Enforcing and Defying Associativity, Commutativity, Totality, and Strong Noninvertibility for One-Way Functions in Complexity Theory
Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist one-way functions (i.e., p-time computable, honest, p-time noninvertible functions) that are total, commutative, and associative but not strongly noninvertible. In this paper we improve the sufficient condition to P ≠ NP.
More generally, in this paper we completely characterize which types of one-way functions stand or fall together with (plain) one-way functions—equivalently, stand or fall together with P ≠ NP. We look at the four attributes used in Rabi and Sherman’s seminal work on algebraic properties of one-way functions (see [RS97,RS93]) and subsequent papers—strongness (of noninvertibility), totality, commutativity, and associativity—and for each attribute, we allow it to be required to hold, required to fail, or “don’t care.” In this categorization there are 34 = 81 potential types of one-way functions. We prove that each of these 81 feature-laden types stand or fall together with the existence of (plain) one-way functions.
KeywordsComputational complexity complexity-theoretic one-way functions associativity commutativity strong noninvertibility
Unable to display preview. Download preview PDF.
- [BDG95]Balcázar, J., Díaz, J., Gabarró, J.: Structural Complexity I, 2nd edn. EATCS Monographs on Theoretical Computer Science. Springer, Heidelberg (1995)Google Scholar
- [Ber77]Berman, L.: Polynomial Reducibilities and Complete Sets. PhD thesis, Cornell University, Ithaca (1977)Google Scholar
- [BFH78]Brassard, G., Fortune, S., Hopcroft, J.: A note on cryptography and NP ∩ coNP − P. Technical Report TR-338, Department of Computer Science, Cornell University, Ithaca (April 1978)Google Scholar
- [HRS04]Hemaspaandra, L., Rothe, J., Saxena, A.: Enforcing and defying associativity, commutativity, totality, and strong noninvertibility for one-way functions in complexity theory. Technical Report TR-854, Department of Computer Science, University of Rochester, Rochester, NY (December 2004); Revised in, Also appears as ACM Computing Research Repository (CoRR) Technical Report cs.CC/050304 (April 2005)Google Scholar
- [HRW97]Hemaspaandra, L., Rothe, J., Wechsung, G.: On sets with easy certificates and the existence of one-way permutations. In: Bongiovanni, G., Bovet, D.P., Di Battista, G. (eds.) CIAC 1997. LNCS, vol. 1203, pp. 264–275. Springer, Heidelberg (1997)Google Scholar
- [RS93]Rabi, M., Sherman, A.: Associative one-way functions: A new paradigm for secret-key agreement and digital signatures. Technical Report CS-TR-3183/UMIACS-TR-93-124, Department of Computer Science, University of Maryland, College Park, Maryland (1993)Google Scholar
- [SS05]Saxena, A., Soh, B.: A novel method for authenticating mobile agents with one-way signature chaining. In: Proceedings of the 7th International Symposium on Autonomous Decentralized Systems, pp. 187–193. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar