3 Exercises 1. Warner's algorithm Refer to Figure 2. This question assumes that N = 1 i.e. a single respondent. (a) Write down the channel when the coin is biased (p=1/3) so that there is a 1/3 probability that coin returns 0, and 2/3 that the coin returns 1. The second coin flip is still fair. (b) Assuming a uniform prior, compute the Bayes' vulnerability. (c) Compute the multiplicative capacity; (d) Now assume that the coin is biased so that there is a p 2 0 probability that coin returns 0, and 1-p that the coin returns 1. The second coin flip is still fair. Write down the channel (based on p); (e) Show that the Bayes' vulnerability is constant as p varies; (f) Given your previous answer, why is p = 0 not a good value to use? 2. The Dining Cryptographers (a) In Section 2.2.1 let NS be the channel. Let * = (1/3, 2/3). Compute Vov (7, NS). (b) Show that the probability of revealing whether or not the NSA paid is always 1, independent of the prior. (c) In Section 2.2.2 let CR be the channel. Compute the Bayes' vulner- ability assuming the uniform prior over cA, cB, cC. 8(d) Show that the optimal value for p is 1/2 to minimise the probability of determining which philosopher paid in the case that the NSA did not pay.: Randomised response with single participant r 0 1 0 3H 1/4 1 U4 3/4 1. In eaeh ease the true response (either 0 or 1) is delivered with probability 3/4. 24 If we aSslune that the adversary knows nothing about the potential tnie answer, then the adversary's prior knowledge is modelled by the uniform distribution over the possible responses 3. Assuming a. uniform prior then the probability of observing a particular value is \"'2. 4. Assuming a uniform prior. we can compute the adversary's posterior dis- tribution for each of the two potential observations using Bayes' theorem. 5. If the observer see 0 then her/his posterior is {Ii/'11., 1/4} i.e. with proba- bility 3,\"! the observer thinks that the true response is 0