The Knowledge-of-Exponent Assumptions and 3-Round Zero-Knowledge Protocols.

Hada and Tanaka [11, 12] showed the existence of 3-round, negligible-error zero-knowledge arguments for NP based on a pair of non-standard assumptions, here called KEA1 and KEA2. In this paper we show that KEA2 is false. This renders vacuous the results of [11, 12]. We recover these results, however, under a suitably modified new assumption called KEA3. What we believe is most interesting is that we show that it is possible to "falsify" assumptions like KEA2 that, due to their nature and quantifier-structure, do not lend themselves easily to "efficient falsification" (Naor [15]).