Many thanks for testing  I added some information on expected number of operation and needed memory. On my test it gives: Avg Mem=188.0 MB (Theoretical 191.0MB) Avg Count=37.19 (Theoretical 2^37.25) Theoretical results are not so bad here, but when you increase the dp, theoretical values become too large because calculation are here done using the asymptotic 2.sqrt(n) + nbKangaroo.2^dp which becomes bad when dp increase. n is the range size. For instance of the test done by PietCoin97 (with dp=20), it gives: Expected operations: 2^42.52 (Measured 2^39.42) Expected RAM: 460.4MB (Measured 58.5MB) It is difficult to get the analytical expression of the theoretical values. If anyone can help to find the analytical expression, it would be great. To find the analytical expression , you have to consider that the extra operation due to dp (nbKangaroo.2^dp) also increase the number of points and increase the probability to find a collision due to the birthday paradox. So the analytical expression should be something like: 2.sqrt(n) + f(nbKangaroo,dp) where f(nbKangaroo,dp) is a function with an asymptotic equal to nbKangaroo.2^dp. Calculation of probability on the birthday paradox are rather complex  Here are 2 loglog plots showing the asymptotic (in blue) for some dp and 40 bit search on a classical birthday paradox with a theoretical average of sqrt(PI/2)*sqrt(n) (instead of 2.sqrt(n) in the kangaroo case): Red plots are experimental plots (averaged on 1000 searches)   Kangaroo v1.3
Start:59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D1000000000000000000
Stop :59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D1FFFFFFFFFFFFFFFFFF
Keys :20
Number of CPU thread: 0
Range width: 2^72
Number of kangaroos: 2^18.58
Suggested DP: 16
Expected operations: 2^37.25
Expected RAM: 191.0MB
DP size: 16 [0xFFFF000000000000]
GPU: GPU #0 GeForce GTX 1050 Ti (6x128 cores) Grid(12x256) (45.0 MB used)
SolveKeyGPU Thread GPU#0: creating kangaroos...
SolveKeyGPU Thread GPU#0: 2^18.58 kangaroos in 1953.4ms
[115.21 MK/s][GPU 115.21 MK/s][Count 2^37.83][Dead 1][40:28 (Avg 23:36)][290.1MB]
Key# 0 Pub: 0x038F63B86D8EE91D4B78FF4680F927DCC7754CF734A386ED5FA45E71DE9328F433
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D102F962838A4EE7364E
[115.23 MK/s][GPU 115.23 MK/s][Count 2^37.29][Dead 1][27:44 (Avg 23:36)][200.6MB]
Key# 1 Pub: 0x0389044AFFFD381B496D63F8C80CDAAB5E57E40DEE6A56C8AFA5194B1FFD83FEBB
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D1F5821BA583622EEE49
[115.22 MK/s][GPU 115.22 MK/s][Count 2^37.72][Dead 3][37:26 (Avg 23:36)][268.8MB]
Key# 2 Pub: 0x0338E88602F88C3268C68552C4C53987F41BB42335A8E36658A80D2F5BDF63615B
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D110DC466D7E3D293A10
[115.21 MK/s][GPU 115.21 MK/s][Count 2^37.20][Dead 0][26:02 (Avg 23:36)][188.6MB]
Key# 3 Pub: 0x026776529C6C8932ABF9DCFDCB2DB2784DCE82164914D4C3294FECFE1B48F3BF27
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D1DE55BDE3E1B2F36A9B
[115.22 MK/s][GPU 115.22 MK/s][Count 2^37.47][Dead 0][31:24 (Avg 23:36)][226.4MB]
Key# 4 Pub: 0x03312940E0EE296C23B1E7888A4D23FED01358C1021FE2091D909B0A8D8AA80DE1
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D11B78CA8A9FBCDBDE3E
[115.22 MK/s][GPU 115.22 MK/s][Count 2^36.81][Dead 0][20:00 (Avg 23:36)][145.6MB]
Key# 5 Pub: 0x020097A3826A7BC1EC5383AE390EEA8C436B2B6D2E3770493F7E993B164C9F233E
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D152477D7EC473A483D4
[115.21 MK/s][GPU 115.21 MK/s][Count 2^37.74][Dead 1][37:54 (Avg 23:36)][272.0MB]
