Exposed ... Mistakes or ...

[brainless]

Dear Friends
2009 after blockchain launch,
first peoples found brain wallet mistakes, words or sentence from dic, easily found by hundreds peoples
later some other strategies, where we listen in passed, some exchanges hacked and some wallets provider services were compromised

still their is lot of strategies exist for pick addresses with balance

here i am posting 1 strategy out of (17), maybe help full to exchanges or corporate or wallet provider, for their own analyzing mistakes, where they maybe could get better secure their services

No Begging - Mod

-----------------

more then 1634 addresses are in used from 2013 to 2021
these addresses are maybe under use of exchange or wallet service provider or maybe hardware wallets

whats mistake inside is all addresses, from owner side, every privatekey is looks difrent from each other
but reality is every private key for those addresses are very close and in series
if any one address prv key exposed, in series all addresses balances could be in high risk

these addresses are in series like 2 ,6, 10, 20, 26 etc

checking......
pick pubkey of any address,
pubkey point * 2, and loop it to till 50000 pubkey generate, convert to address, you will get all these 1634 addresses,
same you can halve till 50000 pubkey too,

Result ...
if some one found privatekey 1 of these addresses, can get all addresses private keys by
privatekey * 2 mod n, and loop to .. Exposed

here posting 1 pubkey, those who want tough experiment, can halve this pubkey in loop to 20m pubkeys, then to address, you will find all 1634 addresses their

and post your experience here, after see how much peoples have experiment mode,
i will post easy solution to check after 24 hours

halve to 20 million pubkey from here
0398ce5d78ed1a3b121cbdbd62daecda243a43cf3f201c3599ee78a391d0190950

Code:

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[1715286838]

1715286838

[1715286838]

1715286838

[1715286838]

1715286838

[1715286838]

1715286838

[brainless]

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[NeuroticFish]

if any one address prv key exposed, in series all addresses balances could be in high risk

I'm not sure I understand where's the problem (as in 2022); similarly with any modern HD wallet, it would be utterly stupid to an user to expose any of his private keys.
Did you manage to get private keys? I guess not. It is written everywhere nowadays that people should avoid brain wallets since they're unsafe. I think that all brain wallets got emptied years ago (the few addresses from your list I've checked were also emptied years ago).

[nc50lc]

I'm not sure I understand where's the problem (as in 2022);

I think he's presuming that there could be a centralized service or exchange that have a broken implementation of HD wallet which produces the child keys in series instead of the standard implementation.

In fact, all of the addresses he presented belong to a single wallet according to walletexplorer: walletexplorer.com/wallet/00113e9c26df0947/addresses
Having a total of 6,023 linked addresses suggests that it's indeed an exchange or service provider.

[brainless]

connection exist here too  26 dec 2020

https://bitcointalk.org/index.php?topic=5303856.msg55941110#msg55941110

[gmaxwell]

Something is wrong here because the order of the group formed by multiplication of 2 mod n is much much larger than 20 million, so you can't generate all these addresses by going 20 million steps in either direction from any of them.

Code:

sage: F = FiniteField (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F)
C = EllipticCurve ([F (0), F (7)])
G = C.lift_x(0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798)
N = FiniteField (C.order())
sage: N(2).multiplicative_order()
1809251394333065553493296640760748560200586941860545380978205674086221273349

[brainless]

Something is wrong here because the order of the group formed by multiplication of 2 mod n is much much larger than 20 million, so you can't generate all these addresses by going 20 million steps in either direction from any of them.

Code:

sage: F = FiniteField (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F)
C = EllipticCurve ([F (0), F (7)])
G = C.lift_x(0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798)
N = FiniteField (C.order())
sage: N(2).multiplicative_order()
1809251394333065553493296640760748560200586941860545380978205674086221273349

yes , will take time aprox 7 hours for 20m generate multiplication steps
but in my first message already tip
pick 1 pubkey from 1634 list, get pubkey
example
https://blockchain.info/q/pubkeyaddr/1121f21h1wyrnm2ipXiKaCULLopeBegt2s
result: 027262bdb402fe5ddc6b1e81cef02bcd673140e6a26857503459a4891920a6a5a9
it could be your starting point,
now take 50k steps by pubkey point * 2, ( you can create script here, same time generate next pubkey and convert address and verify from 1634 list, and can print results)
same 50k steps of pubkey point * 57896044618658097711785492504343953926418782139537452191302581570759080747169
(halv the point)
and same convert and verufy address, print)

up and down side 50k +50k, take few minutes

as for 20m halve required too much time, for this i create an other thread, for discus this issue, where i ask developer, just for basic add/sub/mul conversion creator/generator by pycuda or cuda tools , when said tools available, these job are for seconds for millions point
https://bitcointalk.org/index.php?topic=5409721.0

