 E8C7430A125F8FDA5B3AEA1C2C6CF7E404EE902B7150FE18CE397F2B4759AEC8 (Satoshi/5)*2
03 E1A4232B81AEA764680A74DC0F6CF58B26CB61048F0F70923DFD2F199B3C986C ((Satoshi/5)*2)*2=second
03 678AFDB0FE5548271967F1A67130B7105CD6A828E03909A67962E0EA1F61DEB6 first+second
See? I splitted the public key from the Genesis Block into the sum of two public keys, where the second one is four times bigger than the first one. That means, I reached 1/5th of that key, because (1/5)+(4/5)=1. And you can easily validate that on your side, you need only point doubling, and point addition, nothing else.02 2959350037EACA04CF26538AECA4402432BC2DEC271496446A326B22B7E08742 Satoshi/5=first
02 E8C7430A125F8FDA5B3AEA1C2C6CF7E404EE902B7150FE18CE397F2B4759AEC8 (Satoshi/5)*2
03 E1A4232B81AEA764680A74DC0F6CF58B26CB61048F0F70923DFD2F199B3C986C ((Satoshi/5)*2)*2=second
03 678AFDB0FE5548271967F1A67130B7105CD6A828E03909A67962E0EA1F61DEB6 first+second
Quote
if you divide +n odd key by 5 and then add n/5 to the result, you won't get near the 1/5 of your +n key.
Well, let's see that on some private keys:Code:
SHA-256("garlonicon")=272fc6644fedff1a897d6034bed23f61859e99440ee699033307976590316723
272fc6644fedff1a897d6034bed23f61859e99440ee699033307976590316723 d
a16ff47a7662cc9ee84c4670f2f6d97924556ffe6c267ebe16e623cf335da22e d/5=first
42dfe8f4ecc5993dd0988ce1e5edb2f38dfc031629045d406df9e9119685031b (d/5)*2
85bfd1e9d98b327ba13119c3cbdb65e71bf8062c5208ba80dbf3d2232d0a0636 ((d/5)*2)*2=second
272fc6644fedff1a897d6034bed23f61859e99440ee699033307976590316723 first+second
See? It works. Of course, obviously, if some private key is not divisible by 5, then you won't reach smaller key, you could get some bigger one as well. However, all ECDSA equations will be preserved in a given modulo. 272fc6644fedff1a897d6034bed23f61859e99440ee699033307976590316723 d
a16ff47a7662cc9ee84c4670f2f6d97924556ffe6c267ebe16e623cf335da22e d/5=first
42dfe8f4ecc5993dd0988ce1e5edb2f38dfc031629045d406df9e9119685031b (d/5)*2
85bfd1e9d98b327ba13119c3cbdb65e71bf8062c5208ba80dbf3d2232d0a0636 ((d/5)*2)*2=second
272fc6644fedff1a897d6034bed23f61859e99440ee699033307976590316723 first+second
Just use inversion:
I tried inversion and it doesn't work, if you divide +n odd key by 5 and then add n/5 to the result, you won't get near the 1/5 of your +n key. Code:
n=fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
n/2=7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1
-n/2=7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0
n/4=bfffffffffffffffffffffffffffffff0c0325ad0376782ccfddc6e99c28b0f1
n/5=66666666666666666666666666666665e445f1f5dfb6a67e4cba8c385348e6e7
n/6=d5555555555555555555555555555554463c62c03cbc85871fd9f975582d3661
n/2=7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1
-n/2=7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0
n/4=bfffffffffffffffffffffffffffffff0c0325ad0376782ccfddc6e99c28b0f1
n/5=66666666666666666666666666666665e445f1f5dfb6a67e4cba8c385348e6e7
n/6=d5555555555555555555555555555554463c62c03cbc85871fd9f975582d3661
I always use inversion and I'd never divide by anything other than n/2.
By the way, last time you messed up 1/2 with -1/2
I might have, but I don't recall, as I said I'm like a student learning here.😉
Thanks for nice answers everyone,
Yes, though I thought i was smart and decided to try to keep dividing public key by 2 to the lowest point reaching base point but I Failed Everytime haha.
I think the only way is to guess ODD VS EVEN but that's impossible since private keys are random numbers.
I think it's like guessing 0 or 1 256 times, since there are 5 odd numbers and 5 even numbers in each new value. So the complexity stays the same as before.
Unless of course someone finds a way to know Odd private keys from EVEN, then he can divide by exact reaching his targeting key.
I like your examples guys!
Yes, though I thought i was smart and decided to try to keep dividing public key by 2 to the lowest point reaching base point but I Failed Everytime haha.
I think the only way is to guess ODD VS EVEN but that's impossible since private keys are random numbers.
