Essentially optimal interactive certificates in linear algebra

@inproceedings{Dumas2014EssentiallyOI,
  title={Essentially optimal interactive certificates in linear algebra},
  author={J. Dumas and E. Kaltofen},
  booktitle={ISSAC},
  year={2014}
}
Certificates to a linear algebra computation are additional data structures for each output, which can be used by a---possibly randomized---verification algorithm that proves the correctness of each output. The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size <i>N</i>, meaning <i>N</i> times a factor <i>N</i><sup><i>o</i>(1)</sup>, i.e., a factor <i>N</i><sup><i>η</i>(<i>N</i>)</sup> with lim<sub><i>N</i> → ∞</sub… Expand
Polynomial Time Interactive Proofs for Linear Algebra with Exponential Matrix Dimensions and Scalars Given by Polynomial Time Circuits
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 fieldExpand
Linear Time Interactive Certificates for the Minimal Polynomial and the Determinant of a Sparse Matrix
TLDR
An algorithm is given that computes a certificate for the minimal polynomial of sparse or structured matrices over an abstract field, of sufficiently large cardinality, whose Monte Carlo verification complexity requires a single matrix-vector multiplication and a linear number of extra field operations. Expand
Interactive Certificates for Polynomial Matrices with Sub-Linear Communication
TLDR
The main tools used are reductions to existing linear algebra certificates and a new protocol to verify that a given vector is in the F[x]-linear span of a given matrix. Expand
Interactive certificate for the verification of Wiedemann's Krylov sequence: application to the certification of the determinant, the minimal and the characteristic polynomials of sparse matrices
TLDR
Algorithms that compute certificates for the Krylov sequence of sparse or structured n × n matrices over an abstract field, whose Monte Carlo verification complexity can be made essentially linear, are given. Expand
Certification of Minimal Approximant Bases
TLDR
A certificate is proposed which, for typical instances of the problem, is computed by the prover using O(mømega D/m) additional field operations and allows verification of the approximant basis by a Monte Carlo algorithm with cost bound O( mømega + m D). Expand
Verification protocols with sub-linear communication for polynomial matrix operations
TLDR
New protocols to verify the correctness of various computations on matrices over the ring F[x] of univariate polynomials over a field F are designed and analyzed to minimize the communication cost. Expand
Elimination-based certificates for triangular equivalence and rank profiles
TLDR
Novel certificates for triangular equivalence and rank profiles are given, enabling somebody to verify the row or column rank profiles or the whole rank profile matrix faster than recomputing them, with a negligible overall overhead. Expand
Certificates for Triangular Equivalence and Rank Profiles
TLDR
Novel certificates for triangular equivalence and rank profiles and an interactive protocol, certifying the determinant of dense matrices, faster than the best previously known one are given. Expand
Efficient algorithms and implementation in exact linear algebra. (Algorithmes et implantations efficaces en algèbre linéaire exacte)
TLDR
This paper provides an algorithm that reduces the complexity to matrix multiplication and that allows to outperform in practice the best asymptotic algorithms on a wide range of values and designs a novel approach that aims to minimize the number of synchronizations inherent to fine grained parallel computing. Expand
Selecting Algorithms for Black Box Matrices: Checking For Matrix Properties That Can Simplify Computations
Processes to automate the selection of appropriate algorithms for various matrix computations are described. In particular, processes to check for, and certify, various matrix properties of black-boxExpand
...
1
2
...

References

SHOWING 1-10 OF 28 REFERENCES
Quadratic-time certificates in linear algebra
We present certificates for the positive semidefiniteness of an <i>n</i> by <i>n</i> matrix <i>A</i>, whose entries are integers of binary length log ||<i>A</i>||, that can be verified inExpand
Integer matrix rank certification
TLDR
A Las Vegas algorithm for computing the rank of an 2-n x 2-m integer matrix, with an expected number of log ||<i>nmr</i><sup>ω--2</sup> log ||</i>|| bit operations, is presented. Expand
Every Prime has a Succinct Certificate
  • V. Pratt
  • Mathematics, Computer Science
  • SIAM J. Comput.
  • 1975
TLDR
It remains an open problem whether a prime n can be recognized in only $\log _2^\alpha n$ operations of a Turing machine for any fixed $\alpha $. Expand
Delegating Computation
TLDR
The focus is on the original interactive proof model where no assumptions are made on the computational power or adaptiveness of dishonest provers, and an open question regarding the expressive power of proof systems with such verifiers is settled. Expand
Trading group theory for randomness
  • L. Babai
  • Mathematics, Computer Science
  • STOC '85
  • 1985
TLDR
The aim of this paper is to replace most of the (proven and unproven) group theory of [BS] by elementary combinatorial arguments and defines a new hierarchy of complexity classes “just above <italic>NP</italic””, introducing Arthur vs. Merlin games and proving that it consists precisely of those languages which belong to NP. Expand
Time-Optimal Interactive Proofs for Circuit Evaluation
TLDR
A refinement of a powerful interactive proof protocol originally due to Goldwasser, Kalai, and Rothblum is described, which can more efficiently verify complicated computations as long as that computation is applied independently to many pieces of data. Expand
Random oracles are practical: a paradigm for designing efficient protocols
TLDR
It is argued that the random oracles model—where all parties have access to a public random oracle—provides a bridge between cryptographic theory and cryptographic practice, and yields protocols much more efficient than standard ones while retaining many of the advantages of provable security. Expand
Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients
We present a hybrid symbolic-numeric algorithm for certifying a polynomial or rational function with rational coefficients to be non-negative for all real values of the variables by computing aExpand
Stronger Security Proofs for RSA and Rabin Bits
TLDR
A more efficient algorithm for inverting the RSA/ Rabin function with the help of an oracle that predicts the least-significant bit of x yields provable security guarantees for RSA message bits and for the RSA random number generator for modules N of practical size. Expand
Certifying inconsistency of sparse linear systems
TLDR
An e cient algorithm is given to compute a certi cate of inconsistency for a black box linear system over a eld to certifying that a sparse Diophantine linear system of integer equations has no integer solutions, even when it may have rational solutions. Expand
...
1
2
3
...