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.

ACM Symposium on Principles of Distributed Computing (PODC)Jan 1, 199813 citations

Lower and Upper Bounds on the Randomness Complexity of Private Computations of AND

Eyal Kushilevitz Z, Rafail Ostrovsky, Emmanuel Prouff +3 more

Lecture notes in computer scienceJan 1, 20192 citations

Amortizing Randomness in Private Multiparty Computations

Eyal Kushilevitz Z, Rafail Ostrovsky, Adi Rosén

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 $\ell$ 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 $\ell$ random bits between the computations. Moreover, we prove that our protocols are optimal in the amount of randomness they require.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 $\ell$ 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 $\ell$ random bits between the computations. Moreover, we prove that our protocols are optimal in the amount of randomness they require.

SIAM Journal on Discrete MathematicsJan 1, 20037 citations

Randomness Recycling in Constant-Round Private Computations

Carlo Blundo, Clemente Galdi, Pino Persiano

Lecture notes in computer scienceJan 1, 199910 citations

Lower bounds on the amount of randomness in private computation

Anna Gál, Adi Rosén

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.

ACM Symposium on Theory of Computing (STOC)Jun 9, 200313 citations

\ell

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

\ell

random bits between the computations. Moreover, we prove that our protocols are optimal in the amount of randomness they require.