Key# 6 Pub: 0x03DAE902F3F4E0A62AA7DF422B2BF802CD9ADA747177F853390BEB7F203E4C3F84
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D10D2AB0CDE883E6B3BC
[115.22 MK/s][GPU 115.22 MK/s][Count 2^36.52][Dead 1][16:18 (Avg 23:36)][120.1MB]
Key# 7 Pub: 0x023C4E51FD6EB029CFD4DDCBD93FAD060C7D026D252EC37923355D1B192C69B9E3
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D1B69F2EC54CEEBA50C4
[115.22 MK/s][GPU 115.22 MK/s][Count 2^37.54][Dead 2][33:09 (Avg 23:36)][238.4MB]
Key# 8 Pub: 0x03C3112770BC8455596AF38A2A2E3F556B1F02758945608364C3AFD22FAD806B8D
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D18D06BC6E25E5C7EAEF
[115.21 MK/s][GPU 115.21 MK/s][Count 2^37.15][Dead 0][25:10 (Avg 23:36)][182.5MB]
Key# 9 Pub: 0x0264AE637A90AA93798E7BA6CCFD57CB077BCCAE49D3AB76F6857CA044A423E3AC
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D1C8A4760400224E8C05
[115.22 MK/s][GPU 115.22 MK/s][Count 2^36.17][Dead 0][12:48 (Avg 23:36)][95.9MB]
Key#10 Pub: 0x0359A0CFB0958A2B6FC562B3824D0BE034C7E947780DAF4B8AF403C72D92496BEF
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D197AD08D8E14C058778
[115.22 MK/s][GPU 115.22 MK/s][Count 2^35.52][Dead 0][08:11 (Avg 23:36)][61.8MB]
Key#11 Pub: 0x020B58A06DC49B931A8B9C08E29306B70FBB56F4F3B8BAB7AEBD409B4DC8086C78
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D1C831BC0E9C1450CCCE
[115.22 MK/s][GPU 115.22 MK/s][Count 2^37.26][Dead 1][27:09 (Avg 23:36)][196.5MB]
Key#12 Pub: 0x039E8FE48D08C5465128681D49AE9026240BC612C9BC8A3B6E9D2FA00DBC9021AF
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D17941FCB9F89FAA2C79
[115.22 MK/s][GPU 115.22 MK/s][Count 2^36.50][Dead 0][16:03 (Avg 23:36)][118.4MB]
Key#13 Pub: 0x02131119C5618C26AB5C8DE1B16E60120C918AAC717B06A10ADBCC4BCF44106419
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D166208474EA5CD1D0D9
[115.21 MK/s][GPU 115.21 MK/s][Count 2^36.25][Dead 2][13:33 (Avg 23:36)][101.1MB]
Key#14 Pub: 0x02487FE068AFA2AAFC96EC2941EC8CF4E2600692F8D4F6E4A389AFEA44221E045D
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D1AD9E1D5075941AB176
[115.22 MK/s][GPU 115.22 MK/s][Count 2^37.66][Dead 0][35:54 (Avg 23:36)][258.2MB]
Key#15 Pub: 0x034B202C91CF7AA1563048AFCED10781666C8344E0A23884977F5F3396E3CDFBFE
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D166550C0FED3225AE1D
[115.22 MK/s][GPU 115.22 MK/s][Count 2^37.71][Dead 2][37:14 (Avg 23:36)][267.2MB]
Key#16 Pub: 0x023510C7370A558126EF057C738A4943021E5FB08B41799AE097190DFC5538DB69
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D16DA62AB9D45017CFF3
[115.22 MK/s][GPU 115.22 MK/s][Count 2^35.87][Dead 0][10:26 (Avg 23:36)][78.1MB]
Key#17 Pub: 0x0336CE6F79E48B6493CF2BC2DE87BFE9504D30CD62B9B5F992E1A5933D69076A76
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D1A7F3A8F6001FFF3E8C
[115.21 MK/s][GPU 115.21 MK/s][Count 2^37.62][Dead 3][34:51 (Avg 23:36)][250.4MB]
Key#18 Pub: 0x03292ACD7402F076829CF4DC40B659D24ACCF67F2884DF4CD111847C651F18512A
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D1DF6AE20D48B230C23C
[115.21 MK/s][GPU 115.21 MK/s][Count 2^37.27][Dead 1][27:30 (Avg 23:36)][198.8MB]
Key#19 Pub: 0x036C4AF425D93153FD0593787399A78322699F498C4703E2D1524C15F0137C2D14
Priv: 0x59C0465E6C5C5502E17E79A74FE1FF5365707890EB68D13B97B84AC3FBEDF0AF
Done: Total time 08:39:58
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I doubt again if such range/pubkey adjustments could improve the progress. May be the initial kangaroos allocation is more effecient in the adjusted ranges? But more tests are required in order to prove it (3 or 5 key are not enough...)
