Perfect nonlinear functions and cryptography

Celine Blondeau, Kaisa Nyberg

Research output: Contribution to journalArticleScientificpeer-review

34 Citations (Scopus)
223 Downloads (Pure)


In the late 1980s the importance of highly nonlinear functions in cryptography was first discovered by Meier and Staffelbach from the point of view of correlation attacks on stream ciphers, and later by Nyberg in the early 1990s after the introduction of the differential cryptanalysis method. Perfect nonlinear (PN) and almost perfect nonlinear (APN) functions, which have the optimal properties for offering resistance against differential cryptanalysis, have since then been an object of intensive study by many mathematicians. In this paper, we survey some of the theoretical results obtained on these functions in the last 25 years. We recall how the links with other mathematical concepts have accelerated the search on PN and APN functions. To illustrate the use of PN and APN functions in practice, we discuss examples of ciphers and their resistance to differential attacks. In particular, we recall that in cryptographic applications suboptimal functions are often used.
Original languageEnglish
Pages (from-to)120-147
JournalFinite Fields and Their Applications
Issue numberMarch
Publication statusPublished - 2015
MoE publication typeA1 Journal article-refereed


  • Perfect nonlinear functions
  • PN functions
  • Almost perfect nonlinear functions
  • APN functions
  • Differential uniformity
  • Nonlinearity
  • Differential cryptanalysis


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