Generically speeding-up repeated squaring is equivalent to factoring: Sharp thresholds for all generic-ring delay functions

Lior Rotem*, Gil Segev

*Corresponding author for this work

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

5 Scopus citations


Despite the fundamental importance of delay functions, repeated squaring in RSA groups (Rivest, Shamir and Wagner ’96) is the only candidate offering both a useful structure and a realistic level of practicality. Somewhat unsatisfyingly, its sequentiality is provided directly by assumption (i.e., the function is assumed to be a delay function). We prove sharp thresholds on the sequentiality of all generic-ring delay functions relative to an RSA modulus based on the hardness of factoring in the standard model. In particular, we show that generically speeding-up repeated squaring (even with a preprocessing stage and any polynomial number parallel processors) is equivalent to factoring. More generally, based on the (essential) hardness of factoring, we prove that any generic-ring function is in fact a delay function, admitting a sharp sequentiality threshold that is determined by our notion of sequentiality depth. Moreover, we show that generic-ring functions admit not only sharp sequentiality thresholds, but also sharp pseudorandomness thresholds.

Original languageAmerican English
Title of host publicationAdvances in Cryptology - CRYPTO 2020 - 40th Annual International Cryptology Conference, Proceedings
EditorsDaniele Micciancio, Thomas Ristenpart
Number of pages29
ISBN (Print)9783030568764
StatePublished - 2020
Event40th Annual International Cryptology Conference, CRYPTO 2020 - Santa Barbara, United States
Duration: 17 Aug 202021 Aug 2020

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12172 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference40th Annual International Cryptology Conference, CRYPTO 2020
Country/TerritoryUnited States
CitySanta Barbara

Bibliographical note

Funding Information:
L. Rotem and G. Segev—Supported by the European Union’s Horizon 2020 Framework Program (H2020) via an ERC Grant (Grant No. 714253). L. Rotem—Supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.

Publisher Copyright:
© International Association for Cryptologic Research 2020.


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