Private vs public randomness in presence of faults
Duration and stipend:6 months, 500-600 euros/month Starting date: February-March 2021
Internship institution: LISN, équipe ParSys, physically in LRI (Laboratoire de Recherche en Informa- tique), Université Paris-Saclay (remote supervision in case of COVID lockdown is possible)
Supervisors: Evangelos Bampas (bampas@lri.fr), Joffroy Beauquier (jb@lri.fr), Janna Burman (bur- man@lri.fr)
Keywords:distributed algorithms, fault tolerance, randomized algorithms, public randomness
To apply:contact the supervisors presenting your motivations and relevant skills (preferably including grade transcripts)
A randomized algorithm has access to random values. The algorithm typically uses uniformly random bits as an auxiliary input (the coin), in the hope of achieving good performance in the “average case” over all possible choices of sequences of random bits. Many distributed problems involving strong adversaries are proven to be deterministically unsolvable. It is the case, for instance, for classical crash or byzantine consensus in asynchronous networks. However, randomized solutions exist. The role of randomization here is to make future choices (of the algorithm) unpredictable. If the adversaries (modeling the communication environment and faults) have some knowledge in advance of what the algorithm will choose to do (determinism), then they may easier determine a winning strategy. But the use of randomization makes these predictions impossible and forces the adversaries to make bets and, eventually, lose. Randomization is used, in these examples, for unpredictability.
The branch of the theory that studies unpredictability is randomness. It is not part of probability theory. As expressed by Knuth [1], the probability theory eludes the problem of getting a random sequence: “The mathematical theory of probability and statistics scrupulously avoids the issue [of defining a random behavior]. . . . The axioms of probability theory are set up so that abstract probabilities can be computed readily, but nothing is said about what probability really signifies and how this concept can be applied meaningfully to the actual world.”
The duality of the two notions, randomization and randomness, can be described in the paradigm of private/public coin. The idea of processess flipping coins is an abstraction. In practice, processes use a pseudo-random generator that only approximates coin flipping. Ultimately, the outputs of a pseudo- random generator are predictable and the only implicit assumption is that making an effective correct prediction would cost too much in terms of time and resources. The theory of randomness considers this issue. Surprisingly enough there are no studies of getting private/public randomness in the presence of strong (byzantine) adversaries. Yet, time and amount of resources available to adversaries are crucial parameters for evaluating the degree of failure tolerance of a system.
The subject of this internship is not how to produce a random sequence, as we assume that some random sequences are given. Each process in a distributed system receives independently a random sequence. The problem is how to obtain, in a distributed way, an output random sequence, common to all (correct) processes, in the possible presence of faulty processes. In other words, we consider the problem of obtaining a public random sequence from private random sequences, in a distributed setting with strong adversaries. A starting point of this study will be the famous paper by O’Connor [2], “An unpredictability approach to finite-state randomness”. A goal would be to design distibuted protocols solving the public randomness problem.
References
[1] Donald E. Knuth: The Art of Computer Programming, volume 2: Seminumerical Algorithms (Third edition, p. 149). Reading, Massachusetts: Addison-Wesley, 1997.
[2] M. G. O’Connor: An unpredictability approach to finite-state randomness. Journal of Computer and System Sciences 37(3), 324-336 (1988).
