Progress Towards the Conjecture on APN Functions and Absolutely Irreducible Polynomials

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

Origin of Biomolecular Networks

Biomolecular networks have already found great utility in characterizing complex biological systems arising from pair-wise interactions amongst biomolecules. Here, we review how graph theoretical approaches can be applied not only for a better understanding of various proximate (mechanistic) relations, but also, ultimate (evolutionary) structures encoded in such networks. A central question deals with the evolutionary dynamics by which different topologies of biomolecular networks might have evolved, as well as the biological principles that can be hypothesized from a deeper understanding of the induced network dynamics. We emphasize the role of gene duplication in terms of signaling game theory, whereby sender and receiver gene players accrue benefit from gene duplication, leading to a preferential attachment mode of network growth. Information asymmetry between sender and receiver genes is hypothesized as a key driver of the resulting network topology. The study of the resulting dynamics suggests many mathematical/computational problems, the majority of which are intractable but yield to efficient approximation algorithms, when studied through an algebraic graph theoretic lens