IntroductionWe study the relationship between the number of rounds needed to repeatedly perform a private computation (i.e., where there are many sets of inputs sequentially given to the players on which the players must compute a function privately) and the overall randomness needed for this task.For the XOR function, we show that for k sets of inputs, if instead of using totally fresh (i.e., independent) random bits for each of these k sets of inputs, we re-use the same f! random bits then we can significantly speedup the round-complexity of each computation compared to what is achieved by the naive strategy of partitioning the fJ random bits between the k computations.
IntroductionWe study the relationship between the number of rounds needed to repeatedly perform a private computation (i.e., where there are many sets of inputs sequentially given to the players on which the players must compute a function privately) and the overall randomness needed for this task.For the XOR function, we show that for k sets of inputs, if instead of using totally fresh (i.e., independent) random bits for each of these k sets of inputs, we re-use the same f! random bits then we can significantly speedup the round-complexity of each computation compared to what is achieved by the naive strategy of partitioning the fJ random bits between the k computations.
We study the relationship between the number of rounds needed to repeatedly perform a private computation (i.e., where there are many sets of inputs sequentially given to the players on which the players must compute a function privately) and the overall randomness needed for this task. For the XOR function we show that, by re-using the same ℓ random bits, we can significantly speed up the round-complexity of each computation compared to what is achieved by the naive strategy of partitioning the ℓ random bits between the computations. Moreover, we prove that our protocols are optimal in the amount of randomness they require.
In the field of information-theoretic cryptography, randomness complexity is a key metric for protocols for private computation, that is, the number of random bits needed to realize the protocol. Although some general bounds are known, even for the relatively simple example of 1-private computation of n-party AND, the exact complexity is unknown. We study two settings. First, we consider the model of Goyal, Ishai, and Song (Crypto '22) where helper parties without any inputs are allowed to assist in the computation. In this setting, we show that two random bits always suffice to compute an arbitrary Boolean circuit C 1-privately: a single designated inputless helper flips the two bits and privately distributes the derived one-time bits to the other helper parties and the input parties as they are needed. We give an explicit construction using seven helper parties per AND gate and three helper parties per XOR gate (plus the single global randomness dealer). Moreover, two random bits are necessary already for the AND functionality (by a reduction to the standard no-helper model together with the lower bound of Kushilevitz, Ostrovsky, Prouff, Rosén, Thillard and Vergnaud (TCC '19), and therefore the worst-case helper-party randomness complexity is exactly 2 bits. Second, in the setting without helper parties, we improve the upper bound from Couteau and Rosén (Asiacrypt '22) on the (asymptotic) randomness complexity of n-party AND from 6 to 5 bits. That is, we give a 1-private protocol for computing the AND of n parties' inputs requiring 5 bits of randomness, for all n ≥ 6. Our construction, like that of Couteau and Rosén, uses a single party to flip the 5 bits and distribute the required derived values during the execution. Our approach to both problems is built around a more systematic exploration of techniques for recycling randomness across sub-computations. As part of resolving the second problem, we isolate an exact local-independence combinatorial object called a Sliding-Window Independence Generator, or a SWIG. A (k,m)-SWIG is a linear generator from a k-bit seed to m ≥ k output bits, where every cyclic length-k sliding window chosen from m output bits is perfectly uniform. We give an explicit (k,m)-SWIG for every k ≥ 1 and every m ≥ k and use a (5,n-1)-SWIG in our no-helper AND protocol.
We consider the amount of randomness necessary in information-theoretic private protocols. We prove that at least Ω(log n) random bits are necessary for the t-private computation of the function xor by n players, for any t ≥ 2. In view of the upper bound of O(t2log(n/t))[19], this bound is tight, up to constant factors, for any fixed t. For a class of protocols obeying certain restrictions, we give stronger lower bounds of Ω(t log (n/t)). We note that all known randomness efficient private protocols designed specifically for xor belong to this class. All our lower bounds hold for the "trusted dealer" model as well, and the Ω(t log (n/t)) lower bound for restricted protocols is tight, up to constant factors, for any t ≥ 2 in this model.In comparison, the previous lower bounds on the amount of randomness required by t-private computation of explicit functions did not grow with n for constant values of t, and our results improve the previous lower bounds for xor for any 2 ≤ t = o(log n). Our results also show that already for t = 2, Ω(log n) random bits are necessary, while it is known that for the case of t = 1 a single random bit is sufficient for privately computing xor for any number of players.Our proofs use novel techniques by which we extract random variables from a t-private protocol, and then use the t-privacy property of the protocol to prove properties of these random variables. These properties in turn imply that the number of random bits used by the players is large.
