Proof of stake coin is more resilient to 51% attack. Since 51% hashpower is not enough to execute a 51% attack. The attackers must also have at least 51% stake.
I.e. another insecure copy&paste without any background... %)
1) Only active stake matters. I.e. if there are total 1000-2000 coins participating in generation, it doesn't matter how many coins exists here. Only this 1000-2000 coins should be counted.
2) Coins amount doesn't matter, only coin*day weight. If you have 1 coin with 0.5 days age then you have the weight, which equal to 0.5 coins with 1 day age.
3) Attacker don't need any hashpower, he needs only some coins.
4) Unlike PoW hashing rate, stake inputs could be re-used multiple times. Attacker can retry attempts to generate subsequent blocks until he get success. He can create multiple wallet copies, for example.
Attempts to get successful double spend could be calculated as 1/AttackerWeight^Nconf, AttackerWeight = (% of network weight) / 100, Nconf = 4 (the worst case, 4 subsequent PoS blocks).
So, average required attempts will be:
% Avg # attempts
1 100000000
2 6250000
3 1234567.901
4 390625
5 160000
6 77160.49383
7 41649.31279
8 24414.0625
9 15241.57903
10 10000
11 6830.134554
12 4822.530864
13 3501.277966
14 2603.082049
15 1975.308642
16 1525.878906
17 1197.303672
18 952.5986892
19 767.3360395
20 625
21 514.1890467
22 426.8834096
23 357.3457785
24 301.408179
25 256
26 218.8298729
27 188.1676423
28 162.6926281
29 141.3865211
30 123.4567901
31 108.281241
32 95.36743164
33 84.32264881
34 74.83147951
35 66.63890046
36 59.53741808
37 53.35720891
38 47.95850247
39 43.22565391
40 39.0625
41 35.38869706
42 32.13681542
43 29.25002069
44 26.6802131
45 24.38652644
46 22.33411116
47 20.49314289
48 18.83801119
49 17.34665256
50 16
51 14.78152682
52 13.67686706
53 12.67349861
54 11.76047764
55 10.92821529
56 10.16828925
57 9.473284438
58 8.836657566
59 8.252622002
60 7.716049383
61 7.222385408
62 6.767577565
63 6.348012923
64 5.960464478
65 5.602044746
66 5.270165551
67 4.962503079
68 4.676967469
69 4.411676278
70 4.164931279
71 3.935198148
72 3.72108863
73 3.521344861
74 3.334825557
75 3.160493827
76 2.997406404
77 2.844704104
78 2.701603369
79 2.567388756
80 2.44140625
81 2.323057313
82 2.211793566
83 2.107112049
84 2.008550964
85 1.915685875
86 1.828126293
87 1.745512606
88 1.667513319
89 1.593822573
90 1.524157903
91 1.458258212
92 1.395881947
93 1.336805445
94 1.280821431
95 1.227737663
96 1.177375699
97 1.129569775
98 1.084165785
99 1.041020356
100 1
First column: Attacker's coin*day weight % of the total network weight.
Second column: Average attempts required to get success in the double-spend attack.
5) PoW don't participate in the network protection. If 1 PoW block was generated on the network in the time of your attack then you need less confirmations.
1/AttackerWeight^3
% Avg # attempts
1 1000000
2 125000
3 37037,03704
4 15625
5 8000
6 4629,62963
7 2915,451895
8 1953,125
9 1371,742112
10 1000
11 751,3148009
12 578,7037037
13 455,1661356
14 364,4314869
15 296,2962963
16 244,140625
17 203,5416243
18 171,4677641
19 145,7938475
20 125
21 107,9796998
22 93,91435011
23 82,18952905
24 72,33796296
25 64
26 56,89576695
27 50,80526343
28 45,55393586
29 41,00209111
30 37,03703704
31 33,56718472
32 30,51757813
33 27,82647411
34 25,44270303
35 23,32361516
36 21,43347051
37 19,7421673
38 18,22423094
39 16,85800502
40 15,625
41 14,5093658
42 13,49746248
43 12,5775089
44 11,73929376
45 10,9739369
46 10,27369113
47 9,631777159
48 9,04224537
49 8,499859752
50 8
51 7,538578676
52 7,111970869
53 6,716954264
54 6,350657928
55 6,010518407
56 5,694241983
57 5,39977213
58 5,125261388
59 4,869046981
60 4,62962963
61 4,405655099
62 4,19589809
63 3,999248141
64 3,814697266
65 3,641329085
66 3,478309263
67 3,324877063
68 3,180337879
69 3,044056632
70 2,915451895
71 2,793990685
72 2,679183813
73 2,570581748
74 2,467770912
75 2,37037037
76 2,278028867
77 2,19042216
78 2,107250628
79 2,028237117
80 1,953125
81 1,881676423
82 1,813670724
83 1,748903001
84 1,68718281
85 1,628332994
86 1,572188612
87 1,518595967
88 1,467411721
89 1,41850209
90 1,371742112
91 1,327014973
92 1,284211391
93 1,243229064
94 1,203972145
95 1,16635078
96 1,130280671
97 1,095682682
98 1,062482469
99 1,030610152
100 1
