Trading group theory for randomness

  title={Trading group theory for randomness},
  author={L{\'a}szl{\'o} Babai},
  booktitle={STOC '85},
  • L. Babai
  • Published in STOC '85 1 December 1985
  • Mathematics
In a previous paper [BS] we proved, using the elements of the theory of <italic>nilpotent groups</italic>, that some of the <italic>fundamental computational problems in matriz groups</italic> belong to <italic>NP</italic>. These problems were also shown to belong to <italic>coNP</italic>, assuming an <italic>unproven hypothesis</italic> concerning <italic>finite simple groups</italic>. The aim of this paper is to replace most of the (proven and unproven) group theory of [BS] by elementary… 
Deciding finiteness of matrix groups in Las Vegas polynomial time
It is shown that one can decide whether or not G G is finite, in Las Vegas polynomial time, and structural properties such as solvability andnilpotence are decidable in Monte Carlo polynometric time.
Probabilistic checking of proofs: a new characterization of NP
It is shown that approximating Clique and Independent Set, even in a very weak sense, is NP-hard, and the class NP contains exactly those languages for which membership proofs can be verified probabilistically in polynomial time.
Separating and collapsing results on the relativized probabilistic polynomial-time hierarchy
The probabilistic polynomial-time hierarchy (BPH) is the hierarchy generated by applying the BP-operator to the Meyer-Stockmeyer polynomial-time hierarchy (PH), where the BP-operator is the natural
Essentially optimal interactive certificates in linear algebra
All the authors' certificates are based on interactive verification protocols with the interaction removed by a Fiat-Shamir identification heuristic, and the validity of the verification procedure is subject to standard computational hardness assumptions from cryptography.
Polynomial-time theory of matrix groups
The order of the largest semisimple quotient can be determined in randomized polynomial time (no number theory oracles required and no restriction on parity), and a natural problem is obtained that belongs to BPP and is not known to belong either to RP or to coRP.
Unknotting is in AM ∩ co-AM
Hass, Lagarias, and Pippenger analyzed the computational complexity of various decision problems in knot theory. They proved that the problem whether a given knot is unknotting is in <b>NP</b>, and
Some facets of complexity theory and cryptography: A five-lecture tutorial
  • J. Rothe
  • Computer Science, Mathematics
  • 2002
This tutorial discusses the notion of one-way functions both in a cryptographic and in a complexity-theoretic setting, and considers interactive proof systems and some interesting zero-knowledge protocols.
New (and Old) Proof Systems for Lattice Problems
It is shown that \(\textsf {GapSPP}\) admits SZK proofs for remarkably low approximation factors, improving on prior work by up to roughly \(\sqrt{n}\).
An Efficient Quantum Algorithm for Some Instances of the Group Isomorphism Problem
This paper presents a quantum algorithm solving instances of the nonabelian group isomorphism problem exponentially faster than the best known classical algorithms.
On the communication complexity of zero-knowledge proofs
This paper studies the concrete complexity of the known general methods for constructing zero-knowledge proofs, and establishes that circuit-based methods, which can be applied in either the GMR or the BCC model, have the potential of producing proofs which could be used in practice.


On the Complexity of Matrix Group Problems I
A theory of black box groups is built, and it is proved that for such subgroups, membership and divisor of the order are in NPB, and under a plausible mathematical hypothesis on short presentations of finite simple groups, nom membership and exaact order will also be inNPB.
Two theorems on random polynomial time
  • L. Adleman
  • Computer Science
    19th Annual Symposium on Foundations of Computer Science (sfcs 1978)
  • 1978
Where the traditional method of polynomial reduction has been inapplicable, randomness has been used in demonstrating intractibility by Adleman and Manders, and in showing problems equivalent by Rabin, a new examination of randomness is in order.
Riemann's Hypothesis and tests for primality
  • G. Miller
  • Computer Science, Mathematics
  • 1975
It is shown that primality is testable in time a polynomial in the length of the binary representation of a number, and a partial solution is given to the relationship between the complexity of computing the prime factorization of a numbers, computing the Euler phi function, and computing other related functions.
The knowledge complexity of interactive proof-systems
A computational complexity theory of the “knowledge” contained in a proof is developed and examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity.
Polynomial-time algorithms for permutation groups
It is demonstrated that the normal closure of a subgroup can be computed in polynomial time, and that this proceaure can be used to test a group for solvability.
Relative to a Random Oracle A, PA != NPA != co-NPA with Probability 1
Let A be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to A, and${\bf NP}^A is shown, with probability 1, to contain a-immune set, i.e., a set having no infinite subset in ${\bf P]^A $.
Universal Classes of Hash Functions
A Fast Monte-Carlo Test for Primality
A uniform distribution a from a uniform distribution on the set 1, 2, 3, 4, 5 is a random number and if a and n are relatively prime, compute the residue varepsilon.