Time oblivious Sampling Algorithm for Random Variate Generation

Given a fair random bit generator, how do you generate random numbers with arbitrary probabilities?

Previously we saw the Knuth-Yao algorithm for generating random variates with arbitrary probabilities. We showed how to model the algorithm in FizzBee and looked at the number of random bits used on average and showed that the number of bits used (or time to output) depends on the output generated.

In this post, we will see and analyze a time-oblivious sampling algorithm for random variate generation based on the paper Resistance to Timing Attacks for Sampling and Privacy Preserving Schemes.

The Algorithm

For an algorithm, consider the following probabilities [1/2, 1/3, 1/6].

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Simplifying the algorithm

For this, I am simplifying the code to some extent. In the original code, there are references to LCM of the denominators of the probabilities. But for the sake of simplicity, I am assuming the input is given with such that the denominator is already the LCM. That is, [1/2, 1/3, 1/6] is given as [3/6, 2/6, 1/6]. The, the array P is [3, 2, 1] and Q = 6.

There is another reference to binary search, that is actually not required, as you could do an early return from within the for loop.

And finally, you can see in the graph that after input 11, it is the same as reaching the root. So, we will simplify the graph such that after 11, the graph will go to the root. This will reduce the solution to bounded state space.

FizzBee Spec

Run in playground

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# Time oblivious algorithm for Random Variate Generation
# FizzBee representation of https://drops.dagstuhl.de/storage/00lipics/lipics-vol256-forc2023/LIPIcs.FORC.2023.11/LIPIcs.FORC.2023.11.pdf

P = [3, 2, 1]
Q = 6

REQUIRED_LEVEL = math.ceil(math.log(Q, 2))

action Init:
  value = -1
  random_bits = ''

atomic func RandomBit():
  oneof:
    `zero` return 0
    `one` return 1

atomic action GenRandomValue:
    if value != -1:
        return value

    random_bits = ''
    while True:
        bit = RandomBit()
        random_bits = random_bits + str(bit)
        leaf_nodes = 1 << len(random_bits)
        random_int = int(random_bits, 2)
        if random_int >= Q/2 and len(random_bits) == REQUIRED_LEVEL-1:
            random_bits = ''
            continue

        if leaf_nodes  >= Q:
            value = GetValue(random_int)
            #random_bits = ''
            return value


atomic func GetValue(j):
    cumulative = 0
    for i in range(0, len(P)):
        cumulative = cumulative + P[i]
        if j < cumulative:
            return i

You can run this in the FizzBee playground and see the state graph. This graph is actually the mirror image of the DDG tree shown above. The algorithm takes the left link on 1 and right link on 0.

Probabilistic Model Checking

Now, let us evaluate the actual probabilities of the generated numbers. The probabilistic evaluation is not supported in the online playground yet, until then you can install FizzBee locally following the instructions.

./fizz samples/timeoblivious-sampling/TimeObliviousSampling.fizz
Model checking samples/timeoblivious-sampling/TimeObliviousSampling.json
configFileName: samples/timeoblivious-sampling/fizz.yaml
StateSpaceOptions: options:{max_actions:100  max_concurrent_actions:5}  action_options:{key:"GenRandomValue"  value:{max_actions:1}}
Nodes: 13, elapsed: 3.508375ms
Time taken for model checking: 3.522458ms
Writen graph dotfile: samples/timeoblivious-sampling/out/run_2024-06-02_17-42-33/graph.dot
To generate svg, run: 
dot -Tsvg samples/timeoblivious-sampling/out/run_2024-06-02_17-42-33/graph.dot -o graph.svg && open graph.svg
Max Depth 4
PASSED: Model checker completed successfully
Writen 1 node files and 1 link files to dir samples/timeoblivious-sampling/out/run_2024-06-02_17-42-33

For smaller state spaces, this will generate the graph as in the playground. Also, take note of the paths, the json file path and the output directory path, as you will need them to generate the graph.

Set the probabilities and cost

In this example, there is only one place where the non-determinism happens. It is at choosing the random bits. By default FizzBee assumes, uniform distribution for the random bits. But we can customize when required. Similarly, we can also assign the cost for each step.

