## The Isomorphism Conjecture Fails Relative to a Random Oracle (1996)

Venue: | J. ACM |

Citations: | 41 - 4 self |

### BibTeX

@INPROCEEDINGS{Kurtz96theisomorphism,

author = {Stuart A. Kurtz and Stephen R. Mahaney and James S. Royer},

title = {The Isomorphism Conjecture Fails Relative to a Random Oracle},

booktitle = {J. ACM},

year = {1996},

pages = {157--166}

}

### Years of Citing Articles

### OpenURL

### Abstract

Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NP-complete sets---the k-creative sets---and defined a class of sets (the K k f 's) that are necessarily k-creative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NP-complete sets. Clearly, the Berman--Hartmanis and Joseph--Young conjectures cannot both be correct. We introduce a family of strong one-way functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scramb...

### Citations

875 | Theory of Recursive Functions and Effective Computability - Rogers - 1967 |

309 | Algebraic Methods for Interactive Proof Systems
- Lund, Fortnow, et al.
- 1990
(Show Context)
Citation Context ...at coNP ` P SAT , but coNP R 6` P SAT;R for random R. More recently, Chor, Goldreich, and Hastad [CGH90] showed that coNP R 6` IP R for random R, which together with the Fortnow, Karloff, Lund, Nisan =-=[FKLN92]-=- proof that PH ` IP gives another counterexample. Both counterexamples have a common flavor. Imagine that there are exponentially many boxes, one of which contains a prize. If the prize is placed at r... |

159 |
On isomorphism and density of NP and other complete sets
- Berman, Hartmanis
- 1977
(Show Context)
Citation Context ...Fails Relative to a Random Oracle Stuart A. Kurtz University of Chicago Stephen R. Mahaney y University of Arizona James S. Royer z Syracuse University December 16, 1996 Abstract Berman and Hartmanis =-=[BH77]-=- conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young [JY... |

145 | Complexity Measures for Public-Key Cryptosystems - Grollman, Selman - 1984 |

101 |
Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis
- Mahaney
- 1982
(Show Context)
Citation Context ... on structural predictions of the conjecture. For example, the isomorphism conjecture predicts that there are no sparse NP-complete languages. Mahaney, building on work of P. Berman and Fortune, (cf. =-=[Mah82]) verified-=- this prediction under the hypothesis P 6= NP. Another direction pursued in the years immediately following the conjecture was to "relocate" it to other natural degrees. In their original ar... |

54 |
Relative to a random oracle A, P A 6=NP A 6= co-NP A with probability 1
- Bennett, Gill
- 1981
(Show Context)
Citation Context .... Because of this, the existence of an oracle A relative to which T A holds can only be taken as weak evidence for (unrelativized) T . In 1981, Bennett and Gill made the following dramatic conjecture =-=[BG81]-=-. The Random Oracle Conjecture If the set of oracles A relative to which a complexity theoretic statement T A holds has measure one, i.e., if T holds relative to a random oracle, then the unrelativize... |

46 |
Polynomial Reducibilities and Complete Sets
- Berman
- 1977
(Show Context)
Citation Context ...inal article, Berman and Hartmanis conjectured not only that the NP-complete degree collapsed, but also that the PSPACE-complete degree collapsed. While they could not prove their conjectures, Berman =-=[Ber77]-=- was able to obtain a number of important partial results, e.g., that the complete m-degree for EXP consists of a single 1-li degree. A re-examination of the isomorphism conjecture began with Joseph a... |

32 | Limits on the Provable Consequences of One-Way Functions
- Rudich
- 1989
(Show Context)
Citation Context ...of length 3n + 1 for infinitely many n, then T R cannot accept range(�� R ) relative to a random oracle. The corollary follows immediately. 2 Independently of and simultaneously with our work, Rud=-=ich [Rud88]-=- proved that, relative to a random oracle, there is a one way function f such that no BPP-machine can invert f on a nonsparse set. His proof is essentially the proof of Lemma 4.7, together with the ob... |

30 |
Oracles for structural properties: The isomorphism problem and public-key cryptography
- Homer, Selman
- 1992
(Show Context)
Citation Context ... a collapsing m-degree in NP. As sparse oracles seem less likely to distort structural relationships, we take this as evidence for the proposition that some m-degree in NP collapses. Homer and Selman =-=[HS88]-=- achieved the first breakthrough, producing an oracle relative to which the m-complete degree for \Sigma p 2 collapses, as well as an oracle relative to which it fails to collapse. By their efforts, t... |

