One-Way Functions and the Nonisomorphism of NP-Complete Sets

Abstract The One-way Conjecture states that the existence of nonisomorphic NP-complete sets implies the existence of one-way functions. This paper gives a relativized counterexample to the conjecture by constructing a relativized world that has nonisomorphic NP-complete sets, but lacks one-way functions.

[1]  Juris Hartmanis,et al.  On Isomorphisms and Density of NP and Other Complete Sets , 1977, SIAM J. Comput..

[2]  Juris Hartmanis,et al.  Generalized Kolmogorov complexity and the structure of feasible computations , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[3]  Juris Hartmanis,et al.  Complexity Classes without Machines: On Complete Languages for UP , 1988, Theor. Comput. Sci..

[4]  Stuart A. Kurtz.  A relativized failure of the Berman-Hartmanis conjecture , 1983 .

[5]  Steven Homer,et al.  Oracles for structural properties: the isomorphism problem and public-key cryptography , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

[6]  L. Berman.  Polynomial reducibilities and complete sets. , 1977 .

[7]  Steven Homer,et al.  Complete Problems and Strong Polynomial Reducibilities , 1989, STACS.

[8]  Osamu Watanabe.  On One-One Polynomial Time Equivalence Relations , 1985, Theor. Comput. Sci..

[9]  Leslie G. Valiant,et al.  Relative Complexity of Checking and Evaluating , 1976, Inf. Process. Lett..

[10]  Paul Young,et al.  Reductions Among Polynomial Isomorphism Types , 1985, Theor. Comput. Sci..

[11]  Paul Young,et al.  Some structural properties of polynomial reducibilities and sets in NP , 1983, STOC.

[12]  Stuart A. Kurtz,et al.  The Structure of Complete Degrees , 1990 .

[13]  R. Solovay,et al.  Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question , 1975 .

[14]  Deborah Joseph,et al.  Some Remarks on Witness Functions for Nonpolynomial and Noncomplete Sets in NP , 1985, Theor. Comput. Sci..

[15]  Stuart A. Kurtz,et al.  The isomorphic conjecture fails relative to a random oracle , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

[16]  Stuart A. Kurtz,et al.  The ismorphism conjecture fails relative to a random oracle , 1989, STOC '89.

[17]  Stuart A. Kurtz,et al.  Progress on collapsing degrees , 1987, Computational Complexity Conference.

[18]  Paul Young,et al.  Juris Hartmanis: fundamental contributions to isomorphism problems , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[19]  Stuart A. Kurtz,et al.  Collapsing degrees , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[20]  Ding-Zhu Du,et al.  A note on one-way functions and polynomial-time isomorphisms , 1986, STOC '86.

[21]  Juris Hartmanis,et al.  One-Way Functions, Robustness, and the Non-Isomorphism of NP-Complete Sets , 1987, Proceeding Structure in Complexity Theory.

[22]  Judy Goldsmith,et al.  Three results on the polynomial isomorphism of complete sets , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[23]  Alan L. Selman,et al.  Complexity Measures for Public-Key Cryptosystems , 1988, SIAM J. Comput..

[24]  Eric Allender.  Isomorphisms and 1-L Reductions , 1988, J. Comput. Syst. Sci..