International Association for Cryptologic Research

International Association
for Cryptologic Research


Adi Rosén


Lower and Upper Bounds on the Randomness Complexity of Private Computations of AND
We consider multi-party information-theoretic private protocols, and specifically their randomness complexity. The randomness complexity of private protocols is of interest both because random bits are considered a scarce resource, and because of the relation between that complexity measure and other complexity measures of boolean functions such as the circuit size or the sensitivity of the function being computed [12, 17].More concretely, we consider the randomness complexity of the basic boolean function and, that serves as a building block in the design of many private protocols. We show that and cannot be privately computed using a single random bit, thus giving the first non-trivial lower bound on the 1-private randomness complexity of an explicit boolean function, $$f: \{0,1\}^n \rightarrow \{0,1\}$$. We further show that the function and, on any number of inputs n (one input bit per player), can be privately computed using 8 random bits (and 7 random bits in the special case of $$n=3$$ players), improving the upper bound of 73 random bits implicit in [17]. Together with our lower bound, we thus approach the exact determination of the randomness complexity of and. To the best of our knowledge, the exact randomness complexity of private computation is not known for any explicit function (except for xor, which is trivially 1-random, and for several degenerate functions).
Randomness versus Fault-Tolerance
We investigate the relations between two major requirements of multiparty protocols: {\em fault tolerance} (or {\em resilience}) and {\em randomness}. Fault-tolerance is measured in terms of the maximum number of colluding faulty parties, t, that a protocol can withstand and still maintain the privacy of the inputs and the correctness of the outputs (of the honest parties). Randomness is measured in terms of the total number of random bits needed by the parties in order to execute the protocol. Previously, the upper bound on the amount of randomness required by general constructions for securely computing any non-trivial function f was polynomial both in $n$, the total number of parties, and the circuit-size C(f). This was the state of knowledge even for the special case t=1 (i.e., when there is at most one faulty party). In this paper, we show that for any linear-size circuit, and for any number t < n/3 of faulty parties, O(poly(t) * log n) randomness is sufficient. More generally, we show that for any function f with circuit-size C(f), we need only O(poly(t) * log n + poly(t) * C(f)/n) randomness in order to withstand any coalition of size at most t. Furthermore, in our protocol only t+1 parties flip coins and the rest of the parties are deterministic. Our results generalize to the case of adaptive adversaries as well.