29 |
Some remarks on witness functions for non- polynomial and non-complete sets
- Joseph, Young
- 1985
(Show Context)
Citation Context ...77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young =-=[JY85]-=- gave a structural definition of a class of NP-complete sets---the k-creative sets---and defined a class of sets (the K k f 's) that are necessarily k-creative. They went on to conjecture that certain... |

26 | The isomorphism conjecture holds relative to an oracle
- Fenner, Fortnow, et al.
- 1996
(Show Context)
Citation Context ...ils to collapse. By their efforts, the complete m-degree for \Sigma p 2 became the first natural degree to have both collapsing and noncollapsing relativizations. Recently, Fenner, Fortnow, and Kurtz =-=[FFK92]-=- demonstrated the existence of an oracle relative to which the isomorphism conjecture is true. Although the construction of these two oracles were substantial technical advances, neither is so natural... |

25 | The random oracle hypothesis is false
- Chor, Goldreich, et al.
- 1990
(Show Context)
Citation Context ...dom set. Since 1981, the random oracle conjecture has been refuted twice. Kurtz [Kur83a] pointed out that coNP ` P SAT , but coNP R 6` P SAT;R for random R. More recently, Chor, Goldreich, and Hastad =-=[CGH90]-=- showed that coNP R 6` IP R for random R, which together with the Fortnow, Karloff, Lund, Nisan [FKLN92] proof that PH ` IP gives another counterexample. Both counterexamples have a common flavor. Ima... |

22 |
On the random oracle hypothesis
- Kurtz
- 1982
(Show Context)
Citation Context ...Conjecture is true, or there are essential structural properties of random sets that cannot be captured by any pseudorandom set. Since 1981, the random oracle conjecture has been refuted twice. Kurtz =-=[Kur83a]-=- pointed out that coNP ` P SAT , but coNP R 6` P SAT;R for random R. More recently, Chor, Goldreich, and Hastad [CGH90] showed that coNP R 6` IP R for random R, which together with the Fortnow, Karlof... |

17 |
Juris Hartmanis: Fundamental Contributions to Isomorphism Problems
- Young
- 1990
(Show Context)
Citation Context ...ies and continue their reading with Sections 3 and 4. 1.1 A Brief Survey In this section, we briefly survey the research that led to this work. The reader may wish to consult Young's excellent survey =-=[You90]-=- of structural research on isomorphisms, as well as the surveys by Mahaney [Mah86] and and Kurtz, Mahaney, and Royer [KMR90]. 1.1.1 The Structural Approach In [BH77] Berman and Hartmanis made the foll... |

16 |
One-way functions and the nonisomorphism of NP-complete sets
- Hartmanis, Hemachandra
- 1991
(Show Context)
Citation Context ...njecture relative to Kurtz's oracle is different from that predicted by Joseph and Young, as it is obtained by splitting the m-complete degree for NP into several 1-degrees. Hartmanis and Hemachandra =-=[HH91]-=-, by combining Kurtz's construction with Rackoff's [Rac82] construction of an oracle relative to which P = UP 6= NP, construct an oracle relative to which both conjectures fail. Thus, while the Berman... |

16 |
A note on one-way functions and polynomialtime isomorphisms
- Ko, Long, et al.
- 1987
(Show Context)
Citation Context ...tion. Watanabe [Wat85] conjectured that the existence of a oneway function is an adequate hypothesis for the construction of a noncollapsing 1-li degree, and he was proven correct by Ko, Long, and Du =-=[KLD86]-=-. This validation of a prediction of the Joseph-Young conjecture is an important piece of evidence in its favor. Somewhat later, we [KMR88] showed that there are nontrivial collapsing mdegrees, provid... |

16 |
Relativized questions involving probabilistic algorithms
- Rackoff
- 1982
(Show Context)
Citation Context ...t predicted by Joseph and Young, as it is obtained by splitting the m-complete degree for NP into several 1-degrees. Hartmanis and Hemachandra [HH91], by combining Kurtz's construction with Rackoff's =-=[Rac82]-=- construction of an oracle relative to which P = UP 6= NP, construct an oracle relative to which both conjectures fail. Thus, while the Berman-Hartmanis and the Joseph-Young conjectures cannot both be... |

12 |
Collapsing degrees
- Kurtz, Mahaney, et al.
- 1988
(Show Context)
Citation Context ...g 1-li degree, and he was proven correct by Ko, Long, and Du [KLD86]. This validation of a prediction of the Joseph-Young conjecture is an important piece of evidence in its favor. Somewhat later, we =-=[KMR88]-=- showed that there are nontrivial collapsing mdegrees, providing analogous evidence in favor of the Berman-Hartmanis conjecture. (See [KMR90] for a more complete discussion of this.) 1.1.2 Relativizat... |

