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A New Family of APN Quadrinomials

Budaghyan, Lilya; Helleseth, Tor; Kaleyski, Nikolay Stoyanov
Journal article, Peer reviewed
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https://hdl.handle.net/11250/2756784
Utgivelsesdato
2020
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  • Department of Informatics [740]
  • Registrations from Cristin [5237]
Originalversjon
IEEE Transactions on Information Theory. 2020, 66(11), 7081 - 7087   https://doi.org/10.1109/TIT.2020.3007513
Sammendrag
The binomial B(x) = x 3 +βx 36 (where β is primitive in F 2 2) over F 2 10 is the first known example of an Almost Perfect Nonlinear (APN) function that is not CCZ-equivalent to a power function, and has remained unclassified into any infinite family of APN functions since its discovery in 2006. We generalize this binomial to an infinite family of APN quadrinomials of the form x 3 +a(x 2i+1 )2 k +bx 3·2m +c(x2 i+m+2m ) 2k from which B(x) can be obtained by setting a = β, b = c = 0, i = 3, k = 2. We show that for any dimension n = 2m with m odd and 3 + m,setting(a, b, c)=(β, β 2 , 1) and i =m -2 or i = (m - 2) -1 mod n yields an APN function, and verify that for n = 10 the quadrinomials obtained in this way for i = m - 2 and i = (m - 2) -1 mod n are CCZ-inequivalent to each other, to B(x), and to any other known APN function over F 2 10.
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IEEE
Tidsskrift
IEEE Transactions on Information Theory
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Copyright 2020 IEEE

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