# 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.