Polynomial Time Interactive Proofs for Linear Algebra with Exponential Matrix Dimensions and Scalars Given by Polynomial Time Circuits

Abstract : We present an interactive probabilistic proof protocol that certifies in (log N)O(1) arithmetic and Boolean operations for the verifier the determinant, for example, of an N x N matrix over a field whose entries a(i,j) are given by a single (log NO(1)-depth arithmetic circuit, which contains (log NO(1) field constants and which is polynomial time uniform, for example, which has size (log NO(1). The prover can produce the interactive certificate within a (log NO(1) factor of the cost of computing the determinant. Our protocol is a version of the proofs for muggles protocol by Goldwasser, Kalai and Rothblum [STOC 2008, J. ACM 2015]. An application is the following: suppose in a system of k homogeneous polynomials of total degree ≤ d in the k variables y1,...,yk the coefficient of the term y1e1 ... ykek in the i-th polynomial is the (hypergeometric) value ((i+e1 + ... + ek)!)/((i!)(e1!)...(ek!)), where e! is the factorial of e. Then we have a probabilistic protocol that certifies (projective) solvability or inconsistency of such a system in (k log(d))O(1) bit complexity for the verifier, that is, in polynomial time in the number of variables k and the logarithm of the total degree, log(d).
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-01657873
Contributor : Gilles Villard <>
Submitted on : Thursday, December 7, 2017 - 10:58:56 AM
Last modification on : Wednesday, August 21, 2019 - 10:38:02 AM

Identifiers

Citation

Jean-Guillaume Dumas, Erich Kaltofen, Gilles Villard, Lihong Zhi. Polynomial Time Interactive Proofs for Linear Algebra with Exponential Matrix Dimensions and Scalars Given by Polynomial Time Circuits. ISSAC 2017 - 42nd International Symposium on Symbolic and Algebraic Computation, Jul 2017, Kaiserslautern, Germany. pp.125-132, ⟨10.1145/3087604.3087640⟩. ⟨hal-01657873⟩

Share

Metrics

Record views

1220