On the complexity of hardness amplification

Chi Jen Lu*, Shi-Chun Tsai, Hsin Lung Wu

*Corresponding author for this work

Research output: Contribution to journalArticle

7 Scopus citations


For δ ∈ (0, 1) and k, n ∈ N, we study the task of transforming a hard function f: {0, 1}n → {0, 1}, with which any small circuit disagrees on (1 - δ)/2 fraction of the input, into a harder function f′, with which any small circuit disagrees on (1 - δk)/2 fraction of the input. First, we show that such hardness amplification, when carried out in some black-box way, must require a high complexity. In particular, it cannot be realized by a circuit of depth d and size 2o (kk1/d) or by a nondeterministic circuit of size o(k/log k) (and arbitrary depth) for any δ ∈ (0, 1). This extends the result of Viola, which only works when (1 - δ)/2 is small enough. Furthermore, we show that even without any restriction on the complexity of the amplification procedure, such a black-box hardness amplification must be inherently nonuniform in the following sense. To guarantee the hardness of the resulting function f′, even against uniform machines, one has to start with a function f, which is hard against nonuniform algorithms with Ω (k log (1/δ)) bits of advice. This extends the result of Trevisan and Vadhan, which only addresses the case with (1 - δ)/2 = 2-n. Finally, we derive similar lower bounds for any black-box construction of a pseudorandom generator (PRG) from a hard function. To prove our results, we link the task of hardness amplifications and PRG constructions, respectively, to some type of error-reduction codes, and then we establish lower bounds for such codes, which we hope could find interest in both coding theory and complexity theory.

Original languageEnglish
Pages (from-to)4575-4586
Number of pages12
JournalIEEE Transactions on Information Theory
Issue number10
StatePublished - 9 Oct 2008


  • Complexity theory
  • Computational complexity
  • Construction industry
  • Decoding
  • Encoding
  • Hardness amplification
  • Integrated circuit modeling
  • List-decodable code
  • Noise
  • Pseudorandom generator
  • Transforms

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