The article may be found at the following publishers website: http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WJ0-4BK2DJP-1&_user=47004&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000005018&_version=1&_urlVersion=0&_userid=47004&md5=e381546c6cbad5d8b0a1a3c114c4d3cb We study the ambiguity, or ‘‘many-to-one’’-ness, of two-argument, one-way functions that are strong (that is, hard to invert even if one of their arguments is given), total, and associative. Such powerful one- way functions are the basis of a cryptographic paradigm described by Rabi and Sherman (Inform. Process. Lett. 64(2) (1997) 239) and were shown by Hemaspaandra and Rothe (J. Comput. System Sci. 58(3) (1999) 648) to exist exactly if standard one-way functions exist. Rabi and Sherman (1997) show that no total, associative function deﬁned over a universe having at least two elements is one-to-one. We show that if P[not equal to]UP; then, for every deN+, there is an O(logd n)-to-one, strong, total, associative, one-way function sd : We argue that this bound is tight in the sense that any total, associative function having similar properties to sd but not necessarily strong or one-way must have at least the same order of magnitude of ambiguity as sd has. We demonstrate that the techniques used in proving the above-stated results easily apply to other classes of total, associative functions. We provide a complete characterization for the existence of strong, total, associative, one-way functions whose ambiguity approaches the lower bounds we provide. We say a language is in PolylogP if there exists a polynomial-time Turing machine M accepting the language such that for some d ARþ it holds that M has on each string x at most O ðlog d n Þ accepting paths, where n ¼ jxj: We show that PaPolylogP if and only for some d ARþ there exists an O ðlog d n Þ-to-one, strong, total, associative, one-way function.
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Homan, Christopher, "Tight lower bounds on the ambiguity of strong, total, associative, one-way functions" (2004). Lower Bounds on the Ambiguity of Strong, Total, Associative, One-Way Functions, C. Homan, Journal of Computer and System Sciences, Vol. 68 (3), pp. 657-674. Accessed from
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