I think it's like guessing 0 or 1 256 times, since there are 5 odd numbers and 5 even numbers in each new value. So the complexity stays the same as before.
Unless of course someone finds a way to know Odd private keys from EVEN, then he can divide by exact reaching his targeting key.
I like your examples guys!
Quote
The result is not random at all, did you even read my first post and other posts?
Random in, random out. Unknown in, unknown out. Infinity in, infinity out. It is that simple. If you divide some public key, where you don't know the private key, the result is totally random from your perspective. It may be non-random for the key owner, but not for the attacker.Quote
I have tried 4, 5, 6 etc, I couldn't get any meaningful result, I'm still learning.
Just use inversion:Code:
n=fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
n/2=7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1
-n/2=7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0
n/4=bfffffffffffffffffffffffffffffff0c0325ad0376782ccfddc6e99c28b0f1
n/5=66666666666666666666666666666665e445f1f5dfb6a67e4cba8c385348e6e7
n/6=d5555555555555555555555555555554463c62c03cbc85871fd9f975582d3661
By the way, last time you messed up 1/2 with -1/2. This is a common mistake, been there, done that. If you learn more about inversion, you will know why. And if you want a proof, then halve a point using your method, and then double it. If you reach points, where 02 and 03 prefixes are inverted, you will understand, why you need inversion, instead of relying just on standard division, provided by any BigInteger implementation.n/2=7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1
-n/2=7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0
n/4=bfffffffffffffffffffffffffffffff0c0325ad0376782ccfddc6e99c28b0f1
n/5=66666666666666666666666666666665e445f1f5dfb6a67e4cba8c385348e6e7
n/6=d5555555555555555555555555555554463c62c03cbc85871fd9f975582d3661
Quote
Furthermore, there are ways to accurately divide your key by 2 without any division, but that rock is really big and I'm afraid the glass will shatter
This problem is called "discrete logarithm", that's why you have DL letters in ECDLP. You can always divide a number by two, and finally reach the base point. The problem is: if you try to halve some point in a loop, you will visit a significant portion of the elliptic curve, I guess something above 2^250, maybe n/6 or something like that. So, don't run brute force on that, because you will never reach that answer in any reasonable time. Does dividing a public key by a non division number really result in a random unknown point in the curve? Why is that though? For me, this makes no sense : 
The result is not random at all, did you even read my first post and other posts? Let me give you a solution in order to successfully divide an odd key by 2, I have tried 4, 5, 6 etc, I couldn't get any meaningful result, I'm still learning.
Now the example, you could either add 1 or subtract 1 from your odd private (public key) key and then divide by 2 or you could simply divide the odd key by 2 add n/2 to the result, you will get the same result.
If your key is divisible by 2, 4, 8 etc but the last digit is 1, you could reach key 0x3 by dividing by 2 adding n/2 and dividing the result again by 2 adding n/2 till you reach the lowest point.
Demonstration :
Private key :
Code:
0000000000000000000000000000000000000000000000000000000000008001
Public key :Code:
03a3dd9bdf806a8c86fa43fd7c8af5fe3949b9244d46cfdf42aa716e3168f8e67b
Divided by 2, result:Code:
024d2432b7f67a3b25fe7f6a52164f1bf0235d3f6ce00569f4fc954b823c7c7607
Adding n/2 to above N/2 =
Code:
7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0
Result, private key :
Code:
0000000000000000000000000000000000000000000000000000000000004001
Result, public key :Code:
03d7ec238cd058837f44315fb8db2134f01daea15c10320c9173e7d5f68ff2d9d0
And if you keep dividing you will eventually reach 0x3, dividing 3 by 2 will give you 0x2 then 1, and dividing 1 by 2 will give you half, aka n/2 and take a look at 0.5 = n/2, it is a number around 2^255.
Now to simplify it, imagine you have 23 as your odd key, dividing by 2 adding n/2 will give you 12, it's like adding 1 to 23 = 24/2 = 12. You have 2 options.
Furthermore, there are ways to accurately divide your key by 2 without any division, but that rock is really big and I'm afraid the glass will shatter, you should start with small rocks, learn from Musk throwing a big rock at E-truck's window claiming it won't shatter. Lol
Quote
Does dividing a public key by a non division number really result in a random unknown point in the curve?
Yes, of course. For that reason, ECDSA is secure, and simple tricks like "let's divide it" won't work.Quote
Why is that though?
Because we are in finite field. And the beauty of finite fields is that they are, well, finite.Quote
For me, this makes no sense
It makes perfect sense, if you know, how things are calculated.In real numbers, you have things like sqrt(8). In finite fields, you have for example 18.