[_Counselor]

as for 20m halve required too much time, for this i create an other thread, for discus this issue, where i ask developer, just for basic add/sub/mul conversion creator/generator by pycuda or cuda tools , when said tools available, these job are for seconds for millions point
https://bitcointalk.org/index.php?topic=5409721.0

If you need 50k keys by sequential halving the original, why use expensive division on each step, you just can take "last" key (original / 2^50000), and then generate all keys by fast doubling.

[gmaxwell]

You can't do 1809251394333065553493296640760748560200586941860545380978205674086221273349 operations in several hours either.

I don't have any reason to doubt that you've listed a number of points related by doubling, but the claim that you were able to wrap around and enumerate all of them from any starting position and then moving in one direction (either doubling or halving) can't be true, AFAICT.

[_Counselor]

You can't do 1809251394333065553493296640760748560200586941860545380978205674086221273349 operations in several hours either.

I don't have any reason to doubt that you've listed a number of points related by doubling, but the claim that you were able to wrap around and enumerate all of them from any starting position and then moving in one direction (either doubling or halving) can't be true, AFAICT.

He doesn't talking about mapping all points, which can be generated by doubling/halving.
As I understand it, he found a sequence of keys that are generated in this way and leads to the used addresses.
It looks like a manual generation of something like HD wallet with bad realisation. Something like if you take first address key ("master key") and derive other wallet addresses just by doubling "master key" N times. It is obvious that such an algorithm, unlike a normal HD wallet, allows you to recover the keys to all addresses, if you know any one of them.

[brainless]

You can't do 1809251394333065553493296640760748560200586941860545380978205674086221273349 operations in several hours either.

I don't have any reason to doubt that you've listed a number of points related by doubling, but the claim that you were able to wrap around and enumerate all of them from any starting position and then moving in one direction (either doubling or halving) can't be true, AFAICT.

He doesn't talking about mapping all points, which can be generated by doubling/halving.
As I understand it, he found a sequence of keys that are generated in this way and leads to the used addresses.
It looks like a manual generation of something like HD wallet with bad realisation. Something like if you take first address key ("master key") and derive other wallet addresses just by doubling "master key" N times. It is obvious that such an algorithm, unlike a normal HD wallet, allows you to recover the keys to all addresses, if you know any one of them.

Agree

[gmaxwell]

pick pubkey of any address,
pubkey point * 2, and loop it to till 50000 pubkey generate, convert to address, you will get all these 1634 addresses,
same you can halve till 50000 pubkey too,

What am I failing to understand here, then? Because to me this clearly appears to saying that you can "pick any of the pubkeys" and double it to generate all 1634 addresses.    Or, alternatively, you can do the same by repeated halving.

Because the order is ~2^250, that isn't possible.  

Now, if it said that you could start at the first (in sequence rather than sorted order) and double to generate the reset or the last and halve to generate the reset-- sure, that is plausible.

[NotATether]

A bit off-topic but...

SAGE seems to have represented these finite fields and curves & operations really well. It would be nice if this  functionality could be had in other languages as well - especially C-C++ where there are many applications for this kind of thing because of its raw speed, but are hard to implement because all the crypto libraries for these languages are half-arsed or incomplete (libsecp256k1 is the closest that replicates SAGE stuff but a general-purpose curve and field library would be a nice to-have).

[CrunchyF]

Something is wrong here because the order of the group formed by multiplication of 2 mod n is much much larger than 20 million, so you can't generate all these addresses by going 20 million steps in either direction from any of them.

