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A deterministic algorithm for sparse multivariate polynomial interpolation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
A hard-core predicate for all one-way functions
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
A lower bound for matrix multiplication
SIAM Journal on Computing
Hiding instances in multioracle queries
STACS 90 Proceedings of the seventh annual symposium on Theoretical aspects of computer science
Efficient checking of computations
STACS 90 Proceedings of the seventh annual symposium on Theoretical aspects of computer science
Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields
SIAM Journal on Computing
Checking computations in polylogarithmic time
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Self-testing/correcting for polynomials and for approximate functions
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Highly resilient correctors for polynomials
Information Processing Letters
Random-self-reducibility of complete sets
SIAM Journal on Computing
A direct version of Shamir and Snir's lower bounds on monotone circuit depth
Information Processing Letters
SIAM Journal on Computing
Reducing randomness via irrational numbers
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Checking polynomial identities over any field: towards a derandomization?
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
An exponential lower bound for depth 3 arithmetic circuits
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
On the efficiency of local decoding procedures for error-correcting codes
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Learning Polynomials with Queries: The Highly Noisy Case
SIAM Journal on Discrete Mathematics
Randomness efficient identity testing of multivariate polynomials
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Lower bounds for matrix product, in bounded depth circuits with arbitrary gates
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Matching is as easy as matrix inversion
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Breaking the O(n1/(2k-1)) Barrier for Information-Theoretic Private Information Retrieval
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Exponential lower bound for 2-query locally decodable codes via a quantum argument
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Derandomizing polynomial identity tests means proving circuit lower bounds
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Primality and identity testing via Chinese remaindering
Journal of the ACM (JACM)
Depth-3 Arithmetic Formulae over Fields of Characteristic Zero
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Lower bounds on arithmetic circuits via partial derivatives
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Lower Bounds for Linear Locally Decodable Codes and Private Information Retrieval
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
Better Lower Bounds for Locally Decodable Codes
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
Lower Bounds for Matrix Product
SIAM Journal on Computing
Proving Hard-Core Predicates Using List Decoding
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Multi-linear formulas for permanent and determinant are of super-polynomial size
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Deterministic Polynomial Identity Testing in Non-Commutative Models
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
Interpolation of depth-3 arithmetic circuits with two multiplication gates
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Polynomial Identity Testing for Depth 3 Circuits
Computational Complexity
The ideal membership problem and polynomial identity testing
Information and Computation
The monomial ideal membership problem and polynomial identity testing
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
A quadratic lower bound for three-query linear locally decodable codes over any field
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Three-Query Locally Decodable Codes with Higher Correctness Require Exponential Length
ACM Transactions on Computation Theory (TOCT)
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FSTTCS'06 Proceedings of the 26th international conference on Foundations of Software Technology and Theoretical Computer Science
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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In this work we study two, seemingly unrelated, notions. Locally Decodable Codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial Identity Testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero. We improve known results on locally decodable codes and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results:We show that if E: Fn → Fm is a linear LDC with 2 queries then m = exp(Ω(n)). Previously this was only known for fields of size n [18].We show that from every depth 3 arithmetic circuit (ΣΠΣ circuit), C, with a bounded (constant) top fan-in that computes the zero polynomial, one can construct a locally decodeable code. More formally: Assume that C is minimal (no subset of the multiplication gates sums to zero) and simple (no linear function appears in all the multiplication gates). Denote by d the degree of the polynomial computed by C and by r the rank of the linear functions appearing in C. Then we can construct a linear LDC with 2 queries, that encodes messages of length r/polylog(d) by codewords of length O(d).We prove a structural theorem for ΣΠΣ circuits, with a bounded top fan-in, that compute the zero polynomial. In particular we show that if such a circuit is simple and minimal and of polynomial size then its rank, r, is only polylogarithmic in the number of variables (a priory it could have been linear).We give new PIT algorithms for ΣΠΣ circuits with a bounded top fan-in:A deterministic algorithm that runs in quasi polynomial time.A randomized algorithm that runs in polynomial time and uses only polylogarithmic number of random bits..Moreover, when the circuit is multilinear our deterministic algorithm runs in polynomial time. Previously, deterministic subexponential time algorithms for PIT in bounded depth circuits were known only for depth 2 circuits (in the black box model)[22, 9, 28]. In particular, for the special case of depth 3 circuits with 3 multiplication gates our result resolves an open question asked by Klivans and Spielman [28].