Injective trapdoor functions via derandomization: How strong is Rudich’s black-box barrier?

Lior Rotem*, Gil Segev

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations


We present a cryptographic primitive P satisfying the following properties: Rudich’s seminal impossibility result (PhD thesis ’88) shows that P cannot be used in a black-box manner to construct an injective one-way function.P can be used in a non-black-box manner to construct an injective one-way function assuming the existence of a hitting-set generator that fools deterministic circuits (such a generator is known to exist based on the worst-case assumption that E=DTIME(2O(n)) has a function of deterministic circuit complexity 2Ω(n)).Augmenting P with a trapdoor algorithm enables a non-black-box construction of an injective trapdoor function (once again, assuming the existence of a hitting-set generator that fools deterministic circuits), while Rudich’s impossibility result still holds. The primitive P and its augmented variant can be constructed based on any injective one-way function and on any injective trapdoor function, respectively, and they are thus unconditionally essential for the existence of such functions. Moreover, P can also be constructed based on various known primitives that are secure against related-key attacks, thus enabling to base the strong structural guarantees of injective one-way functions on the strong security guarantees of such primitives. Our application of derandomization techniques is inspired mainly by the work of Barak, Ong and Vadhan (CRYPTO ’03), which on one hand relies on any one-way function, but on the other hand only results in a non-interactive perfectly-binding commitment scheme (offering significantly weaker structural guarantees compared to injective one-way functions), and does not seem to enable an extension to public-key primitives. The key observation underlying our approach is that Rudich’s impossibility result applies not only to one-way functions as the underlying primitive, but in fact to a variety of “unstructured” primitives. We put forward a condition for identifying such primitives, and then subtly tailor the properties of our primitives such that they are both sufficiently unstructured in order to satisfy this condition, and sufficiently structured in order to yield injective one-way and trapdoor functions. This circumvents the basic approach underlying Rudich’s long-standing evidence for the difficulty of constructing injective one-way functions (and, in particular, injective trapdoor functions) based on seemingly weaker or unstructured assumptions.

Original languageAmerican English
Title of host publicationTheory of Cryptography - 16th International Conference, TCC 2018, Proceedings
EditorsAmos Beimel, Stefan Dziembowski
PublisherSpringer Verlag
Number of pages27
ISBN (Print)9783030038069
StatePublished - 2018
Event16th Theory of Cryptography Conference, TCC 2018 - Panaji, India
Duration: 11 Nov 201814 Nov 2018

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11239 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference16th Theory of Cryptography Conference, TCC 2018

Bibliographical note

Publisher Copyright:
© International Association for Cryptologic Research 2018.


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