To refer to the step for assigning probabilities and cost, we can use the labels. The labels are defined by prefixing with with backtick enclosed string. For example, we have zero and one as labels for the random bits. To reference them, it will be namespaced to the function or action name. So, to refer to the zero label in the RandomBit function, it will be RandomBit.zero.

Now, create a file perf_model.yaml with the following content.

configs:
  RandomBit.zero:
    counters:
      toss:
        numeric: 1
  RandomBit.one:
    counters:
      toss:
        numeric: 1

Here, any time the zero label is taken, the toss counter is incremented by 1. Similarly, for the one label. In FizzBee cost/reward are referred to as counter.

For analyzing random number generation algorithms, it is customary to use the number of coin tosses or the random bits used as the cost.

Run the model checker

Metrics(mean={'toss': 3.666666567325592}, histogram=[(0.75, {'toss': 3.0}), (0.9375, {'toss': 5.0}), (0.984375, {'toss': 7.0}), (0.99609375, {'toss': 9.0}), (0.9990234375, {'toss': 11.0}), (0.999755859375, {'toss': 13.0}), (0.99993896484375, {'toss': 15.0}), (0.9999847412109375, {'toss': 17.0}), (0.9999961853027344, {'toss': 19.0}), (0.9999990463256836, {'toss': 21.0}), (0.9999997615814209, {'toss': 23.0}), (0.9999999403953552, {'toss': 25.0})])
     7: 0.166667 state: {'random_bits': '"000"', 'value': '0'} / returns: {}
     8: 0.166667 state: {'random_bits': '"001"', 'value': '0'} / returns: {}
     9: 0.166667 state: {'random_bits': '"010"', 'value': '0'} / returns: {}
    10: 0.166667 state: {'random_bits': '"011"', 'value': '1'} / returns: {}
    11: 0.166667 state: {'random_bits': '"100"', 'value': '1'} / returns: {}
    12: 0.166667 state: {'random_bits': '"101"', 'value': '2'} / returns: {}

You can actually see the generated probabilities and the mean and the histogram for the time to complete.

If you sum up the probabilities for each return value, you will see the probabilities match.

You will also see the cost of average number of tosses to reach each of these states.

Cost to reach terminal states:
     7: {'toss': 3.666666567325592}
     8: {'toss': 3.666666567325592}
     9: {'toss': 3.666666567325592}
    10: {'toss': 3.666666567325592}
    11: {'toss': 3.666666567325592}
    12: {'toss': 3.666666567325592}

That is, on average, it takes 3.67 tosses to generate the output and the output is independent of the time taken to generate the output.

This fixes the side channel vulnerability in the Knuth-Yao algorithm.

Reducing the possible output space

In the above implementation, we made the generated graph to match the DDG tree. But in practice, we can reduce the number of output states, by keeping only one output state per output value.

Just remove the col=0 from the Init action. This will make the col variable to be a local variable.

action Init:
  value = -1
-  random_bits='''

Run the model checker again, and you will see the number of nodes reduced.

Now, run the performance model checker. You will see the output probabilities for each return value summed as you would want.

Metrics(mean={'toss': 3.666666567325592}, histogram=[(0.75, {'toss': 3.0}), (0.9375, {'toss': 5.0}), (0.984375, {'toss': 7.0}), (0.99609375, {'toss': 9.0}), (0.9990234375, {'toss': 11.0}), (0.999755859375, {'toss': 13.0}), (0.99993896484375, {'toss': 15.0}), (0.9999847412109375, {'toss': 17.0}), (0.9999961853027344, {'toss': 19.0}), (0.9999990463256836, {'toss': 21.0}), (0.9999997615814209, {'toss': 23.0}), (0.9999999403953552, {'toss': 25.0})])
     7: 0.500000 state: {'value': '0'} / returns: {}
     8: 0.333333 state: {'value': '1'} / returns: {}
     9: 0.166667 state: {'value': '2'} / returns: {}

Two Dice Problem

We previously solved the two dice problem using a simpler approach that is non-optimal. It took a mean of 7.333333 tosses to generate the output.