Many thanks for the tests They looks very good ! The translation you did is equivalent to what the code do so the result should be equal. The new spreading of kangaroos gives more chance to a key in the middle of the range to be found. At first, a key close to the end of range was harder to find and a key near to the beginning was easier. It more logical to have a spreading from the middle. By expending the range of the odolvlobo's test to 80bit, all the 16 keys are very close to each others and near the end of the range (0xCB5E.....) We should make tests of uniformly distributed key in the range. |
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Confirm that v1.3 has better perfomance! @Jean_Luc Thank you for job!
First key in range 2^80 found in [01:15:13] with 8 dead kangaroo.
Many thanks for testing, you should to decrease a bit your grid size, you have a too small dp for a 2^80 range and a large RAM usage. Could you try with -g 136,128 or -g 68,256 or even -g 68,128, thanks. This GPU should be quite good at 2^88 range with nominal settings |
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Hi there, If someone find something with this tool, please inform us New release 1.3 is ready. If you have any idea on how to optimize it, I would appreciate any tips. Thanks JL Edit: There is a description of the method used on the REAME. |
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They should be spread with a -(k2-k1)/2 translation, the keys you solved are close to the upper bound, it explains the factor of 2.
I added the translation, looks like better. I published a new release https://github.com/JeanLucPons/Kangaroo/releasesThanks to test it  I redid the test above: [115.21 MKey/s][GPU 115.21 MKey/s][Count 2^33.12][Dead 1][01:32][352.0MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^32.87][Dead 0][01:19][297.9MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^32.24][Dead 1][52s][195.1MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^32.62][Dead 0][01:07][251.1MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^33.00][Dead 0][01:27][325.7MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^32.92][Dead 1][01:22][307.3MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^33.24][Dead 1][01:42][382.4MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^33.09][Dead 1][01:32][345.0MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^32.82][Dead 0][01:17][288.6MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^32.78][Dead 0][01:14][279.1MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^33.31][Dead 2][01:47][401.0MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^34.06][Dead 5][02:59][672.8MB]
[105.71 MKey/s][GPU 105.71 MKey/s][Count 2^29.98][Dead 0][12s][44.4MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^33.59][Dead 0][02:09][485.6MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^32.92][Dead 0][01:22][307.3MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^32.96][Dead 1][01:24][316.6MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^33.41][Dead 2][01:54][428.9MB]
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Jean_Luc, can you edit code for geting good resualt ? Please.... I very hope Kangaroo will can brute bitcoin... ??
I'm sorry, I do my maximum, but this algorithm is known from long time and it cannot break Secpk1. May be a flash of genius will hit a day me but for the moment nothing better than O(sqrt(n))... |
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Thanks for the reading and the tests The dead kangaroos are well handled, they are detected, deleted and recreated. Was quite a nightmare with the GPU and coalesced memory access  The only things that might happen is that a kangaroo enter in a cycle without having a distinguished point in the cycle but it is rather unlikely. Handling this in the GPU (with a Floyd cycle detection) will just kill GPU performance. |
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Yes there is a problem with the spreading of wild kangaroo i think. They should be spread with a -(k2-k1)/2 translation, the keys you solved are close to the upper bound, it explains the factor of 2. I will try this ASAP. Many thanks for the tests |
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Jean luc does it work also with such an key space ?
Start:8000000000000000000000001
Stop :FFFFFFFFFFFFFFFFFFFFFFFFE
or should i give in all 64 bits ?
and are my setting ok or can i get higher speed ?
Yes it should work. Your setup looks good. |
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I solved 20 times the following key (64bit range) and there was a strange 24 dead kangaroos event !? It seems that from times to times the algorithm has hard time to solve the key... Just a matter of probability ? My Mersenne twister fails ? I will make further tests...
