Biblio

In STOC 1988, Ben-Or, Goldwasser, and Wigderson (BGW) established an important milestone in the fields of cryptography and distributed computing by showing that every functionality can be computed with perfect (information-theoretic and error-free) security at the presence of an active (aka Byzantine) rushing adversary that controls up to n/3 of the parties. We study the round complexity of general secure multiparty computation in the BGW model. Our main result shows that every functionality can be realized in only four rounds of interaction, and that some functionalities cannot be computed in three rounds. This completely settles the round-complexity of perfect actively-secure optimally-resilient MPC, resolving a long line of research. Our lower-bound is based on a novel round-reduction technique that allows us to lift existing three-round lower-bounds for verifiable secret sharing to four-round lower-bounds for general MPC. To prove the upper-bound, we develop new round-efficient protocols for computing degree-2 functionalities over large fields, and establish the completeness of such functionalities. The latter result extends the recent completeness theorem of Applebaum, Brakerski and Tsabary (TCC 2018, Eurocrypt 2019) that was limited to the binary field.

Suppose that you have n truly random bits x=(x1,…,xn) and you wish to use them to generate m≫ n pseudorandom bits y=(y1,…, ym) using a local mapping, i.e., each yi should depend on at most d=O(1) bits of x. In the polynomial regime of m=ns, stextgreater1, the only known solution, originates from (Goldreich, ECCC 2000), is based on Random Local Functions: Compute yi by applying some fixed (public) d-ary predicate P to a random (public) tuple of distinct inputs (xi1,…,xid). Our goal in this paper is to understand, for any value of s, how the pseudorandomness of the resulting sequence depends on the choice of the underlying predicate. We derive the following results: (1) We show that pseudorandomness against F2-linear adversaries (i.e., the distribution y has low-bias) is achieved if the predicate is (a) k=Ω(s)-resilience, i.e., uncorrelated with any k-subset of its inputs, and (b) has algebraic degree of Ω(s) even after fixing Ω(s) of its inputs. We also show that these requirements are necessary, and so they form a tight characterization (up to constants) of security against linear attacks. Our positive result shows that a d-local low-bias generator can have output length of nΩ(d), answering an open question of Mossel, Shpilka and Trevisan (FOCS, 2003). Our negative result shows that a candidate for pseudorandom generator proposed by the first author (computational complexity, 2015) and by O’Donnell and Witmer (CCC 2014) is insecure. We use similar techniques to refute a conjecture of Feldman, Perkins and Vempala (STOC 2015) regarding the hardness of planted constraint satisfaction problems. (2) Motivated by the cryptanalysis literature, we consider security against algebraic attacks. We provide the first theoretical treatment of such attacks by formalizing a general notion of algebraic inversion and distinguishing attacks based on the Polynomial Calculus proof system. We show that algebraic attacks succeed if and only if there exist a degree e=O(s) non-zero polynomial Q whose roots cover the roots of P or cover the roots of P’s complement. As a corollary, we obtain the first example of a predicate P for which the generated sequence y passes all linear tests but fails to pass some polynomial-time computable test, answering an open question posed by the first author (Question 4.9, computational complexity 2015).

Suppose that you have n truly random bits x=(x1,…,xn) and you wish to use them to generate m≫ n pseudorandom bits y=(y1,…, ym) using a local mapping, i.e., each yi should depend on at most d=O(1) bits of x. In the polynomial regime of m=ns, stextgreater1, the only known solution, originates from (Goldreich, ECCC 2000), is based on Random Local Functions: Compute yi by applying some fixed (public) d-ary predicate P to a random (public) tuple of distinct inputs (xi1,…,xid). Our goal in this paper is to understand, for any value of s, how the pseudorandomness of the resulting sequence depends on the choice of the underlying predicate. We derive the following results: (1) We show that pseudorandomness against F2-linear adversaries (i.e., the distribution y has low-bias) is achieved if the predicate is (a) k=Ω(s)-resilience, i.e., uncorrelated with any k-subset of its inputs, and (b) has algebraic degree of Ω(s) even after fixing Ω(s) of its inputs. We also show that these requirements are necessary, and so they form a tight characterization (up to constants) of security against linear attacks. Our positive result shows that a d-local low-bias generator can have output length of nΩ(d), answering an open question of Mossel, Shpilka and Trevisan (FOCS, 2003). Our negative result shows that a candidate for pseudorandom generator proposed by the first author (computational complexity, 2015) and by O’Donnell and Witmer (CCC 2014) is insecure. We use similar techniques to refute a conjecture of Feldman, Perkins and Vempala (STOC 2015) regarding the hardness of planted constraint satisfaction problems. (2) Motivated by the cryptanalysis literature, we consider security against algebraic attacks. We provide the first theoretical treatment of such attacks by formalizing a general notion of algebraic inversion and distinguishing attacks based on the Polynomial Calculus proof system. We show that algebraic attacks succeed if and only if there exist a degree e=O(s) non-zero polynomial Q whose roots cover the roots of P or cover the roots of P’s complement. As a corollary, we obtain the first example of a predicate P for which the generated sequence y passes all linear tests but fails to pass some polynomial-time computable test, answering an open question posed by the first author (Question 4.9, computational complexity 2015).