Now, let us model the same problem using the Knuth-Yao algorithm.

For this, you just need to update the probabilities. The probabilities for the two dice problem are [1/36, 2/36, 3/36, 4/36, 5/36, 6/36, 5/36, 4/36, 3/36, 2/36, 1/36].

For this, just change the values for P and Q.

Q=36
P=[1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1]

Now, run the model checker.

./fizz samples/timeoblivious-sampling/TimeObliviousSampling.fizz
Model checking samples/timeoblivious-sampling/TimeObliviousSampling.json
configFileName: samples/timeoblivious-sampling/fizz.yaml
StateSpaceOptions: options:{max_actions:100  max_concurrent_actions:5}  action_options:{key:"GenRandomValue"  value:{max_actions:1}}
Nodes: 61, elapsed: 26.9405ms
Time taken for model checking: 26.959208ms
Writen graph dotfile: samples/timeoblivious-sampling/out/run_2024-06-02_17-52-51/graph.dot
To generate svg, run: 
dot -Tsvg samples/timeoblivious-sampling/out/run_2024-06-02_17-52-51/graph.dot -o graph.svg && open graph.svg
Max Depth 7
PASSED: Model checker completed successfully
Writen 1 node files and 1 link files to dir samples/timeoblivious-sampling/out/run_2024-06-02_17-52-51

Now, run the performance model checker.

Metrics(mean={'toss': 9.888887849609615}, histogram=[(0.5625, {'toss': 6.0}), (0.80859375, {'toss': 11.0}), (0.916259765625, {'toss': 16.0}), (0.9633636474609375, {'toss': 21.0}), (0.9839715957641602, {'toss': 26.0}), (0.9929875731468201, {'toss': 31.0}), (0.9969320632517338, {'toss': 36.0}), (0.9986577776726335, {'toss': 41.0}), (0.9994127777317772, {'toss': 46.0}), (0.9997430902576525, {'toss': 51.0}), (0.999887601987723, {'toss': 56.0}), (0.9999508258696288, {'toss': 61.0}), (0.9999784863179626, {'toss': 66.0}), (0.9999784863179628, {'toss': 67.0}), (0.9999784863179628, {'toss': 69.0}), (0.9999905877641088, {'toss': 71.0}), (0.9999905877641087, {'toss': 74.0}), (0.9999958821467975, {'toss': 76.0}), (0.9999958821467978, {'toss': 77.0}), (0.9999981984392241, {'toss': 81.0}), (0.9999992118171604, {'toss': 86.0}), (0.9999992118171606, {'toss': 87.0}), (0.9999992118171606, {'toss': 89.0}), (0.9999996551700079, {'toss': 91.0}), (0.9999996551700079, {'toss': 93.0}), (0.9999996551700079, {'toss': 95.0}), (0.9999998491368783, {'toss': 96.0})])
    50: 0.027778 state: {'value': '0'} / returns: {}
    51: 0.055556 state: {'value': '1'} / returns: {}
    52: 0.083333 state: {'value': '2'} / returns: {}
    53: 0.111111 state: {'value': '3'} / returns: {}
    54: 0.138889 state: {'value': '4'} / returns: {}
    55: 0.166667 state: {'value': '5'} / returns: {}
    56: 0.138889 state: {'value': '6'} / returns: {}
    57: 0.111111 state: {'value': '7'} / returns: {}
    58: 0.083333 state: {'value': '8'} / returns: {}
    59: 0.055556 state: {'value': '9'} / returns: {}
    60: 0.027778 state: {'value': '10'} / returns: {}

As you can see, this algorith takes on average 9.89 tosses to generate the output, where as the Knuth-Yao algorithm takes 3.67 tosses. But, this algorithm is secure from side channels attacks.

Conclusion

In this post, we saw the time oblivious sampling algorithm for random variate generation. And, how to analyze algorithms for probability and performance characteristics typically necessary for cryptographic algorithms.

Learn more about how to use FizzBee by going through the tutorials.