Abstract We observe that on the binary finite fields the classification of 2-to-1 binomials is equivalent to the classification of o-monomials, which is a well-studied and elusive problem in finite geometry. This connection implies a complete classification of 2-to-1 binomials $$b=x^d+ux^e$$ b = x d + u x e for a large set of values of ( d , e ). Further, we show that a number of the known infinite families of 2-to-1 maps can be traced back to o-polynomials or to difference maps of APN maps. We also provide some connections between 2-to-1 maps and hyperovals in non-desarguesian planes.

Abstract Brawley and Carlitz introduced diamond products of elements of finite fields and associated composed products of polynomials in 1987. Composed products yield a method to construct irreducible polynomials of large composite degrees from irreducible polynomials of lower degrees. We show that the composed product of two irreducible polynomials of degrees m and n is again irreducible if and only if m and n are coprime and the involved diamond product satisfies a special cancellation property, the so-called conjugate cancellation. This completes the characterization of irreducible composed products, considered in several previous papers. More generally, we give precise criteria when a diamond product satisfies conjugate cancellation. For diamond products defined via bivariate polynomials, we prove simple criteria that characterize when conjugate cancellation holds. We also provide efficient algorithms to check these criteria. We achieve stronger results as well as more efficient algorithms in the case that the polynomials are bilinear. Lastly, we consider possible constructions of normal elements using composed products and the methods we developed.

Abstract We suggest a construction of the minimal polynomial $$m_{\beta ^k}$$ m β k of $$\beta ^k\in {\mathbb {F}}_{q^n}$$ β k ∈ F q n over $${\mathbb {F}}_q$$ F q from the minimal polynomial $$f= m_\beta $$ f = m β for all positive integers k whose prime factors divide $$q-1$$ q - 1 . The computations of our construction are carried out in $${\mathbb {F}}_q$$ F q . The key observation leading to our construction is that for $$k \mid q-1$$ k ∣ q - 1 holds $$\begin{aligned} m_{\beta ^k}(X^k) = \prod _{j=1}^{\frac{k}{t}} \zeta _{k}^{-jn} f (\zeta _{k}^j X), \end{aligned}$$ m β k ( X k ) = ∏ j = 1 k t ζ k - j n f ( ζ k j X ) , where $$t= \max \{m\in {\mathbb {N}}: m \mid \gcd (n,k), f(X) = g (X^m), g \in {\mathbb {F}}_q[X]\}$$ t = max { m ∈ N : m ∣ gcd ( n , k ) , f ( X ) = g ( X m ) , g ∈ F q [ X ] } and $$\zeta _{k}$$ ζ k is a primitive k-th root of unity in $${\mathbb {F}}_q$$ F q . The construction allows to construct a large number of irreducible polynomials over $${\mathbb {F}}_q$$ F q

Abstract We consider image sets of differentially d -uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution. Further, we focus on a particularly interesting case of APN maps on binary fields $$\mathbb {F}_{2^n}$$ F 2 n . We show that APN maps with the minimal image size are very close to being 3-to-1. We prove that for n even the image sets of several important families of APN maps are minimal, and as a consequence they have the classical Walsh spectrum. Finally, we present upper bounds on the image size of APN maps. For a non-bijective almost bent map f , these results imply $$\frac{2^n+1}{3}+1 \le |{\text {Im}}(f)| \le 2^n-2^{(n-1)/2}$$ 2 n + 1 3 + 1 ≤ | Im ( f ) | ≤ 2 n - 2 ( n - 1 ) / 2 .

Abstract We study the cycle structure of permutations $$F(x)=x+\gamma f(x)$$ F ( x ) = x + γ f ( x ) on $$\mathbb {F}_{q^n}$$ F q n , where $$f :\mathbb {F}_{q^n} \rightarrow \mathbb {F}_q$$ f : F q n → F q . We show that for a 1-homogeneous function f the cycle structure of F can be determined by calculating the cycle structure of certain induced mappings on parallel lines of $$\gamma \mathbb {F}_q$$ γ F q . Using this observation we describe explicitly the cycle structure of two families of permutations over $$\mathbb {F}_{q^2}$$ F q 2 : $$x+\gamma {{\,\mathrm{Tr}\,}}(x^{2q-1})$$ x + γ Tr ( x 2 q - 1 ) , where $$q\equiv -1 \pmod 3$$ q ≡ - 1 ( mod 3 ) and $$\gamma \in \mathbb {F}_{q^2}$$ γ ∈ F q 2 , with $$\gamma ^3=-\frac{1}{27}$$ γ 3 = - 1 27 and $$x+\gamma {{\,\mathrm{Tr}\,}}\left( x^{\frac{2^{2s-1}+3\cdot 2^{s-1}+1}{3}}\right) $$ x + γ Tr x 2 2 s - 1 + 3 · 2 s - 1 + 1 3 , where $$q=2^s$$ q = 2 s , s odd and $$\gamma \in \mathbb {F}_{q^2}$$ γ ∈ F q 2 , with $$\gamma ^{(q+1)/3}=1$$ γ ( q + 1 ) / 3 = 1 .

In this extended abstract we present results on the inverses modulo 2n- 1 of the known APN exponents. In particular, we describe explicitly the inverses of the Welch and Dobbertin exponents and give the main ideas of their proofs. Further, we observe that the inverse of the Dobbertin exponent defines an APN function on F2nof algebraic degree n+3/2, which is the first example of such a function.

Based on a computer search, Anne Canteaut conjectured that the exponent $2^{2r}+2^r+1$ in ${\bf F}_{2^{6r}}$ and the exponent $(2^{r}+1)^2$ in ${\bf F}_{2^{4r}}$ yield bent monomial functions. These conjectures are proved in [A. Canteaut, P. Charpin, and G. Kyureghyan, A new class of monomial bent functions, in Proceedings of the 2006 IEEE International Symposium on Information Theory, (ISIT 06 Seattle), IEEE Press, Piscataway, NJ, 2006, pp. 903–906] and [N. G. Leander, IEEE Trans. Inform. Theory, 52 (2006), pp. 738–743]. Both exponents are of binary weight 3 and define functions from the Maiorana–McFarland class $\mathcal{M}$ of bent functions to the subfield. In this paper we show that these are the only such exponents. Our proof is based on the classification of the permutation binomials $X^{2^k+2}+ \nu X$ of finite fields of even characteristics. We also extend the result of Leander, determining all bent monomial functions with the exponent $(2^{r}+1)^2$.

We study the Boolean functions on F2En, n = 6r, of the form x rarr Tr (lambdaxd) with d = 22r+ 2r+ 1. Our main result is the characterization of those lambda for which they are bent

A new almost-perfect nonlinear function (APN) on F(2/sup 10/) which is not equivalent to any of the previously known APN mappings is constructed. This is the first example of an APN mapping which is not equivalent to a power mapping.