Code:

sage: F = FiniteField (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F)
C = EllipticCurve ([F (0), F (7)])
G = C.lift_x(0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798)
N = FiniteField (C.order())
sage: N(2).multiplicative_order()
1809251394333065553493296640760748560200586941860545380978205674086221273349

yes , will take time aprox 7 hours for 20m generate multiplication steps
but in my first message already tip
pick 1 pubkey from 1634 list, get pubkey
example
https://blockchain.info/q/pubkeyaddr/1121f21h1wyrnm2ipXiKaCULLopeBegt2s
result: 027262bdb402fe5ddc6b1e81cef02bcd673140e6a26857503459a4891920a6a5a9
it could be your starting point,
now take 50k steps by pubkey point * 2, ( you can create script here, same time generate next pubkey and convert address and verify from 1634 list, and can print results)
same 50k steps of pubkey point * 57896044618658097711785492504343953926418782139537452191302581570759080747169
(halv the point)
and same convert and verufy address, print)

up and down side 50k +50k, take few minutes

as for 20m halve required too much time, for this i create an other thread, for discus this issue, where i ask developer, just for basic add/sub/mul conversion creator/generator by pycuda or cuda tools , when said tools available, these job are for seconds for millions point
https://bitcointalk.org/index.php?topic=5409721.0

this following python script (you need to have bit and gmpy2 installed) can do the work in 2 seconds...

Code:

from bit.format import public_key_to_address,public_key_to_coords,coords_to_public_key
from bit.utils import bytes_to_hex,hex_to_bytes
import gmpy2

def point_double_gmp(point):

	x1, y1 = point

	m = (3 *gmpy2.powmod(x1, 2,P)) * gmpy2.invert(2 * y1,P)
	x3 = gmpy2.powmod(m, 2,P) - x1 - x1
	y3 = y1 + m * (x3 - x1)
	result = (int(x3%P),int(-y3%P))
	return result

P=0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f

start_pubk='0398ce5d78ed1a3b121cbdbd62daecda243a43cf3f201c3599ee78a391d0190950'
start_coord=public_key_to_coords(hex_to_bytes(start_pubk))


pt=start_coord
for m in range(1,50001):
	pt=point_double_gmp(pt)
	x,y=pt
	mulmod="2^%d mod P "%m
	pubk=coords_to_public_key(x,y,compressed=True)
	add=public_key_to_address(pubk)

	print("pubk * %s ==> pubk:%s add:%s"%(mulmod,bytes_to_hex(pubk),add))

[brainless]

Something is wrong here because the order of the group formed by multiplication of 2 mod n is much much larger than 20 million, so you can't generate all these addresses by going 20 million steps in either direction from any of them.

Code:

sage: F = FiniteField (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F)
C = EllipticCurve ([F (0), F (7)])
G = C.lift_x(0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798)
N = FiniteField (C.order())
sage: N(2).multiplicative_order()
1809251394333065553493296640760748560200586941860545380978205674086221273349

yes , will take time aprox 7 hours for 20m generate multiplication steps
but in my first message already tip
pick 1 pubkey from 1634 list, get pubkey
example
https://blockchain.info/q/pubkeyaddr/1121f21h1wyrnm2ipXiKaCULLopeBegt2s
result: 027262bdb402fe5ddc6b1e81cef02bcd673140e6a26857503459a4891920a6a5a9
it could be your starting point,
now take 50k steps by pubkey point * 2, ( you can create script here, same time generate next pubkey and convert address and verify from 1634 list, and can print results)
same 50k steps of pubkey point * 57896044618658097711785492504343953926418782139537452191302581570759080747169
(halv the point)
and same convert and verufy address, print)

up and down side 50k +50k, take few minutes

as for 20m halve required too much time, for this i create an other thread, for discus this issue, where i ask developer, just for basic add/sub/mul conversion creator/generator by pycuda or cuda tools , when said tools available, these job are for seconds for millions point
https://bitcointalk.org/index.php?topic=5409721.0

this following python script (you need to have bit and gmpy2 installed) can do the work in 2 seconds...

Code:

from bit.format import public_key_to_address,public_key_to_coords,coords_to_public_key
from bit.utils import bytes_to_hex,hex_to_bytes
import gmpy2

def point_double_gmp(point):

	x1, y1 = point

	m = (3 *gmpy2.powmod(x1, 2,P)) * gmpy2.invert(2 * y1,P)
	x3 = gmpy2.powmod(m, 2,P) - x1 - x1
	y3 = y1 + m * (x3 - x1)
	result = (int(x3%P),int(-y3%P))
	return result

P=0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f

start_pubk='0398ce5d78ed1a3b121cbdbd62daecda243a43cf3f201c3599ee78a391d0190950'
start_coord=public_key_to_coords(hex_to_bytes(start_pubk))


pt=start_coord
for m in range(1,50001):
	pt=point_double_gmp(pt)
	x,y=pt
	mulmod="2^%d mod P "%m
	pubk=coords_to_public_key(x,y,compressed=True)
	add=public_key_to_address(pubk)

	print("pubk * %s ==> pubk:%s add:%s"%(mulmod,bytes_to_hex(pubk),add))

script about double point, could you post multiplication point script too ?

