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   <subfield code="a">Certifying Permutations: Noninteractive zero-knowledge based on any trapdoor permutation</subfield>
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   <subfield code="a">In cryptographic protocols it is often necessary to verify/certify the &quot;tools” in use. This work demonstrates certain subtleties in treating a family of trapdoor permutations in this context, noting the necessity to &quot;check” certain properties of these functions. The particular case we illustrate is that of noninteractive zero-knowledge. We point out that the elegant recent protocol of Feige, Lapidot, and Shamir for proving NP statements in noninteractive zero-knowledge requires an additional certification of the underlying trapdoor permutation, and suggest a method for certifying permutations which fills this gap.</subfield>
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