One-way functions and Pseudorandom generators
Combinatorica - Theory of Computing
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Journal of Computer and System Sciences
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Probabilistic Algorithms for Deciding Equivalence of Straight-Line Programs
Journal of the ACM (JACM)
Cook's versus Valiant's hypothesis
Theoretical Computer Science - Selected papers in honor of Manuel Blum
Introduction to Circuit Complexity: A Uniform Approach
Introduction to Circuit Complexity: A Uniform Approach
Uniform constant-depth threshold circuits for division and iterated multiplication
Journal of Computer and System Sciences - Complexity 2001
Hiding Instances in Multioracle Queries
STACS '90 Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Straight-line complexity and integer factorization
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
Randomness vs. Time: De-Randomization under a Uniform Assumption
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Testing polynomials which are easy to compute (Extended Abstract)
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
The complexity of constructing pseudorandom generators from hard functions
Computational Complexity
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
On Defining Integers And Proving Arithmetic Circuit Lower Bounds
Computational Complexity
Improving exhaustive search implies superpolynomial lower bounds
Proceedings of the forty-second ACM symposium on Theory of computing
Interpolation in Valiant’s Theory
Computational Complexity
Non-uniform ACC Circuit Lower Bounds
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
Permanent does not have succinct polynomial size arithmetic circuits of constant depth
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Proving lower bounds via pseudo-random generators
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
Permanent does not have succinct polynomial size arithmetic circuits of constant depth
Information and Computation
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Suppose f is a univariate polynomial of degree r = r(n) that is computed by a size n arithmetic circuit. It is a basic fact of algebra that a nonzero univariate polynomial of degree r can vanish on at most r points. This implies that for checking whether f is identically zero, it suffices to query f on an arbitrary test set of r + 1 points. Could this brute-force method be improved upon by a single point? We develop a framework where such a marginal improvement implies that Permanent does not have polynomial size arithmetic circuits. More formally, we formulate the following hypothesis for any field of characteristic zero: There is a fixed depth d and some function s(n) = O(n), such that for arbitrarily small ε 0, there exists a hitting set Hn ⊂ Z of size at most 2s(nε) against univariate polynomials of degree at most 2s(nε) computable by size n constant-free1 arithmetic circuits, where Hn can be encoded by uniform TC0 circuits of size 2O(nε) and depth d. We prove that the hypothesis implies that Permanent does not have polynomial size constant-free arithmetic circuits. Our hypothesis provides a unifying perspective on several important complexity theoretic conjectures, as it follows from these conjectures for different degree ranges as determined by the function s(n). We will show that it follows for s(n) = n from the widely-believed assumption that poly size Boolean circuits cannot compute the Permanent of a 0,1-matrix over Z. The hypothesis can also be easily derived from the Shub-Smale τ-conjecture [21], for any s(n) with s(n) = ω(log n) and s(n) = O(n). This implies our result strengthens a theorem by Bürgisser [4], who derives the same lower bound from the τ-conjecture. For s(n) = 0, the hypothesis follows from the statement that (n!) is ultimately hard, a statement that is known to imply P ≠ NP over C [21]. We apply our randomness-to-hardness theorem to prove the following unconditional result for Permanent: either Permanent does not have uniform constant-depth threshold circuits of sub-exponential size, or Permanent does not have polynomial-size constant-free arithmetic circuits. Turning to the Boolean world, we give a simplified proof of the following strengthening of Allender's lower bound [2] for the (0,1)-Permanent: either the (0,1)-Permanent is not simultaneously in polynomial time and sub-polynomial space, or logarithmic space does not have uniform constant-depth threshold circuits of polynomial size.