# Algebraic methods for interactive proof systems

@article{Lund1990AlgebraicMF, title={Algebraic methods for interactive proof systems}, author={Carsten Lund and Lance Fortnow and Howard J. Karloff and Noam Nisan}, journal={Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science}, year={1990}, pages={2-10 vol.1} }

An algebraic technique for the construction of interactive proof systems is proposed. The technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system. For the proof, a method is developed for reducing the problem of verifying the value of a low-degree polynomial at two points to verifying the value at one new point. The results have implications for program checking, verification, and self-correction.<<ETX>>

## 837 Citations

### The Complexity of Space Boundes Interactive Proof Systems

- Computer Science, MathematicsComplexity Theory: Current Research
- 1992

An early motivation for the study of interactive proof systems was to extend the notion of NP as the class of problems with efficient “proofs of membership”. Informally, a prover can convince a…

### Designing programs that check their work

- Computer ScienceSTOC '89
- 1989

This paper defines the concept of a program checker, designs program checkers for a few specific and carefully chosen problems in the class P of problems solvable in polynomial time and characterizes the problems that can be checked.

### On the use of interactive proofs for formal programveri

- Computer Science
- 1997

This work studies the feasibility of using interactive proofs in formal program veriication, and proposes approaches to minimize the overhead involved, and demonstrates the ideas on two nite state mutual exclusion protocols.

### On Doubly-Efficient Interactive Proof Systems

- Mathematics, Computer ScienceFound. Trends Theor. Comput. Sci.
- 2018

An interactive proof system is called doubly-efficient if the prescribed prover strategy can be implemented in polynomial-time and the verifier???s strategy can be implemented in almost-linear time.…

### Interactive Proofs with Space Bounded Provers

- Mathematics, Computer ScienceCRYPTO
- 1991

This result shows that one can easily transform checkers into adaptive checkers under the assumption that the program being checked has space bounded by a fixed polynomial, and shows that any multiple prover interactive protocol in which the provers need only a poynomial amount of space can be easily transformed.

### Introduction to Interactive Proofs & The Sumcheck Protocol

- Computer Science
- 2017

Traditional mathematical proofs are static objects: a prover P writes down a sequence of mathematical statements, and then at some later time a verifier V checks that these statements are consistent…

### Interactive proof systems with polynomially bounded strategies

- Mathematics[1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference
- 1992

A main result of the paper is that interactive proof systems in which the Prover is restricted to a polynomial size strategy are equivalent to MA, Merlin-Arthur games, defined by Babai and Moran.

### Probabilistic Proof Systems

- Computer Science, Mathematics
- 1994

This exposition is an expanded version of a survey written for the proceedings of the International Congress of Mathematicians held in Zurich in 1994, and it is hope that this exposition may be accessible to a broad audience of computer scientists and mathematians.

### On the Power of Multi-Prover Interactive Protocols

- Computer Science, MathematicsTheor. Comput. Sci.
- 1994

### A taxonomy of proof systems

- Computer Science
- 1998

The main objective of this essay is to try to clarify the notions of NP interactive proof systems and probabilistically checkable proofs PCP which are closely related to multi prover interactive proofs MIP.

## References

SHOWING 1-10 OF 44 REFERENCES

### Designing programs that check their work

- Computer ScienceSTOC '89
- 1989

This paper defines the concept of a program checker, designs program checkers for a few specific and carefully chosen problems in the class P of problems solvable in polynomial time and characterizes the problems that can be checked.

### On Completeness and Soundness in Interactive Proof Systems

- Computer ScienceAdv. Comput. Res.
- 1989

It is shown that any language having an interactive proof system has one (of the Arthur-Merlin type) with perfect completeness, and only languages in NP have interactive proofs with perfect soundness.

### Interactive proof systems: Provers that never fail and random selection

- Computer Science, Mathematics28th Annual Symposium on Foundations of Computer Science (sfcs 1987)
- 1987

It is shown that any language having a bounded interactive proof has one with perfect completeness and only languages in NP have interactive proofs with perfect soundness, and a new protocol for proving approximately lower bounds and "random selection" is presented.

### PSPACE is provable by two provers in one round

- Computer Science, Mathematics[1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference
- 1991

It is shown that every language in PSPACE, or equivalently every language accepted by an unbounded round interactive proof system, has a one-round, two-prover interactive proof with exponentially…

### The knowledge complexity of interactive proof-systems

- Mathematics, Computer ScienceSTOC '85
- 1985

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.

### The complexity of theorem-proving procedures

- Mathematics, Computer ScienceSTOC
- 1971

It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a…

### Interactive Proof Systems and Alternating Time-Space Complexity

- Computer ScienceTheor. Comput. Sci.
- 1991

### On the power of multi-power interactive protocols

- Computer Science, Mathematics[1988] Proceedings. Structure in Complexity Theory Third Annual Conference
- 1988

A simple characterization of the power of the multipower model in terms of probabilistic oracle Turing machines is given. It is shown that languages that have single-prover interactive protocols also…

### Multi-prover interactive proofs: how to remove intractability assumptions

- Computer Science, MathematicsSTOC '88
- 1988

It is proved that all NP languages have perfect zero-knowledge proof-systems in this model, without making any intractability assumptions, and its properties and applicability to cryptography are examined.

### Power of interaction

- Computer Science, MathematicsACM Distinguished dissertations
- 1992

A comparison with the Blum-Luby-Rubinfield model uniform vs. nonuniform complexity recent developments further research.