Time space tradeoffs for attacks against one-way functions and PRGs

Anindya De*, Luca Trevisan, Madhur Tulsiani

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

41 Scopus citations


We study time space tradeoffs in the complexity of attacks against one-way functions and pseudorandom generators. Fiat and Naor [7] show that for every function f: [N]→[N], there is an algorithm that inverts f everywhere using (ignoring lower order factors) time, space and advice at most N 3/4. We show that an algorithm using time, space and advice at most max {ε5/4N3/4, √εN} exists that inverts f on at least an ε fraction of inputs. A lower bound of ω̃(√ εN)also holds, making our result tight in the "low end" of ε ≤ 3√1/N. (Both the results of Fiat and Naor and ours are formulated as more general trade-offs between the time and the space and advice length of the algorithm. The results quoted above correspond to the interesting special case in which time equals space and advice length.) We also show that for every length-increasing generator G:[N] →[2N] there is a algorithm that achieves distinguishing probability ε between the output of G and the uniform distribution and that can be implemented in polynomial (in logN) time and with advice and space O(ε 2 •NlogN). We prove a lower bound of S•TΩ(ε 2 N) where T is the time used by the algorithm and S is the amount of advice. This lower bound applies even when the distinguisher has oracle access to G. We prove stronger lower bounds in the common random string model, for families of one-way permutations and of pseudorandom generators.

Original languageEnglish (US)
Title of host publicationAdvances in Cryptology - CRYPTO 2010 - 30th Annual Cryptology Conference, Proceedings
Number of pages17
StatePublished - 2010
Event30th Annual International Cryptology Conference, CRYPTO 2010 - Santa Barbara, CA, United States
Duration: Aug 15 2010Aug 19 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6223 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other30th Annual International Cryptology Conference, CRYPTO 2010
Country/TerritoryUnited States
CitySanta Barbara, CA


  • One-way functions
  • pseudorandom generators
  • random permutations
  • time-space tradeoffs

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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