[CrunchyF]

Ok but a multiplication of what by what ?...sorry i d'ont understand

[j2002ba2]

A bit off-topic but...

SAGE seems to have represented these finite fields and curves & operations really well. It would be nice if this  functionality could be had in other languages as well - especially C-C++ where there are many applications for this kind of thing because of its raw speed, but are hard to implement because all the crypto libraries for these languages are half-arsed or incomplete (libsecp256k1 is the closest that replicates SAGE stuff but a general-purpose curve and field library would be a nice to-have).

I have used PARI/GP C library
The following is with the provided gp shell (some empty lines omitted):

Code:

gp > ellgroup(ellinit([0,7]), 2^256 - 2^32 - 977)
%1 = [115792089237316195423570985008687907852837564279074904382605163141518161494337]
gp > factor(%1[1] - 1)
%2 =
[                                2 6]
[                                3 1]
[                              149 1]
[                              631 1]
[               107361793816595537 1]
[            174723607534414371449 1]
[341948486974166000522343609283189 1]

The easiest for me was doing pari_init_opts, then (after recording avma) using gp_read_str with corresponding string (instead of interfacing each and every function), and then pari_sprintf("%Ps",...) to get the result as text (then recovering avma), and parse it to something meaningful.

Couldn't find a function returning the multiplicative order of point though.

[brainless]

Ok but a multiplication of what by what ?...sorry i d'ont understand

for multiplication or halve the point (50k loop)

Code:

from fastecdsa.curve import secp256k1
from fastecdsa.point import Point
from fastecdsa import keys, curve

def c2ux(point):

 p_hex = 'FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F'
 p = int(p_hex, 16)
 compressed_key_hex = point
 x_hex = compressed_key_hex[2:66]
 x = int(x_hex, 16)
 prefix = compressed_key_hex[0:2]

 y_square = (pow(x, 3, p)  + 7) % p
 y_square_square_root = pow(y_square, (p+1) * pow(4, p - 2, p) % p , p)
 if (prefix == "02" and y_square_square_root & 1) or (prefix == "03" and not y_square_square_root & 1):
     y = (-y_square_square_root) % p
 else:
     y = y_square_square_root

 return x_hex

def c2uy(point):

 p_hex = 'FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F'
 p = int(p_hex, 16)
 compressed_key_hex = point
 x_hex = compressed_key_hex[2:66]
 x = int(x_hex, 16)
 prefix = compressed_key_hex[0:2]

 y_square = (pow(x, 3, p)  + 7) % p
 y_square_square_root = pow(y_square, (p+1) * pow(4, p - 2, p) % p , p)
 if (prefix == "02" and y_square_square_root & 1) or (prefix == "03" and not y_square_square_root & 1):
     y = (-y_square_square_root) % p
 else:
     y = y_square_square_root

 computed_y_hex = format(y, '064x')


 return computed_y_hex


def cpub(x,y):
 prefix = '02' if y % 2 == 0 else '03'
 c = prefix+ hex(x)[2:].zfill(64)
 return c

with open('in.txt') as f:
  for line in f:
    line=line.strip()

    xs = int(c2ux(line),16)
    ys = int(c2uy(line),16)
    S = Point(xs, ys, curve=secp256k1)

    xsorg = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
    ysorg = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
    Sorg = Point(xsorg, ysorg, curve=secp256k1)
    for i in range(0,50000):
      S=S*57896044618658097711785492504343953926418782139537452191302581570759080747169
      xx=S.x
      yy=S.y
      pub04=cpub(xx,yy)
      print(pub04,file=open("out.txt", "a"))
      

[NotATether]

I have used PARI/GP C library
The following is with the provided gp shell (some empty lines omitted):

Code:

gp > ellgroup(ellinit([0,7]), 2^256 - 2^32 - 977)
%1 = [115792089237316195423570985008687907852837564279074904382605163141518161494337]
gp > factor(%1[1] - 1)
%2 =
[                                2 6]
[                                3 1]
[                              149 1]
[                              631 1]
[               107361793816595537 1]
[            174723607534414371449 1]
[341948486974166000522343609283189 1]

The easiest for me was doing pari_init_opts, then (after recording avma) using gp_read_str with corresponding string (instead of interfacing each and every function), and then pari_sprintf("%Ps",...) to get the result as text (then recovering avma), and parse it to something meaningful.