Code:
18*18=324
324=79*4+8
324 modulo 79=8
See? In finite fields, you can have "18" as a result of "sqrt(8)", and it is perfectly valid, because "18*18" gives you "8" in this field. 324=79*4+8
324 modulo 79=8
Does dividing a public key by a non division number really result in a random unknown point in the curve? Why is that though? For me, this makes no sense :
@digaran i didn't understand anything you wrote here, but could you please help me at least compile the ecctools that Alberto wrote? I will be greatful
I believe there is a makefile for the ecctools at the top level of the repository.
Then again, it's been some time since I last checked it so it's possible that you compile the programs by running GCC directly. Each (?) of the programs are self-contained in a single file.
I have thought about this topic.
We ( I hope) we no need check all subranges.
here my experiment. at the moment without explanation. ok? will do later.
but for setup :
- target pubkey
- control pubkey
We ( I hope) we no need check all subranges.
here my experiment. at the moment without explanation. ok? will do later.
but for setup :
- target pubkey
- control pubkey
Code:
private1 10101010101010
private2 20101050101010
0 81054462466121336796499689506081535496986294995352433067823614199163723147046 81054462466121336796499689506081535496986294995352433067823614199263723547046
0 diferential 100000400000 100000400000
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1 diferential -2339234126006387786334767373912887027330051803617674836012225517909457403926 113452855111309807637236217634775020825507512475457229546592937623608704090411
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2 diferential 18904830895888358436501385307540882914748990086379576225731455206880516570504 18904830895888358436501385307540882914748990086379576225731455206880516570504
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3 diferential 9549863029881748076170802887314466627038149631263909639802487681877786927368 9549863029881748076170802887314466627038149631263909639802487681877786927368
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6 diferential 12318307365671935683358615426456160409876336625433500466234591823672145265355 12318307365671935683358615426456160409876336625433500466234591823672145265355
7 32371981937314205172181135593826726926599749153289758214491766039672830203804 67234116331344887665299281617947817462937940549140267060867514082387976030056
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8 diferential -65447702612396110456800991526649687047256014592520598129298570471184177801147 50344386624920084966769993482038220805581549686554306253306592670333983693190
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9 diferential -68711789217748072009591573521638978286299213967802690512754712193758029458178 47080300019568123413979411487048929566538350311272213869850450947760132036159
10 102926301544281062598729764452167029202522279359177692784537922792572821440533 77194726158210796949047323339125271901891709519383269588403442094568785997347
10 diferential -25731575386070265649682441113041757300630569839794423196134480698004035443186 90060513851245929773888543895646150552206994439280481186470682443514126051151
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11 diferential 112360000000 112360000000
12 7894915175271558778879839886955993717238924837209652571541261123400113398489 92107343711501519086931465347819926701120789767445946667981379771890594939825
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13 diferential -98489823029441361624646584949918450357585974214385550854169908878877516213574 17302266207874833798924400058769457495251590064689353528435254262640645280763
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14 diferential -18849874992121241115465044071181752441159603487291263504145026557805281638613 96942214245194954308105940937506155411677960791783640878460136583712879855724
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15 diferential 20433898100702858015924291472121395503441923108072041949871499378032617204883 20433898100702858015924291472121395503441923108072041949871499378032617204883
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17 diferential -51618160262418063020146101750860874584999878052117728459715554653327372714343 64173928974898132403424883257827033267837686226957175922889608488190788779994
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36 diferential 156250625000 156250625000
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81 diferential -12188640972349073202481156316703990300298690976744726777116332961738751736246 103603448264967122221089828691983917552538873302330177605488830179779409758091
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82 diferential -12865787693035132824841220556520878650315284919897211598067240348502015721593 102926301544281062598729764452167029202522279359177692784537922793016145772744
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83 diferential -13622598733801905343949527648080930335627948738714694633247666251355075469922 102169490503514290079621457360606977517209615540360209749357496890163086024415
84 101318078082651670995624611882601919371232868744190541334779517749459704438858 101318078082651670995624611882601919371232868744190541334779517750084706938858
84 diferential 625002500000 625002500000
85 38597363079105398474523661669562635950945854759691634794201721047846121171513 1340070006734
85 diferential -38597363079105398474523661669562635950945854759691634794201721046506051164779 77194726158210796949047323339125271901891709519383269588403442095012110329558
86 49625181101706940895816136432294817651216098976746387592545069918514998504788 66166908135609254527754848576393090201621465302328516790060093225160453003979
86 diferential 16541727033902313631938712144098272550405366325582129197515023306645454499191 16541727033902313631938712144098272550405366325582129197515023306645454499191
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88 diferential 38597363079105398474523661669562635950945854759691634794201721048006057164779 38597363079105398474523661669562635950945854759691634794201721048006057164779
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93 diferential -82708635169511568159693560720491362752026831627910645987575116528227252495955 33083454067804627263877424288196545100810732651164258395030046613290908998382