Key# 0 Pub: 0x0259A3BFDAD718C9D3FAC7C187F1139F0815AC5D923910D516E186AFDA28B221DC
Priv: 0x49DCCFD96DC5DF56487436F5A1B18C4F5D34F65DDB48CB5EBB3EF3883C1866D4
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^33.63][Dead 3][02:12][502.4MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^34.07][Dead 0][02:59][675.8MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^31.43][Dead 0][29s][113.4MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^33.70][Dead 1][02:19][525.5MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^35.20][Dead 24][06:31][1472.8MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^34.44][Dead 7][03:52][872.4MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^33.51][Dead 1][02:02][459.6MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^34.27][Dead 7][03:27][779.1MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^33.68][Dead 3][02:17][516.3MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^33.85][Dead 3][02:34][581.7MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^32.33][Dead 1][54s][206.9MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^32.40][Dead 0][57s][216.1MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^33.45][Dead 2][01:57][441.1MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^32.97][Dead 1][01:24][319.5MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^34.66][Dead 6][04:29][1012.6MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^32.12][Dead 0][47s][178.9MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^31.66][Dead 0][34s][131.5MB]
[115.23 MKey/s][GPU 115.23 MKey/s][Count 2^33.32][Dead 3][01:47][403.6MB]
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^34.00][Dead 2][02:52][647.1MB]
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^33.68][Dead 2][02:17][516.5MB]
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During the night, I solved the key with the same config file as Etar in 2^41.59 iterations (09:06:32) and 1 dead kangaroo. It seems normal. I will make further tests to see if changing the jumps can improve things a bit... C:\C++\Kangaroo\VC_CUDA10\x64\Release>Kangaroo.exe -t 0 -gpu ..\..\in.txt
Kangaroo v1.2
Start:49DCCFD96DC5DF56487436F5A1B18C4F5D34F65DDB4800000000000000000000
Stop :49DCCFD96DC5DF56487436F5A1B18C4F5D34F65DDB48FFFFFFFFFFFFFFFFFFFF
Keys :1
Number of CPU thread: 0
Range width: 2^80
Number of random walk: 2^18.58 (Max DP=19)
DP size: 19 [0xFFFFE00000000000]
GPU: GPU #0 GeForce GTX 1050 Ti (6x128 cores) Grid(12x256) (45.0 MB used)
SolveKeyGPU Thread GPU#0: creating kangaroos...
SolveKeyGPU Thread GPU#0: 2^18.58 kangaroos in 1376.4ms
[115.21 MKey/s][GPU 115.21 MKey/s][Count 2^41.59][Dead 1][09:06:28][485.7MB]
Key# 0 Pub: 0x0259A3BFDAD718C9D3FAC7C187F1139F0815AC5D923910D516E186AFDA28B221DC
Priv: 0x49DCCFD96DC5DF56487436F5A1B18C4F5D34F65DDB48CB5EBB3EF3883C1866D4
Done: Total time 09:06:32
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Yes I remarked that this first key of this set often generate dead kangaroos (even in 64bit range)
The fact that the startrange is not zero should not impact, it is just a translation. It is equivalent to what MrFreeDragon said.
All kangaroos are uniformly distributed in the range and make random jumps form G up to 2*sqrt(k2-k1).G
Even if kangaroos goes out of the range it is not really a problem.
I will try to change the calculation on random jump to see if it improve something.
[18.00 MKey/s][Count 2^34.40][20:47][Dead 10][20.2MB]
Key# 0 Pub: 0x0259A3BFDAD718C9D3FAC7C187F1139F0815AC5D923910D516E186AFDA28B221DC
Priv: 0x49DCCFD96DC5DF56487436F5A1B18C4F5D34F65DDB48CB5EBB3EF3883C1866D4
[115.23 MKey/s][GPU 115.23 MKey/s][Count 2^33.52][Dead 5][02:02][463.9MB]
Key# 0 Pub: 0x0259A3BFDAD718C9D3FAC7C187F1139F0815AC5D923910D516E186AFDA28B221DC
Priv: 0x49DCCFD96DC5DF56487436F5A1B18C4F5D34F65DDB48CB5EBB3EF3883C1866D4
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My gpu have only 11gb of memory
I dont know how hashtable can be 13 gb in this reason.
No, i did not try other keys. Only from example.
