The Random Oracle Hypothesis is False

• Authors: R. Chang, B. Chor, O. Goldreich, J. Hartmanis, J. Hastad, D. Ranjan and P. Rohatgi.
• Cite as: Journal of Computer and System Sciences, 49(1):24-39, August 1994.
• Previous incarnations:
  • J. Hartmanis, R. Chang, D. Ranjan and P. Rohatgi. "Structural Complexity Theory: Recent Surprises." In Proceedings of the 2nd Scandinavian Workshop on Algorithm Theory, Springer-Verlag Lecture Notes in Computer Science #447, 1-12, July 1990.
  • J. Hartmanis, R. Chang, D. Ranjan and P. Rohatgi. "Structural Complexity Theory: Recent Surprises." Technical Report 90-1117, Department of Computer Science, Cornell University, April 1990.
  • B. Chor, O. Goldreich and J. Hastad. "The Random Oracle Hypothesis is False." Technical Report 631, Department of Computer Science, Technion, 1990.
• Most readable version: journal version.
• Status: complete.
• Online:
  • Article in Elsevier's Science Direct: DOI 10.1016/S0022-0000(05)80084-4.
  • Final journal submission available as a PDF file (with embedded Type 1 fonts): roh.pdf.

Abstract:

The Random Oracle Hypothesis, attributed to Bennett and Gill, essentially states that the relationships between complexity classes which hold for almost all relativized worlds must also hold in the unrelativized case. This paper shows that the Random Oracle Hypothesis is false by showing that for almost all oracles A, IP(A) is not equal to PSPACE(A). If the Random Oracle Hypothesis were true, it would contradict Shamir's result that IP = PSPACE. In fact, it is shown that for almost all oracles A, coNP(A) is not a subset of IP(A). These results extend to the multi-prover proof systems of Ben-Or, Goldwasser, Kilian and Wigderson. In addition, this paper shows that the Random Oracle Hypothesis is sensitive to small changes in the definition. A class IPP, which is similar to IP, is defined. Surprisingly, the IPP = PSPACE result holds for all oracle worlds.