94 38597363079105398474523661669562635950945854759691634794201721048856222181614 3350175016835
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95 2020202020202 4020210020202
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96 diferential 2500010000000 2500010000000
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97 diferential -77194726158210796949047323339125271901891709519383269588403442091012094329558 38597363079105398474523661669562635950945854759691634794201721050506067164779
98 5050505050505 10050525050505
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private2 20101050101010
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0 diferential 100000400000 100000400000
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7 32371981937314205172181135593826726926599749153289758214491766039672830203804 67234116331344887665299281617947817462937940549140267060867514082387976030056
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8 diferential -65447702612396110456800991526649687047256014592520598129298570471184177801147 50344386624920084966769993482038220805581549686554306253306592670333983693190
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10 102926301544281062598729764452167029202522279359177692784537922792572821440533 77194726158210796949047323339125271901891709519383269588403442094568785997347
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11 diferential 112360000000 112360000000
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20 diferential 125000500000 125000500000
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36 diferential 156250625000 156250625000
37 113954119566882605020022239214899210902792523576232445582881271663241698773808 27569545056503856053231186906830454250675610542636881995858372176871007500255
37 diferential -86384574510378748966791052308068756652116913033595563587022899486370691273553 29407514726937446456779932700619151200720651245479340795582263655147470220784
38 48557972905971307758271703390740090389899623729934637321737649059509245305706 100851174497017331497948922426921726194406910823710400591301271123581964045084
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42 95827935920537541040196677248569303050624191127510265695949100531085737272917 63885290613691694026797784832379535367082794085006843797299400354287624446893
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I guess I was speaking arabic right? 😅 that is math right there, follow up on it to get some where.
About compiling, I'm the worst person to ask, because I have the same problem as you, when I for example open visual studio it's like an alien world to me. In case you didn't know, I used WP's ready to use for windows tools.
Nono, you werent speaking arabic. You just explain a lot of good things but the way you explain them, nobody but einstein could understand it.
Your examples are good, but as i said the way you explain things are catastrophic.
Thank you for nothing. We probably need help of Wandering around the jungle, i think he is the only and other few who can compile codes. I asked him but unfortunately he ignored me
@digaran i didn't understand anything you wrote here, but could you please help me at least compile the ecctools that Alberto wrote? I will be greatful
I want to do some mathematics, maybe i can get near to #130
I guess I was speaking arabic right? 😅 that is math right there, follow up on it to get some where.I want to do some mathematics, maybe i can get near to #130
About compiling, I'm the worst person to ask, because I have the same problem as you, when I for example open visual studio it's like an alien world to me. In case you didn't know, I used WP's ready to use for windows tools.
@digaran i didn't understand anything you wrote here, but could you please help me at least compile the ecctools that Alberto wrote? I will be greatful
I want to do some mathematics, maybe i can get near to #130
I want to do some mathematics, maybe i can get near to #130
So no, key division does not decrease search time.
Hmm, i dont know if i agree with this statement. When you create 1 million key with the help of subtracting the public key, You have like created a million clones of the targeted public key, and with that, you increase the chance. When you find 1 of the 1 million pubkey, you get back to the original key. So the chances is higher. And so is very similar to dividing.
And if you are dealing with prime numbers, look at N of secp256k1, if you just subtract 1 from it you can divide it by dozens of already known divisor numbers.
@digaran, you are fucking smart with coming up with this idea, but it still wont help us a lot finding the targeted priviate key.
Teach me more tips like that though!
I'm the last person you'd want to teach you something, I just share what I know, they might be incorrect sometimes.
What you need to know about public keys, ignore their size and char length, it's all smoke and mirrors, they are simply representations of private keys, and private keys are simply ordinary numbers but very big ones.
Hexadecimal keys can make our work easier, for instance, if you have a large number like 3637343230303931363537383831353435
By the time you can figure out how to extract a certain number out of that, you'd be spending a few minutes, but if you convert to hex you will get this
b355b9d94ef43E58a7f80fe3b85b
And in seconds I can select my target and then by guess work I'd take b as my first char assuming we know the first char is b, so I'd go with b000000000000E00000000000000 then I subtract from target to get
355b9d94ef43058a7f80fe3b85b
Then I go with this
bffffffffffffeffffffffffffff subtracting target from it to reach
caa4626b10bc0a75807f01c47a4 and then I add the last result to this + 355b9d94ef43058a7f80fe3b85b to reach =
ffffffffffff0ffffffffffffff
And just like that I now know the E I guessed is really there otherwise I'd see something else instead of 0 in the middle of fs = ffffffffffff0ffffffffffffff . 😉
Fun fact, 99% of times when I share something I know with others, I learn something new then and later on, this is why we need to share our knowledge, we gain more than we already have by sharing.
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