We consider the amount of randomness necessary in information-theoretic private protocols. We prove that at least Ω(log n) random bits are necessary for the t-private computation of the function xor by n players, for any t ≥ 2. In view of the upper bound of O(t2log(n/t))[19], this bound is tight, up to constant factors, for any fixed t. For a class of protocols obeying certain restrictions, we give stronger lower bounds of Ω(t log (n/t)). We note that all known randomness efficient private protocols designed specifically for xor belong to this class. All our lower bounds hold for the "trusted dealer" model as well, and the Ω(t log (n/t)) lower bound for restricted protocols is tight, up to constant factors, for any t ≥ 2 in this model.In comparison, the previous lower bounds on the amount of randomness required by t-private computation of explicit functions did not grow with n for constant values of t, and our results improve the previous lower bounds for xor for any 2 ≤ t = o(log n). Our results also show that already for t = 2, Ω(log n) random bits are necessary, while it is known that for the case of t = 1 a single random bit is sufficient for privately computing xor for any number of players.Our proofs use novel techniques by which we extract random variables from a t-private protocol, and then use the t-privacy property of the protocol to prove properties of these random variables. These properties in turn imply that the number of random bits used by the players is large.
In a network of n players, each player i having private input zi, we show how the players can collaboratively evaluate a function f(zl, ..., zn) in a way that does not compromise the privacy of the players' inputs, and yet requires only a constant number of rounds of interaction.The underlying model of computation is a complete network of private channels, with broadcast, and a majority of the players must behave honestly.Our solution assumes the existence of a one-way function.share bi to player i.For some parameter t, t < n/2, we require that no t players get information about b from their pieces; and yet, b is recoverable, and is known to be recoverable, given the cooperation of the n -t good players--even if the t bad players try to obstruct b's recovery, or try to alter the recovered value.The value b which a player has effectively "committed to" is independent of the values that honest players may concurrently be committing to.After the sharing stage, a computation stage follows, in which each player, given his own shares of xl, ..., x,~, computes his own share of f(zt,..., z,~).To accomplish this, the function f to be evaluated is represented by a
We show that any multiparty functionality can be evaluated using a 2-round protocol with perfect correctness and perfect semihonest security, provided that the majority of parties are honest. This settles the round complexity of information-theoretic semihonest multiparty computation, resolving a longstanding open question [Y. Ishai and E. Kushilevitz, Randomizing polynomials: A new representation with applications to round-efficient secure computation, in Proceedings of the 41st Annual Symposium on Foundations of Computer Science FOCS 2000, IEEE Computer Society, 2000, pp. 294--304]. The protocol is efficient for NC1 functionalities. Furthermore, given black-box access to a one-way function, the protocol can be made efficient for any polynomial functionality, at the cost of only guaranteeing computational security. Our results are based on a new notion of multiparty randomized encoding which extends and relaxes the standard notion of randomized encoding of functions [Y. Ishai and E. Kushilevitz, Randomizing polynomials: A new representation with applications to round-efficient secure computation, in Proceedings of the 41st Annual Symposium on Foundations of Computer Science FOCS 2000, IEEE Computer Society, 2000, pp. 294--304]. The property of a multiparty randomized encoding (MPRE) is that if the functionality g is an encoding of the functionality f, then for any (permitted) coalition of players, their respective outputs and inputs in g allow them to simulate their respective inputs and outputs in f, without learning anything else, including the other outputs of f. We further introduce a new notion of effective degree, and show that the round complexity of a functionality f is characterized by the degree of its MPRE. We construct degree-2 MPREs for general functionalities in several settings under different assumptions, and use these constructions to obtain 2-round protocols. Our constructions also give rise to new protocols in the client-server model with optimal round complexity.