1. Anonymity – Generalizing Mixes
R. Newman

2. Topics Defining anonymity Need for anonymity Defining privacy
Threats to anonymity and privacy
Mechanisms to provide anonymity
Applications of anonymity technology

3. Mix Generics
Mix must make input messages unlinkable with output messages
Messages must all be same length
Messages must all be encrypted so as to appear random
Can't hide source/destination addresses along a single hop in path, but must hide sender and receiver, as well as distance along path
Mix must randomize order of output
Batching Strategy
How does Mix collect messages for mixing
How does Mix select and forward messages

4. Mix Triggers Timed mix Threshold mix Hybrid mix Pool mix
Mix gathers messages for period T, then sends
Threshold mix
Mix gathers N messages, then sends
Classic Chaum Mix
Hybrid mix
Mix sends when N messages or period T reached
Pool mix
Mix keeps pool of messages of size P, when pool reaches size N+P, N randomly chosen messages are sent
Continuous mix
Mix attaches random delay D from some distribution to each msg M, sends M when delay is reached

5. Mix Generics Function p: N -> [0,1] Threshold mix Timed mix
p depends on number of messages, outputs fraction of messages that mix flushes
When to execute p?
Threshold mix
Mix runs p when N messages are in batch
p always returns 1 (all messages are sent)
Timed mix
Mix runs p every T seconds
p(n) = 1 always (flush all messages)

6. Mix Generics Function p: N -> [0,1] Threshold Pool mix
p depends on number of messages, outputs fraction of messages that mix flushes
When to execute p?
Threshold Pool mix
Mix runs p when pool reaches size n = N+P
p(n) = N/(N+P) = 1 – P/(N+P)
(always leave P messages in pool)
Timed Pool mix
Mix runs p every T seconds
p(n) = (n-P)/n = 1 – P/n

7. Measuring Anonymity
Anonymity Set Size = number of input messages that could correspond to a given output message
Threshold Mix:
Anonymity set size is always N (threshold)
Timed Mix:
Anonymity set size depends on batch size
Pool Mix:
Message may remain in pool for some time
Anonymity set size = number of messages that ever entered Mix!
Not a good model!

8. Measuring Anonymity
Anonymity Set Size = number of input messages that could correspond to a given output message
Change to ”effective” AS size
Use Entropy as measure
H = - Sum [pi log pi ], pi is probability of ith item
If N items, all have prob p = 1/N, then H = log N
i.e., number of bits to specify which item out of N
Here, pi is the probability that a particular input message i corresponds to an output message
Effective AS size is 2H

9. Measuring Anonymity Timed Pool Mix Mix fires every T seconds
ni msgs in pool in ith round
ni – P messages are randomly selected
Prob(msg is flushed) = pi = (ni–P)/ni
Prob(msg that arrived in round r leaves in round i) =
pr P [j = i to r-1] (1 – pj)

10. Measuring Anonymity Threshold Pool Mix
Mix fires whenever n = N+P
N messages are randomly selected
Prob(msg is flushed) = N/(N+P) = 1 – P/(N+P)
Prob(msg that arrived in round r leaves in round i) =
[N/(N+P)] Prod [j = i to r-1] [N/(N+P)]
Gain anonymity at cost of increasing delay
Delay increased probabilistically according to exponential distribution

11. Binomial Mix Treat p(n) not as a (hard) fraction
Treat as a probability instead
Bias for coin to toss for each message in pool
Decide whether to flush that message or not
Now size of pool varies
Let s = number of messages flushed
On average, s = n p(n)
But follows binomial distribution
Variance = n p (1-p)
Attacker gains less info about size of pool
Practically can’t guess n with prob > 15%

12. Mix Padding
In addition to padding messages to some constant length (and segmenting longer messages), mix may introduce dummy messages into traffic
Dummy messages especially useful in timed mixes (may not have many messages to send)
Strong resistance from network guys
Increased network load!
Research question: how much does this form of padding help, and what is the relationship between increase in anonymity and cost of padding?

13. Next Attacks on Mix networks Cost measures of Optimization
Rerouting
Message padding
Optimization
How to preserve anonymity at low (least?) cost
Information leakage
How much information is revealed?
How?
How to prevent?
Treat as (covert) communication channel