Couldn't find a function returning the multiplicative order of point though.

Amazing stuff, I'll definitely check it out!

[brainless]

A bit off-topic but...

SAGE seems to have represented these finite fields and curves & operations really well. It would be nice if this  functionality could be had in other languages as well - especially C-C++ where there are many applications for this kind of thing because of its raw speed, but are hard to implement because all the crypto libraries for these languages are half-arsed or incomplete (libsecp256k1 is the closest that replicates SAGE stuff but a general-purpose curve and field library would be a nice to-have).

I have used PARI/GP C library
The following is with the provided gp shell (some empty lines omitted):

Code:

gp > ellgroup(ellinit([0,7]), 2^256 - 2^32 - 977)
%1 = [115792089237316195423570985008687907852837564279074904382605163141518161494337]
gp > factor(%1[1] - 1)
%2 =
[                                2 6]
[                                3 1]
[                              149 1]
[                              631 1]
[               107361793816595537 1]
[            174723607534414371449 1]
[341948486974166000522343609283189 1]

The easiest for me was doing pari_init_opts, then (after recording avma) using gp_read_str with corresponding string (instead of interfacing each and every function), and then pari_sprintf("%Ps",...) to get the result as text (then recovering avma), and parse it to something meaningful.

Couldn't find a function returning the multiplicative order of point though.

are you trying to find magic numbers, which part i mention here before a year
https://bitcointalk.org/index.php?topic=5244940.msg57373246#msg57373246
here is full list of numbers

Code:

