A fast parallel algorithm to compute the rank of a matrix over an arbitrary field
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
The parallel complexity of exponentiating polynomials over finite fields
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Completeness theorems for non-cryptographic fault-tolerant distributed computation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
An approximation algorithm for the number of zeros of arbitrary polynomials over GF[q]
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Approximating the number of zeroes of a GF[2] polynomial
Journal of Algorithms
Fast Parallel Computation of Polynomials Using Few Processes
Proceedings on Mathematical Foundations of Computer Science
Perfect Constant-Round Secure Computation via Perfect Randomizing Polynomials
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
The Computational Complexity of ({\it XOR, AND\/})-Counting Problems
The Computational Complexity of ({\'it XOR, AND\'/})-Counting Problems
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
COMPUTATIONALLY PRIVATE RANDOMIZING POLYNOMIALS AND THEIR APPLICATIONS
Computational Complexity
Verifying and decoding in constant depth
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
A (de)constructive approach to program checking
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Fast parallel matrix and GCD computations
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
Efficient multi-party computation over rings
EUROCRYPT'03 Proceedings of the 22nd international conference on Theory and applications of cryptographic techniques
Algorithms for modular counting of roots of multivariate polynomials
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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Randomizing polynomials represent a function f(x) by a low-degree randomized mapping p(x, r) over a finite field F such that, for any input x, the output distribution of p(x, r) depends only on the value of f(x). We study the class of functions f which admit an efficient representation by constant-degree randomizing polynomials. It is known that this class contains NC1 as well as log-space classes contained in NC2. Whether it contains all polynomial-time computable functions is a wide open question. A positive answer would have major and unexpected consequences, including the existence of efficient constant-round multiparty protocols with unconditional security, and the equivalence of (polynomial-time) cryptography and cryptography in NC0. We obtain evidence for the limited power of randomizing polynomials by showing that a useful subclass of constant-degree randomizing polynomials cannot efficiently capture functions beyond NC. Concretely, we consider randomizing polynomials over fields F of a small characteristic in which each monomial has degree (at most) 2 in the random inputs r and constant degree in x. This subclass captures most constructions of randomizing polynomials from the literature. Our main result is that all functions f which can be efficiently represented by such randomizing polynomials over fields of a small characteristic are in non-uniform NC. (The same holds over arbitrary fields given a quadratic residuosity oracle.) This result is obtained in two steps: (1) we observe that computing f as above reduces to counting roots of degree-2 multivariate polynomials; (2) we design parallel algorithms for the latter problem. These parallel root counting algorithms may be of independent interest. On the flip side, our main result provides an avenue for obtaining new parallel algorithms via the construction of randomizing polynomials. This gives an unexpected application of cryptography to algorithm design. We provide several examples for the potential usefulness of this approach.