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Perfect nonlinear binomials and their semifields. (English) Zbl 1194.12003

The authors show that the binomial \[ F(x) = x^{p^s+1}-u^{p^k-1}x^{p^k+p^{2k+s}}\tag{b} \] is a perfect nonlinear (PN) mapping on \(\text{GF}(p^n)\) if \(p\) is an odd prime, \(n=3k\), \(\gcd(3,k) = 1\), \(k \equiv s \bmod 3\), \(n/\gcd(s,n)\) is odd and \(u\) is a primitive element of \(\text{GF}(p^n)\). The APN-ness of binomials of this shape for \(p=2\) had been investigated in [L. Budaghyan, C. Carlet and G. Leander, IEEE Trans. Inf. Theory. 54 , No. 9, 4218–4229 (2008; Zbl 1177.94135)].
Every PN Dembowski-Ostrom polynomial over \(\text{GF}(p^n)\), i.e. a PN polynomial of the form \(\sum_{i,j=0}^{n-1}a_{i,j}x^{p^i+p^j}, a_{i,j}\in \text{GF}(p^n)\), defines a commutative (pre)semifield of order \(p^n\) and vice versa, moreover if \(n\) is odd then the presemifields corresponding to PN Dembowski-Ostrom polynomials \(F\) and \(G\) are isotopic if and only if \(F\) and \(G\) are EA-equivalent, see R. S. Coulter and M. Henderson [Adv. Math. 217, 282–304 (2008; Zbl 1194.12007)]. The authors show that (b) is not EA-equivalent to a monomial PN Dembowski-Ostrom polynomial from which one can conclude that the corresponding semifield is not isotopic to a finite field and the twisted field of Albert, and if \(p \geq 5\) it is not isotopic to any semifield known so far. For a further new semifield see J. Bierbrauer [Des. Codes Cryptography 54, No. 3, 189–200 (2010; Zbl 1269.12006)].

MSC:

12E20 Finite fields (field-theoretic aspects)
11T06 Polynomials over finite fields
12K10 Semifields
51E15 Finite affine and projective planes (geometric aspects)
94A60 Cryptography
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