The complexity of promise problems with applications to public-key cryptography*

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A “promise problem” is a formulation of partial decision problem. Complexity issues about promise problems arise from considerations about cracking problems for public-key cryptosystems. Using a notion of Turing reducibility between promise problems, this paper disproves a conjecture made by Even and Yacobi (1980), that would imply nonexistence of public-key cryptosystems with NP-hard cracking problems. In its place a new conjecture is raised having the same consequence. In addition, the new conjecture implies that NP-complete sets cannot be accepted by Turing machines that have at most one accepting computation for each input word.

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This research was done while the second author visited the Computer Science Department, Technion, Haifa, Israel, with funds provided by the United States-Israel Educational Foundation (Fulbright Award), and while the third author visited the Electrical Engineering and Computer Science Department, University of California at San Diego, La Jolla, California. Some of the results of this paper were presented by the second and third authors at the 8 th International Colloquium on Automata, Languages, and Programming, Aarhus, Denmark, July 1982. This research was supported in part by the National Science Foundation under Grants MCS77-23493 A02 and MCS81-20263.