Fast Probabilistic Algorithms for Verification of Polynomial Identities

@article{Schwartz1980FastPA,
  title={Fast Probabilistic Algorithms for Verification of Polynomial Identities},
  author={J. Schwartz},
  journal={Journal of the ACM (JACM)},
  year={1980},
  volume={27},
  pages={701 - 717}
}
  • J. Schwartz
  • Published 1980
  • Computer Science
  • Journal of the ACM (JACM)
The s tar thng success o f the Rabm-S t ra s sen -So lovay p n m a h t y algori thm, together wi th the intr iguing foundat tonal posstbthty that axtoms of randomness may constttute a useful fundamenta l source o f m a t h e m a u c a l truth independent of the standard axmmaUc structure of mathemaUcs, suggests a wgorous search for probabdisuc algonthms In dlustratmn of this observaUon, vanous fast probabdlsttc algonthms, with probability of correctness guaranteed a prion, are presented for… Expand
Non-deterministic exponential time has two-prover interactive protocols
TLDR
It is shown that the class of languages having tow-prover interactive proof systems is nondeterministic exponential time and that to prove membership in languages inEXP, the honest provers need the power ofEXP only. Expand
Non-deterministic exponential time has two-prover interactive protocols
TLDR
It is shown that the class of languages having tow-prover interactive proof systems is nondeterministic exponential time and that to prove membership in languages inEXP, the honest provers need the power ofEXP only. Expand
Proof verification and the hardness of approximation problems
TLDR
It is proved that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP = P, and there exists a positive ε such that approximating the maximum clique size in an N-vertex graph to within a factor of Nε is NP-hard. Expand
Efficient Checking of Polynomials and Proofs and the Hardness of Appoximation Problems
  • M. Sudan
  • Mathematics, Computer Science
  • Lecture Notes in Computer Science
  • 1995
TLDR
Results from coding theory are used as a starting point and several algorithmic techniques including pairwise independent sampling to give efficient randomized algorithms for error-detection and error-correction for some well-known codes. Expand
Probabilistic Search Algorithms with Unique Answers and Their Cryptographic Applications
TLDR
A new type of probabilistic search algorithm, which is guaranteed to run in expected polynomial time, and to produce a correct and unique solution with high probability is introduced, called the Bellagio algorithm. Expand
Reducing randomness via irrational numbers
TLDR
A general methodology for testing whether a given polynomial with integer coefficients is identically zero is proposed, which can decrease the error probability by increasing the precision of the approximations instead of using more random bits. 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
On the hardness of computing the permanent of random matrices
TLDR
It is shown that unless the polynomial-time hierarchy collapses to its second level, noPolynomial time algorithm can compute the permanent of every matrix with probability at least 13n3/p, nor can it compute the Permanent of at least a 49n^3 /\sqrt p -fraction of the matrices. Expand
Use of algebraically independent numbers in computation
TLDR
A Probabilistic zero recognition test for polynomials is got which is somewhat more expensive computationally than the usual probabilistic method of choosing random integers in a large interval and evaluating, but which depends on the ability to choose a random point in the unit cube and to approximate a polyno­ mial at that point. Expand
Fast computation of the Smith normal form of an integer matrix
TLDR
An even faster, more space efficient algorithm which requires an expected number of 0 and returns the correct result with probability at least 1 – c for a user specified tolerance e >0, and is of the Monte Carlo type. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 44 REFERENCES
A NEW DECISION METHOD FOR ELEMENTARY ALGEBRA
A. Tarski [4] has given a decision method for elementary algebra. In essence this comes to giving an algorithm for deciding whether a given finite set of polynomial inequalities has a solution. BelowExpand
The Calculation of Multivariate Polynomial Resultants
TLDR
An efficient algorithm is presented for the exact calculation of resultants of multivariate polynomials with integer coefficients over GF(p) using modular homomorphisms and the Chinese remainder theorem, and other algorithms are compared. Expand
Fast computation of GCDs
  • R. Moenck
  • Mathematics, Computer Science
  • STOC
  • 1973
TLDR
An integer greatest common divisor (GCD) algorithm due to Schönhage is generalized to hold in all euclidean domains which possess a fast multiplication algorithm and a new faster algorithm for multivariate polynomial GCD's can be derived. Expand
A Decision Method For Elementary Algebra And Geometry
By a decision method for a class K of sentence (or other expressions) is meant a method by means of which, given any sentence θ, one can always decide in a finite number of steps whether θ is in K;Expand
The Design and Analysis of Computer Algorithms
TLDR
This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs. Expand
A note on monte carlo primality tests and algorithmic information theory
Solovay and Strassen, and Miller and Rabin have discovered fast algorithms for testing primality which use coin-flipping and whose con
Proof, Completeness, Transcendentals, and Sampling
  • P. Davis
  • Mathematics, Computer Science
  • JACM
  • 1977
This paper considers informally the relationship between computer aided mathematical proof, formal algebraic languages, computation with transcendental numbers, and proof by sampling.
Die Frage der endlich vielen Schritte in der Theorie der Polynomideale
Die Ringbereiche, in denen die in der vorliegenden Arbeit auftretenden Ideale definiert sind, sollen Polynombereiche sein. Ein Ideal soll gegeben heisen, wenn eine Basis des Ideals bekannt ist, esExpand
A note on Monte Carlo pnmahty tests and algonthm , c reformation theory
  • Commun Pure Appl Math .
  • 1978
Probabdtsttc algorithms In Algorithms and Complexzty
  • New D~reclions and Recent Result. ~ J.F. Traub, FEd,
  • 1976
...
1
2
3
4
5
...