1
2
3
4
6
8
12
16
24
32
48
64
96
149
192
298
447
596
631
894
1192
1262
1788
1893
2384
2524
3576
3786
4768
5048
7152
7572
9536
10096
14304
15144
20192
28608
30288
40384
60576
94019
121152
188038
282057
376076
564114
752152
1128228
1504304
2256456
3008608
4512912
6017216
9025824
18051648
107361793816595537
214723587633191074
322085381449786611
429447175266382148
644170762899573222
858894350532764296
1288341525799146444
1717788701065528592
2576683051598292888
3435577402131057184
5153366103196585776
6871154804262114368
10306732206393171552
15996907278672735013
20613464412786343104
31993814557345470026
47990721836018205039
63987629114690940052
67745291898271783847
95981443672036410078
127975258229381880104
135490583796543567694
174723607534414371449
191962887344072820156
203235875694815351541
255950516458763760208
270981167593087135388
349447215068828742898
383925774688145640312
406471751389630703082
511901032917527520416
524170822603243114347
541962335186174270776
698894430137657485796
767851549376291280624
812943502779261406164
1023802065835055040832
1048341645206486228694
1083924670372348541552
1397788860275314971592
1535703098752582561248
1625887005558522812328
2096683290412972457388
2167849340744697083104
2795577720550629943184
3071406197505165122496
3251774011117045624656
4193366580825944914776
4335698681489394166208
5591155441101259886368
6503548022234091249312
8386733161651889829552
10094048492842495793203
11182310882202519772736
13007096044468182498624
16773466323303779659104
20188096985684991586406
26033817522627741345901
30282145478527487379609
33546932646607559318208
40376193971369983172812
52067635045255482691802
60564290957054974759218
78101452567883224037703
80752387942739966345624
104135270090510965383604
110250596354215468384319
121128581914109949518436
156202905135766448075406
161504775885479932691248
208270540181021930767208
220501192708430936768638
242257163828219899036872
312405810271532896150812
323009551770959865382496
330751789062646405152957
416541080362043861534416
441002385416861873537276
484514327656439798073744
624811620543065792301624
646019103541919730764992
661503578125292810305914
833082160724087723068832
882004770833723747074552
969028655312879596147488
1249623241086131584603248
1323007156250585620611828
1666164321448175446137664
1764009541667447494149104
1938057310625759192294976
2499246482172263169206496
2646014312501171241223656
3528019083334894988298208
4998492964344526338412992
5292028625002342482447312
7056038166669789976596416
10584057250004684964894624
16427338856778104789263531
21168114500009369929789248
32854677713556209578527062
49282016570334314367790593
65709355427112419157054124
98564033140668628735581186
131418710854224838314108248
197128066281337257471162372
262837421708449676628216496
394256132562674514942324744
525674843416899353256432992
788512265125349029884649488
1051349686833798706512865984
1577024530250698059769298976
3154049060501396119538597952
341948486974166000522343609283189
683896973948332001044687218566378
1025845460922498001567030827849567
1367793947896664002089374437132756
2051690921844996003134061655699134
2735587895793328004178748874265512
4103381843689992006268123311398268
5471175791586656008357497748531024
8206763687379984012536246622796536
10942351583173312016714995497062048
16413527374759968025072493245593072
21884703166346624033429990994124096
32827054749519936050144986491186144
50950324559150734077829197783195161
65654109499039872100289972982372288
101900649118301468155658395566390322
152850973677452202233487593349585483
203801298236602936311316791132780644
215769495280698746329598817457692259
305701947354904404466975186699170966
407602596473205872622633582265561288
431538990561397492659197634915384518
611403894709808808933950373398341932
647308485842096238988796452373076777
815205192946411745245267164531122576
863077981122794985318395269830769036
1222807789419617617867900746796683864
1294616971684192477977592904746153554
1630410385892823490490534329062245152
1726155962245589970636790539661538072
2445615578839235235735801493593367728
2589233943368384955955185809492307108
3260820771785646980981068658124490304
3452311924491179941273581079323076144
4891231157678470471471602987186735456
5178467886736769911910371618984614216
6904623848982359882547162158646152288
9782462315356940942943205974373470912
10356935773473539823820743237969228432
13809247697964719765094324317292304576
18758639927001554244874997690913623113
20713871546947079647641486475938456864
32149654796824113203110223801196146591
37517279854003108489749995381827246226
41427743093894159295282972951876913728
56275919781004662734624993072740869339
64299309593648226406220447602392293182
75034559708006216979499990763654492452
96448964390472339609330671403588439773
112551839562009325469249986145481738678
128598619187296452812440895204784586364
150069119416012433958999981527308984904
192897928780944679218661342807176879546
225103679124018650938499972290963477356
257197238374592905624881790409569172728
300138238832024867917999963054617969808
385795857561889358437322685614353759092
450207358248037301876999944581926954712
514394476749185811249763580819138345456
600276477664049735835999926109235939616
771591715123778716874645371228707518184
900414716496074603753999889163853909424
1028788953498371622499527161638276690912
1200552955328099471671999852218471879232
1543183430247557433749290742457415036368
1800829432992149207507999778327707818848
2057577906996743244999054323276553381824
2795037349123231582486374655946129843837
3086366860495114867498581484914830072736
3601658865984298415015999556655415637696
5590074698246463164972749311892259687674
6172733720990229734997162969829660145472
8385112047369694747459123967838389531511
11180149396492926329945498623784519375348
11836701793937980728516123542966496184303
16770224094739389494918247935676779063022
22360298792985852659890997247569038750696
23673403587875961457032247085932992368606
33540448189478778989836495871353558126044
35510105381813942185548370628899488552909
44720597585971705319781994495138077501392
47346807175751922914064494171865984737212
67080896378957557979672991742707116252088
71020210763627884371096741257798977105818
89441195171943410639563988990276155002784
94693614351503845828128988343731969474424
