The Power of Negations in Cryptography

  • Siyao Guo
  • Tal Malkin
  • Igor C. Oliveira
  • Alon Rosen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9014)


The study of monotonicity and negation complexity for Bool-ean functions has been prevalent in complexity theory as well as in computational learning theory, but little attention has been given to it in the cryptographic context. Recently, Goldreich and Izsak (2012) have initiated a study of whether cryptographic primitives can be monotone, and showed that one-way functions can be monotone (assuming they exist), but a pseudorandom generator cannot.

In this paper, we start by filling in the picture and proving that many other basic cryptographic primitives cannot be monotone. We then initiate a quantitative study of the power of negations, asking how many negations are required. We provide several lower bounds, some of them tight, for various cryptographic primitives and building blocks including one-way permutations, pseudorandom functions, small-bias generators, hard-core predicates, error-correcting codes, and randomness extractors. Among our results, we highlight the following.

  • Unlike one-way functions, one-way permutations cannot be monotone.

  • We prove that pseudorandom functions require logn − O(1) negations (which is optimal up to the additive term).

  • We prove that error-correcting codes with optimal distance parameters require logn − O(1) negations (again, optimal up to the additive term).

  • We prove a general result for monotone functions, showing a lower bound on the depth of any circuit with t negations on the bottom that computes a monotone function f in terms of the monotone circuit depth of f. This result addresses a question posed by Koroth and Sarma (2014) in the context of the circuit complexity of the Clique problem.


Boolean Function Monotone Function Pseudorandom Generator Boolean Circuit Cryptographic Primitive 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© International Association for Cryptologic Research 2015

Authors and Affiliations

  • Siyao Guo
    • 1
  • Tal Malkin
    • 2
  • Igor C. Oliveira
    • 2
  • Alon Rosen
    • 3
  1. 1.Department of Computer Science and EngineeringChinese University of Hong KongChina
  2. 2.Department of Computer ScienceColumbia UniversityUSA
  3. 3.Efi Arazi School of Computer ScienceIDC HerzliyaIsrael

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