10 |
A relativized failure of the Berman-Hartmanis conjecture
- Kurtz
- 1983
(Show Context)
Citation Context ...uffices. In contrast, and in spite of widely perceived similarities between NP and EXP, progress has not been made in obtaining an oracle relative to which the complete degree for NP collapses. Kurtz =-=[Kur83b]-=- provided the first example of an oracle relative to which P 6= NP and yet the isomorphism conjecture fails. The failure of the isomorphism conjecture relative to Kurtz's oracle is different from that... |

9 |
Reductions among polynomial isomorphism types
- Mahaney, Young
- 1985
(Show Context)
Citation Context .... 2.3 Padding and Pairing A padding function hh\Delta; \Deltaii is a polynomial-time computable, one-one function from pairs of strings to strings that is polynomial-time invertible in both arguments =-=[MY85]-=-. A language A is paddable [BH77] if and only if for all x and y, x 2 A () hhx; yii 2 A: If a padding function hh\Delta; \Deltaii is also onto, then we say hh \Delta; \Deltaii is a polynomialtime pair... |

5 | Every polynomial-time 1-degree collapses iff P - Fenner, Kurtz, et al. - 1989 |

5 |
A survey of one way functions in complexity theory, Mathematical Systems Theory 25
- Selman
- 1992
(Show Context)
Citation Context ...es the failure. The K k f languages are not merely m-complete for NP, they are 1-li complete. Thus, the Joseph-Young conjecture predicts that the 1-li complete degree for NP fails to collapse. Selman =-=[Sel92]-=- observed that the the Joseph-Young conjecture entails the following, simpler, conjecture: The Encrypted Complete Set Conjecture There exists a one-way functionsf such that f(SAT) is not is isomorphic... |

3 |
The structure of complete degrees, Complexity Theory Retrospective
- Kurtz, Mahaney, et al.
- 1990
(Show Context)
Citation Context ...t led to this work. The reader may wish to consult Young's excellent survey [You90] of structural research on isomorphisms, as well as the surveys by Mahaney [Mah86] and and Kurtz, Mahaney, and Royer =-=[KMR90]-=-. 1.1.1 The Structural Approach In [BH77] Berman and Hartmanis made the following conjecture: The Isomorphism Conjecture All NP-complete languages are polynomialtime isomorphic to one another. That is... |

3 |
On one-one polynomial time equivalence relations. Theoretical Computer Science 38:157--165
- Watanabe
- 1985
(Show Context)
Citation Context ...e polynomialtime isomorphic, and thus every 1-li degree collapses. Therefore, a minimal hypothesis for the construction of a noncollapsing 1-li degree is the existence of a one-way function. Watanabe =-=[Wat85]-=- conjectured that the existence of a oneway function is an adequate hypothesis for the construction of a noncollapsing 1-li degree, and he was proven correct by Ko, Long, and Du [KLD86]. This validati... |

2 |
On some natural complete operators. Theoretical Computer Science, 37(1):1–30
- Ko
- 1985
(Show Context)
Citation Context ...t polynomial-time invertible. Our definition of one-way function requires totality, which is not the case in all presentations, e.g., [GS88]. Berman [Ber77], Grollmann and Selman [GS84, GS88], and Ko =-=[Ko85]-=- show that the existence of one-way functions is equivalent to P 6= UP. Ko, Long, and Du [KLD86] show by a simple padding construction that if one-way functions exist, then length-increasing one-way f... |

1 |
Progress on collapsing degrees (extended abstract
- Kurtz, Mahaney, et al.
- 1987
(Show Context)
Citation Context ...Joseph-Young conjecture holds; and we expected that the techniques of [KMR88] could be used to construct an oracle relative to which the Berman-Hartmanis conjecture holds. We achieved limited success =-=[KMR87]-=- by constructing a sparse oracle relative to which there is a collapsing m-degree in NP. As sparse oracles seem less likely to distort structural relationships, we take this as evidence for the propos... |

1 |
Sparse sets and reducibilities, Studies in Complexity Theory
- Mahaney
- 1986
(Show Context)
Citation Context ...ection, we briefly survey the research that led to this work. The reader may wish to consult Young's excellent survey [You90] of structural research on isomorphisms, as well as the surveys by Mahaney =-=[Mah86]-=- and and Kurtz, Mahaney, and Royer [KMR90]. 1.1.1 The Structural Approach In [BH77] Berman and Hartmanis made the following conjecture: The Isomorphism Conjecture All NP-complete languages are polynom... |

1 | On the p-isomorphism conjecture, Theoretical Computer Science 83 - Watanabe - 1991 |