Vanguard Rotation Statistics
Sybil rotation counts for a given number of Guards
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.
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).
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.
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
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:
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 ***.
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:
P(R=r) = ProbMaxXX(X=r)*r / \sum_{i=1}^N ProbMaxXX(X=i)*i
or in Python:
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:
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:
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.