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# Vanguard Rotation Statistics
## Sybil rotation counts for a given number of Guards {#SybilTable}
The probability of Sybil success for Guard discovery can be modeled as
the probability of choosing 1 or more malicious middle nodes for a
sensitive circuit over some period of time.
```text
P(At least 1 bad middle) = 1 - P(All Good Middles)
= 1 - P(One Good middle)^(num_middles)
= 1 - (1 - c/n)^(num_middles)
```
c/n is the adversary compromise percentage
In the case of Vanguards, num_middles is the number of Guards you rotate
through in a given time period. This is a function of the number of vanguards
in that position (v), as well as the number of rotations (r).
```text
P(At least one bad middle) = 1 - (1 - c/n)^(v*r)
```
Here's detailed tables in terms of the number of rotations required for
a given Sybil success rate for certain number of guards.
```text
1.0% Network Compromise:
Sybil Success One Two Three Four Five Six Eight Nine Ten Twelve Sixteen
10% 11 6 4 3 3 2 2 2 2 1 1
15% 17 9 6 5 4 3 3 2 2 2 2
25% 29 15 10 8 6 5 4 4 3 3 2
50% 69 35 23 18 14 12 9 8 7 6 5
60% 92 46 31 23 19 16 12 11 10 8 6
75% 138 69 46 35 28 23 18 16 14 12 9
85% 189 95 63 48 38 32 24 21 19 16 12
90% 230 115 77 58 46 39 29 26 23 20 15
95% 299 150 100 75 60 50 38 34 30 25 19
99% 459 230 153 115 92 77 58 51 46 39 29
5.0% Network Compromise:
Sybil Success One Two Three Four Five Six Eight Nine Ten Twelve Sixteen
10% 3 2 1 1 1 1 1 1 1 1 1
15% 4 2 2 1 1 1 1 1 1 1 1
25% 6 3 2 2 2 1 1 1 1 1 1
50% 14 7 5 4 3 3 2 2 2 2 1
60% 18 9 6 5 4 3 3 2 2 2 2
75% 28 14 10 7 6 5 4 4 3 3 2
85% 37 19 13 10 8 7 5 5 4 4 3
90% 45 23 15 12 9 8 6 5 5 4 3
95% 59 30 20 15 12 10 8 7 6 5 4
99% 90 45 30 23 18 15 12 10 9 8 6
10.0% Network Compromise:
Sybil Success One Two Three Four Five Six Eight Nine Ten Twelve Sixteen
10% 2 1 1 1 1 1 1 1 1 1 1
15% 2 1 1 1 1 1 1 1 1 1 1
25% 3 2 1 1 1 1 1 1 1 1 1
50% 7 4 3 2 2 2 1 1 1 1 1
60% 9 5 3 3 2 2 2 1 1 1 1
75% 14 7 5 4 3 3 2 2 2 2 1
85% 19 10 7 5 4 4 3 3 2 2 2
90% 22 11 8 6 5 4 3 3 3 2 2
95% 29 15 10 8 6 5 4 4 3 3 2
99% 44 22 15 11 9 8 6 5 5 4 3
```
The rotation counts in these tables were generated with:
## Skewed Rotation Distribution {#MaxDist}
In order to skew the distribution of the third layer guard towards
higher values, we use max(X,X) for the distribution, where X is a
random variable that takes on values from the uniform distribution.
Here's a table of expectation (arithmetic means) for relevant
ranges of X (sampled from 0..N-1). The table was generated with the
following python functions:
```text
def ProbMinXX(N, i): return (2.0*(N-i)-1)/(N*N)
def ProbMaxXX(N, i): return (2.0*i+1)/(N*N)
def ExpFn(N, ProbFunc):
exp = 0.0
for i in range(N): exp += i*ProbFunc(N, i)
return exp
```
The current choice for second-layer Vanguards-Lite guards is noted with **,
and the current choice for third-layer Full Vanguards is noted with ***.
```text
Range Min(X,X) Max(X,X)
22 6.84 14.16**
23 7.17 14.83
24 7.51 15.49
25 7.84 16.16
26 8.17 16.83
27 8.51 17.49
28 8.84 18.16
29 9.17 18.83
30 9.51 19.49
31 9.84 20.16
32 10.17 20.83
33 10.51 21.49
34 10.84 22.16
35 11.17 22.83
36 11.50 23.50
37 11.84 24.16
38 12.17 24.83
39 12.50 25.50
40 12.84 26.16
40 12.84 26.16
41 13.17 26.83
42 13.50 27.50
43 13.84 28.16
44 14.17 28.83
45 14.50 29.50
46 14.84 30.16
47 15.17 30.83
48 15.50 31.50***
```
The Cumulative Density Function (CDF) tells us the probability that a
guard will no longer be in use after a given number of time units have
passed.
Because the Sybil attack on the third node is expected to complete at any
point in the second node's rotation period with uniform probability, if we
want to know the probability that a second-level Guard node will still be in
use after t days, we first need to compute the probability distribution of
the rotation duration of the second-level guard at a uniformly random point
in time. Let's call this P(R=r).
For P(R=r), the probability of the rotation duration depends on the selection
probability of a rotation duration, and the fraction of total time that
rotation is likely to be in use. This can be written as:
```text
P(R=r) = ProbMaxXX(X=r)*r / \sum_{i=1}^N ProbMaxXX(X=i)*i
```
or in Python:
```text
def ProbR(N, r, ProbFunc=ProbMaxXX):
return ProbFunc(N, r)*r/ExpFn(N, ProbFunc)
```
For the full CDF, we simply sum up the fractional probability density for
all rotation durations. For rotation durations less than t days, we add the
entire probability mass for that period to the density function. For
durations d greater than t days, we take the fraction of that rotation
period's selection probability and multiply it by t/d and add it to the
density. In other words:
```text
def FullCDF(N, t, ProbFunc=ProbR):
density = 0.0
for d in range(N):
if t >= d: density += ProbFunc(N, d)
# The +1's below compensate for 0-indexed arrays:
else: density += ProbFunc(N, d)*(float(t+1))/(d+1)
return density
```
Computing this yields the following distribution for our current parameters:
```text
t P(SECOND_ROTATION <= t)
1 0.03247
2 0.06494
3 0.09738
4 0.12977
5 0.16207
10 0.32111
15 0.47298
20 0.61353
25 0.73856
30 0.84391
35 0.92539
40 0.97882
45 1.00000
```
This CDF tells us that for the second-level Guard rotation, the
adversary can expect that 3.3% of the time, their third-level Sybil
attack will provide them with a second-level guard node that has only
1 day remaining before it rotates. 6.5% of the time, there will
be only 2 day or less remaining, and 9.7% of the time, 3 days or less.
Note that this distribution is still a day-resolution approximation.
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