arXiv subjects

Shanthanu S. Rai

Publications and source records attributed to Shanthanu S. Rai.

4 recordsLinked to original sources

arXiv2026

Explicit Nonlinear Functions beyond the Fourier bound

We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb F_2^n \to \mathbb F_2^m$. Concretely, we want an $F$ and an $A = A(m,n)> 0$ as small as possible, so that for every affine map $L: \mathbb F_2^n \to \mathbb F_2^m$ (of the form $L(x) = M x + b $) we have: $$\mathrm{agree}(F, L) := |\{ x \in \mathbb F_2^n \mid F(x) = L(x) \}| \leq A.$$ Such questions have been studied by Nyberg (1991,1993), Carlet and Ding (2004,2007), Liu, Mesnager and Chen (2017), Nagy (2025), and Biryukov, Turecek, and Udovenko (2026). There is a classical method of constructing such functions from bent-functions and Fourier analytic ideas; the best bound achievable by this method is: $$ A(m,n) = Θ(2^{n-m} + 2^{n/2}),$$ and in particular, is never smaller than $2^{n/2}$. In this work, we show how to construct highly nonlinear functions beyond this Fourier bound. Concretely, we show how to construct for every $γ>0$, a function $F: \mathbb F_2^n \to \mathbb F_2^m$ with $m = O_γ(n)$, achieving $$ A(m,n) \leq (1 + γ)^n.$$ Surprisingly, we even achieve the same quantitative behavior for the much harder question of having low agreement with $m$-tuples of degree $d$ polynomials $Q: \mathbb F_2^n \to \mathbb F_2^m$, with $m = O_{γ, d}(n)$. Here the previously best bounds were of the form $A(m,n) = O( 2^{-\frac{n}{2^{d+1}}} \cdot 2^n )$ of Ben-Sasson and Kopparty (2010), based on Gowers-norm-type arguments. All our results generalize to all finite fields $\mathbb F_q$ in place of $\mathbb F_2$. Our methods are based on a new connection to classical results on counting solutions to systems of polynomial equations via algebraic methods. This connection brings us to basic questions in combinatorics, about graphs and hypergraphs with simultaneously a small number of edges and independent sets.

cs.CR↗

arXiv2026

Deterministic Algorithms for Low Individual Degree Factors of Sparse Polynomials

We study factoring algorithms for general sparse polynomials and sparse polynomials of bounded individual degree and prove the following results. 1. We give a deterministic polynomial-time algorithm which takes as input an $n$-variate $s$-sparse polynomial $f$ of bounded individual degree $d$ and outputs a list of circuits which contains all factors of $f$, although there might be additional spurious circuits in the list. The algorithm runs in time $\operatorname{poly}(n, s^d)$. Additionally, every circuit in the list has constant depth. Our algorithm works over all fields of characteristic 0 or sufficiently large characteristic. Our result generalizes a recent result of Chuyoon and Shpilka that gives a $\operatorname{poly}(n, s^d)$-time algorithm for recovering all sparse factors of $f$ (without spurious factors). As a corollary, we can also recover all factors of $f$ in time $\operatorname{poly}(n, s^{d^2 \log n})$, and recover the algorithmic result of Bhargava, Saraf and Volkovich and its improvement by Chuyoon and Shpilka. Both the above consequences follow from known interpolation and divisibility testing techniques. 2. We give a deterministic quasipolynomial-time algorithm which takes as input a general $n$-variate $s$-sparse polynomial $f$ of (unbounded) individual degree $D$ and outputs a list of polynomials which contains all factors of $f$ that have bounded individual degree $d$. The algorithm runs in time $\operatorname{poly}(D^{d \log s}, s^{d^2 \log n})$ and works over arbitrary fields. The list may again contain spurious elements. Our result strengthens results of Dutta, Sinhababu and Thierauf and Kumar, Ramanathan and Saptharishi which give algorithms to recover all factors of $f$ of bounded total degree. A consequence of our algorithm is a new upper bound on the total number of bounded individual degree factors of a sparse polynomial.

cs.CC↗

arXiv2026

Modular composition & polynomial GCD in the border of small, shallow circuits

Modular composition is the problem of computing the coefficient vector of the polynomial $f(g(x)) \bmod h(x)$, given as input the coefficient vectors of univariate polynomials $f$, $g$, and $h$ over an underlying field $\mathbb{F}$. While this problem is known to be solvable in nearly-linear time over finite fields due to work of Kedlaya & Umans, no such near-linear-time algorithms are known over infinite fields, with the fastest known algorithm being from a recent work of Neiger, Salvy, Schost & Villard that takes $O(n^{1.43})$ field operations on inputs of degree $n$. In this work, we show that for any infinite field $\mathbb{F}$, modular composition is in the border of algebraic circuits with division gates of nearly-linear size and polylogarithmic depth. Moreover, this circuit family can itself be constructed in near-linear time. Our techniques also extend to other algebraic problems, most notably to the problem of computing greatest common divisors of univariate polynomials. We show that over any infinite field $\mathbb{F}$, the GCD of two univariate polynomials can be computed (piecewise) in the border sense by nearly-linear-size and polylogarithmic-depth algebraic circuits with division gates, where the circuits themselves can be constructed in near-linear time. While univariate polynomial GCD is known to be computable in near-linear time by the Knuth--Schönhage algorithm, or by constant-depth algebraic circuits from a recent result of Andrews & Wigderson, obtaining a parallel algorithm that simultaneously achieves polylogarithmic depth and near-linear work remains an open problem of great interest. Our result shows such an upper bound in the setting of border complexity.

cs.CC↗

arXiv2025

Closure under factorization from a result of Furstenberg

We show that algebraic formulas and constant-depth circuits are closed under taking factors. In other words, we show that if a multivariate polynomial over a field of characteristic zero has a small constant-depth circuit or formula, then all its factors can be computed by small constant-depth circuits or formulas respectively. Our result turns out to be an elementary consequence of a fundamental and surprising result of Furstenberg from the 1960s, which gives a non-iterative description of the power series roots of a bivariate polynomial. Combined with standard structural ideas in algebraic complexity, we observe that this theorem yields the desired closure results. As applications, we get alternative (and perhaps simpler) proofs of various known results and strengthen the quantitative bounds in some of them. This includes a unified proof of known closure results for algebraic models (circuits, branching programs and VNP), an extension of the analysis of the Kabanets-Impagliazzo hitting set generator to formulas and constant-depth circuits, and a (significantly) simpler proof of correctness as well as stronger guarantees on the output in the subexponential time deterministic algorithm for factorization of constant-depth circuits from a recent work of Bhattacharjee, Kumar, Ramanathan, Saptharishi & Saraf.

cs.CC↗