The hash table is centralized and managed by the CPU. You have 7 dead kangaroo, a dead kangaroo is a collision in the same herd, and the kangaroo is re created. There is the same probability to get a dead kangaroo or to solve the key. Having 7 dead kangaroo without solving the key is like throwing 7 times a coin and having consecutively nine time "heads", bad luck ! On this key, i got 20 dead kangaroos ! ~2^35 , ~4 times more iterations needed compare to the average 2^33... [114.85 MKey/s][GPU 114.85 MKey/s][Count 2^34.98][Dead 20][05:41][1269.3MB] Key#11 Pub: 0x03D4E6FA664BD75A508C0FF0ED6F2C52DA2ADD7C3F954D9C346D24318DBD2ECFC6 Priv: 0x49DCCFD96DC5DF56487436F5A1B18C4F5D34F65DDB48CB5EE3579364DE939B0C |
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2h and nothing, speed drop to 267Mkey, GPU usage drop to 25%
That's strange, may be a bug somewhere, the gpu usage drops may be because the HashTable becomes too big. I'll try Did you try with others key ? |
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My Hi-End GPU can`t find any key for 1h))
That should work. For 80bit search, 2^41 is an average time, with a probability of ~50% to find the key. I will try this night. Many thanks to MrFreeDragon for testing the software |
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My low cost hardware has solved the 80bit key is 03:52:25. I will now work on endomorphism and symetry optimization. Kangaroo v1.2
Start:25FEEE926526B0B4F0085358DF14702F7F6F04E8EC2200000000000000000000
Stop :25FEEE926526B0B4F0085358DF14702F7F6F04E8EC22FFFFFFFFFFFFFFFFFFFF
Keys :1
Number of CPU thread: 0
Range width: 2^80
Number of random walk: 2^18.58 (Max DP=19)
DP size: 19 [0xFFFFE00000000000]
GPU: GPU #0 GeForce GTX 1050 Ti (6x128 cores) Grid(12x256) (45.0 MB used)
SolveKeyGPU Thread GPU#0: creating kangaroos...
SolveKeyGPU Thread GPU#0: 2^18.58 kangaroos in 1320.9ms
[115.22 MKey/s][GPU 115.22 MKey/s][Count 2^40.35][Dead 1][03:52:22][209.5MB]
Key# 0 Pub: 0x02E9CE716922FFB1CC2306E55D4E5A4F4A9B9D050E4ABB3EB95B246E7998A2508D
Priv: 0x25FEEE926526B0B4F0085358DF14702F7F6F04E8EC2243F5E7A6482FFC1F8DC4
Done: Total time 03:52:25
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Range 2^80, DP - by default, speed 1100Mkeys/s but not a single key was found in 1 hour
I published a new release 1.2 with faster kangaroo creation. I'm trying to attack a 80bit key but on my hardware (115MK/s), it will take time in.txt 25FEEE926526B0B4F0085358DF14702F7F6F04E8EC2200000000000000000000
25FEEE926526B0B4F0085358DF14702F7F6F04E8EC22FFFFFFFFFFFFFFFFFFFF
02E9CE716922FFB1CC2306E55D4E5A4F4A9B9D050E4ABB3EB95B246E7998A2508D
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Range 2^64, DP-12, speed 900Mkeys/s, Time to find 16 keys is 15m30s (the time is about the same as for weaker cards, although the speed is higher)
Yes having a too large DP size and too much threads create an overhead. You need more iterations. Range 2^80, DP - by default, speed 1100Mkeys/s but not a single key was found in 1 hour
I will try to make a test with a 80bit key today and improve the creation of kangaroo which is slow. |
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Yes with a large number of thread and a small range (64bits) you face an overhead due to the DP method.
You can try to set by hand the dpSize using the -d option.
C:\C++\Kangaroo\VC_CUDA10\x64\Release>Kangaroo.exe -t 0 -d 10 -gpu ..\..\in.txt
You will get a warning but ignore it.
As MrFreeDragon said, such a GPU will be much more efficient on 80bit search or more.
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A solved wild point become P1 + d1.G = (k1 + d1).G A collision with a new wild point P2 + d2.G will give P2 + d2.G = (k1 + d1).G => P2 = (k1 + d1 - d2).G I think solved wild can be reused. We can say that a wild kangaroo trapped becomes a tame one |
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