shares: increase default amount of sampled shares to N from 16

[Wondertan]

16 samples = 1 in ~100 failure rate
32 samples = 1 in ~10,000
64 samples = 1 in ~100,000,000
100 samples = 1 in ~3T
ETH is going to use 75 samples

cc @musalbas

[musalbas]

Failure rate calculation (for k = 1024):

0 samples: 1 in 1
1 samples: 1 in 1
2 samples: 1 in 1
3 samples: 1 in 2
4 samples: 1 in 3
5 samples: 1 in 4
6 samples: 1 in 5
7 samples: 1 in 7
8 samples: 1 in 10
9 samples: 1 in 13
10 samples: 1 in 17
11 samples: 1 in 23
12 samples: 1 in 31
13 samples: 1 in 42
14 samples: 1 in 56
15 samples: 1 in 75
16 samples: 1 in 100
17 samples: 1 in 134
18 samples: 1 in 179
19 samples: 1 in 239
20 samples: 1 in 319
21 samples: 1 in 426
22 samples: 1 in 568
23 samples: 1 in 758
24 samples: 1 in 1,012
25 samples: 1 in 1,350
26 samples: 1 in 1,802
27 samples: 1 in 2,404
28 samples: 1 in 3,207
29 samples: 1 in 4,279
30 samples: 1 in 5,710
31 samples: 1 in 7,618
32 samples: 1 in 10,165
33 samples: 1 in 13,562
34 samples: 1 in 18,094
35 samples: 1 in 24,142
36 samples: 1 in 32,210
37 samples: 1 in 42,976
38 samples: 1 in 57,338
39 samples: 1 in 76,501
40 samples: 1 in 102,069
41 samples: 1 in 136,181
42 samples: 1 in 181,694
43 samples: 1 in 242,417
44 samples: 1 in 323,435
45 samples: 1 in 431,530
46 samples: 1 in 575,750
47 samples: 1 in 768,170
48 samples: 1 in 1,024,899
49 samples: 1 in 1,367,428
50 samples: 1 in 1,824,433
51 samples: 1 in 2,434,172
52 samples: 1 in 3,247,692
53 samples: 1 in 4,333,096
54 samples: 1 in 5,781,251
55 samples: 1 in 7,713,393
56 samples: 1 in 10,291,272
57 samples: 1 in 13,730,701
58 samples: 1 in 18,319,618
59 samples: 1 in 24,442,191
60 samples: 1 in 32,610,982
61 samples: 1 in 43,509,859
62 samples: 1 in 58,051,239
63 samples: 1 in 77,452,484
64 samples: 1 in 103,337,807
65 samples: 1 in 137,874,251
66 samples: 1 in 183,953,109
67 samples: 1 in 245,431,965
68 samples: 1 in 327,457,655
69 samples: 1 in 436,897,151
70 samples: 1 in 582,912,445
71 samples: 1 in 777,727,460
72 samples: 1 in 1,037,651,774
73 samples: 1 in 1,384,445,438
74 samples: 1 in 1,847,141,076
75 samples: 1 in 2,464,474,742
76 samples: 1 in 3,288,127,030
77 samples: 1 in 4,387,053,172
78 samples: 1 in 5,853,254,833
79 samples: 1 in 7,809,475,600
80 samples: 1 in 10,419,487,903
81 samples: 1 in 13,901,779,479
82 samples: 1 in 18,547,910,610
83 samples: 1 in 24,746,820,527
84 samples: 1 in 33,017,471,544
85 samples: 1 in 44,052,308,230
86 samples: 1 in 58,775,313,574
87 samples: 1 in 78,418,255,584
88 samples: 1 in 104,626,598,691
89 samples: 1 in 139,594,557,912
90 samples: 1 in 186,249,235,018
91 samples: 1 in 248,495,027,305
92 samples: 1 in 331,549,278,710
93 samples: 1 in 442,353,366,798
94 samples: 1 in 590,171,619,364
95 samples: 1 in 787,411,421,867
96 samples: 1 in 1,050,647,294,382
97 samples: 1 in 1,401,680,556,293
98 samples: 1 in 1,870,265,625,984
99 samples: 1 in 2,495,069,045,634
100 samples: 1 in 3,329,833,365,893

Script:

def th2(s,k):
	p = c(0,k)
	for i in range(1,s):
		p *= c(i,k)
	return 1-p

def c(i,k):
	return 1 - (k+1)**2/(4*k**2-i)

if __name__ == '__main__':
	for x in range(100+1):
		print("{} samples: 1 in {:,}".format(x, int(1/(1-th2(x,1024)))))

Theorem 2 in p19 in https://arxiv.org/pdf/1809.09044.pdf:

Note: this assumes a stronger adversary in practice that withholds data "arranged as a square".

[renaynay]

We should perform tests once we have #638 in place such that we can pick a DefaultSampleAmount that is reasonable. Eventually, we could probably even make this variable configurable such that users can choose the "reliability" of their light node. WDYT @liamsi @Wondertan @musalbas

[musalbas]

Yes, it should be a configurable parameter.