### Progress Towards the Conjecture on APN Functions and Absolutely Irreducible Polynomials

**2016-01-30**

1602.02576 | math.NT

Almost Perfect Nonlinear (APN) functions are very useful in cryptography,
when they are used as S-Boxes, because of their good resistance to differential
cryptanalysis. An APN function $f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$
is called exceptional APN if it is APN on infinitely many extensions of
$\mathbb{F}_{2^n}$. Aubry, McGuire and Rodier conjectured that the only
exceptional APN functions are the Gold and the Kasami-Welch monomial functions.
They established that a polynomial function of odd degree is not exceptional
APN provided the degree is not a Gold number $(2^k+1)$ or a Kasami-Welch number
$(2^{2k}-2^k+1)$. When the degree of the polynomial function is a Gold number,
several partial results have been obtained [1, 7, 8, 10, 17]. One of the
results in this article is a proof of the relatively primeness of the
multivariate APN polynomial conjecture, in the Gold degree case. This helps us
extend substantially previous results. We prove that Gold degree polynomials of
the form $x^{2^k+1}+h(x)$, where $deg(h)$ is any odd integer (with the natural
exceptions), can not be exceptional APN.
We also show absolute irreducibility of several classes of multivariate
polynomials over finite fields and discuss their applications.

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