Paper 2020/1444

On known constructions of APN and AB functions and their relation to each other

Marco Calderini, Lilya Budaghyan, and Claude Carlet


This work is dedicated to APN and AB functions which are optimal against differential and linear cryptanlysis when used as S-boxes in block ciphers. They also have numerous applications in other branches of mathematics and information theory such as coding theory, sequence design, combinatorics, algebra and projective geometry. In this paper we give an overview of known constructions of APN and AB functions, in particular, those leading to infinite classes of these functions. Among them, the bivariate construction method, the idea first introduced in 2011 by the third author of the present paper, turned out to be one of the most fruitful. It has been known since 2011 that one of the families derived from the bivariate construction contains the infinite families derived by Dillon's hexanomial method. Whether the former family is larger than the ones it contains has stayed an open problem which we solve in this paper. Further we consider the general bivariate construction from 2013 by the third author and study its relation to the recently found infinite families of bivariate APN functions.

Available format(s)
Secret-key cryptography
Publication info
Preprint. MINOR revision.
almost perfect nonlinearalmost bentBoolean functionsdifferential uniformity
Contact author(s)
marco calderini @ uib no
2020-11-19: received
Short URL
Creative Commons Attribution


      author = {Marco Calderini and Lilya Budaghyan and Claude Carlet},
      title = {On known constructions of APN and AB functions and their relation to each other},
      howpublished = {Cryptology ePrint Archive, Paper 2020/1444},
      year = {2020},
      note = {\url{}},
      url = {}
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