134161792757915115959345983485414232504176
142040421527255768742193482515597954211636
178882390343886821279127977980552310005568
189387228703007691656257976687463938948848
268323585515830231918691966970828465008352
284080843054511537484386965031195908423272
378774457406015383312515953374927877897696
536647171031660463837383933941656930016704
568161686109023074968773930062391816846544
757548914812030766625031906749855755795392
1136323372218046149937547860124783633693088
1763668567296759128548902407902007931461147
2272646744436092299875095720249567267386176
3527337134593518257097804815804015862922294
5291005701890277385646707223706023794383441
7054674269187036514195609631608031725844588
10582011403780554771293414447412047588766882
14109348538374073028391219263216063451689176
21164022807561109542586828894824095177533764
28218697076748146056782438526432126903378352
42328045615122219085173657789648190355067528
56437394153496292113564877052864253806756704
84656091230244438170347315579296380710135056
112874788306992584227129754105728507613513408
169312182460488876340694631158592761420270112
338624364920977752681389262317185522840540224
36712202954417214842724336778420075919000906527493
73424405908834429685448673556840151838001813054986
110136608863251644528173010335260227757002719582479
146848811817668859370897347113680303676003626109972
220273217726503289056346020670520455514005439164958
293697623635337718741794694227360607352007252219944
440546435453006578112692041341040911028010878329916
587395247270675437483589388454721214704014504439888
881092870906013156225384082682081822056021756659832
1174790494541350874967178776909442429408029008879776
1762185741812026312450768165364163644112043513319664
2349580989082701749934357553818884858816058017759552
3524371483624052624901536330728327288224087026639328
5470118240208165011565926179984591311931135072596457
7048742967248105249803072661456654576448174053278656
10940236480416330023131852359969182623862270145192914
16410354720624495034697778539953773935793405217789371
21880472960832660046263704719938365247724540290385828
23165400064237262565759056507183067904889572018848083
32820709441248990069395557079907547871586810435578742
43760945921665320092527409439876730495449080580771656
46330800128474525131518113014366135809779144037696166
59746473235060985162263246577491932178200490877270861
65641418882497980138791114159815095743173620871157484
69496200192711787697277169521549203714668716056544249
87521891843330640185054818879753460990898161161543312
92661600256949050263036226028732271619558288075392332
119492946470121970324526493154983864356400981754541722
131282837764995960277582228319630191486347241742314968
138992400385423575394554339043098407429337432113088498
175043783686661280370109637759506921981796322323086624
179239419705182955486789739732475796534601472631812583
185323200513898100526072452057464543239116576150784664
238985892940243940649052986309967728712801963509083444
262565675529991920555164456639260382972694483484629936
277984800770847150789108678086196814858674864226176996
350087567373322560740219275519013843963592644646173248
358478839410365910973579479464951593069202945263625166
370646401027796201052144904114929086478233152301569328
477971785880487881298105972619935457425603927018166888
525131351059983841110328913278520765945388966969259872
555969601541694301578217356172393629717349728452353992
716957678820731821947158958929903186138405890527250332
741292802055592402104289808229858172956466304603138656
955943571760975762596211945239870914851207854036333776
1050262702119967682220657826557041531890777933938519744
1111939203083388603156434712344787259434699456904707984
1433915357641463643894317917859806372276811781054500664
1482585604111184804208579616459716345912932609206277312
1911887143521951525192423890479741829702415708072667552
2223878406166777206312869424689574518869398913809415968
2867830715282927287788635835719612744553623562109001328
3451644609571352122298099419570277117828546230808364367
3823774287043903050384847780959483659404831416145335104
4447756812333554412625738849379149037738797827618831936
5735661430565854575577271671439225489107247124218002656
6903289219142704244596198839140554235657092461616728734
8902224512024086789177223740046297894551873140713358289
10354933828714056366894298258710831353485638692425093101
11471322861131709151154543342878450978214494248436005312
13806578438285408489192397678281108471314184923233457468
17804449024048173578354447480092595789103746281426716578
20709867657428112733788596517421662706971277384850186202
26706673536072260367531671220138893683655619422140074867
27613156876570816978384795356562216942628369846466914936
35608898048096347156708894960185191578207492562853433156
37700024611323481637388108590397409204444509743557913291
41419735314856225467577193034843325413942554769700372404
53413347072144520735063342440277787367311238844280149734
55226313753141633956769590713124433885256739692933829872
71217796096192694313417789920370383156414985125706866312
75400049222646963274776217180794818408889019487115826582
82839470629712450935154386069686650827885109539400744808
106826694144289041470126684880555574734622477688560299468
110452627506283267913539181426248867770513479385867659744
113100073833970444912164325771192227613333529230673739873
142435592192385388626835579840740766312829970251413732624
150800098445293926549552434361589636817778038974231653164
165678941259424901870308772139373301655770219078801489616
213653388288578082940253369761111149469244955377120598936
220905255012566535827078362852497735541026958771735319488
226200147667940889824328651542384455226667058461347479746
284871184384770777253671159681481532625659940502827465248
301600196890587853099104868723179273635556077948463306328
331357882518849803740617544278746603311540438157602979232
427306776577156165880506739522222298938489910754241197872
452400295335881779648657303084768910453334116922694959492
569742368769541554507342319362963065251319881005654930496
603200393781175706198209737446358547271112155896926612656
662715765037699607481235088557493206623080876315205958464
854613553154312331761013479044444597876979821508482395744
904800590671763559297314606169537820906668233845389918984
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