arXiv subjects

Heeralal Janwa

Publications and source records attributed to Heeralal Janwa.

6 recordsLinked to original sources

arXiv2026

A new criterion for the absolute irreducibility of multivariate polynomials over finite fields

A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry and its applications in coding theory, cryptography, and other fields. Among the applications of absolute irreducibility are bounding the number of rational points via the Weil conjectures and establishing exceptional APN and permutation properties of functions over finite fields. In this article, we present a new criterion for the absolute irreducibility of hypersurfaces defined by multivariate polynomials over finite fields. Our criterion does not require testing for irreducibility in the ground or extension fields, assuming that the leading form is square-free. We just require multivariate GCD computations and the square-free property. Since almost all polynomials are known to be square-free, our absolute irreducibility criterion is valid for almost all multivariate polynomials.

math.AG↗

arXiv2018

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.

q-bio.MN↗

arXiv2016

On the absolute irreducibility of hyperplane sections of generalized Fermat varieties in $\Bbb{P}^3$ and the conjecture on exceptional APN functions: the Kasami-Welch degree case

Let $f$ be a function on a finite field $F$. The decomposition of the generalized Fermat variety $X$ defined by the multivariate polynomial of degree $n$, $ϕ(x,y,z)=f(x)+f(y)+f(z)$ in $\Bbb{P}^3(\overline{\mathbb{F}}_2)$, plays a crucial role in the study of almost perfect non-linear (APN) functions and exceptional APN functions. Their structure depends fundamentally on the Fermat varieties corresponding to the monomial functions of exceptional degrees $n=2^k+1$ and $n=2^{2k}-2^k+1$ (Gold and Kasami-Welch numbers, respectively). Very important results for these have been obtained by Janwa, McGuire and Wilson in [12,13]. In this paper we study $X$ related to the Kasami-Welch degree monomials and its decomposition into absolutely irreducible components. We show that, in this decomposition, the components intersect transversally at a singular point. This structural fact implies that the corresponding generalized Fermat hypersurfaces, related to Kasami-Welch degree polynomial families, are absolutely irreducible. In particular, we prove that if $f(x)=x^{2^{2k}-2^k+1}+h(x)$, where ${\rm deg}(h)\equiv 3{\pmod 4}$, then the corresponding APN multivariate hypersurface is absolutely irreducible, and hence $f(x)$ is not exceptional APN function. We also prove conditional result in the case when ${\rm deg}(h)\equiv 5{\pmod 8}$. Since for odd degree $f(x)$, the conjecture needs to be resolved only for the Gold degree and the Kasami-Welch degree cases our results contribute substantially to the proof of the conjecture on exceptional APN functions---in the hardest case: the Kasami-Welch degree.

math.AG↗

arXiv2016

On Parameters of Subfield Subcodes of Extended Norm-Trace Codes

In this article we describe how to find the parameters of subfield subcodes of extended Norm--Trace codes. With a Gröbner basis of the ideal of the $\mathbb{F}_{q^r}$ rational points of the Norm--Trace curve one can determine the dimension of the subfield subcodes or the dimension of the trace code. We also find a BCH--like bound from the minimum distance of the original supercode.

math.AG↗

arXiv2016

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.

math.NT↗

arXiv2012

On the Conjecture on APN Functions

An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field $\mathbb{F}$ is called exceptional APN, if it is also APN on infinitely many extensions of $\mathbb{F}$. In this article we consider the most studied case of $\mathbb{F}=\mathbb{F}_{2^n}$. A conjecture of Janwa-Wilson and McGuire-Janwa-Wilson (1993/1996), settled in 2011, was that the only exceptional monomial APN functions are the monomials $x^n$, where $n=2^i+1$ or $n={2^{2i}-2^i+1}$ (the Gold or the Kasami exponents respectively). A subsequent conjecture states that any exceptional APN function is one of the monomials just described. One of our result is that all functions of the form $f(x)=x^{2^k+1}+h(x)$ (for any odd degree $h(x)$, with a mild condition in few cases), are not exceptional APN, extending substantially several recent results towards the resolution of the stated conjecture.

cs.IT↗