arXiv subjects

Indranath Sengupta

Publications and source records attributed to Indranath Sengupta.

42 records · Page 3Linked to original sources

arXiv2016

Minimal Graded Free Resolution for Monomial Curves in $\mathbb{A}^{4}$ defined by almost arithmetic sequences

Let $\mm=(m_0,m_1,m_2,n)$ be an almost arithmetic sequence, i.e., a sequence of positive integers with ${\rm gcd}(m_0,m_1,m_2,n) = 1$, such that $m_0<m_1<m_2$ form an arithmetic progression, $n$ is arbitrary and they minimally generate the numerical semigroup $Γ= m_0\N + m_1\N + m_2\N + n\N$. Let $k$ be a field. The homogeneous coordinate ring $k[Γ]$ of the affine monomial curve parametrically defined by $X_0=t^{m_0},X_{1}=t^{m_1},X_2=t^{m_3},Y=t^{n}$ is a graded $R$-module, where $R$ is the polynomial ring $k[X_0,X_1,X_3, Y]$ with the grading $°{X_i}:=m_i, °{Y}:=n$. In this paper, we construct a minimal graded free resolution for $k[Γ]$.

math.AC↗

arXiv2011

GB-hash : Hash Functions Using Groebner Basis

In this paper we present an improved version of HF-hash, viz., GB-hash : Hash Functions Using Groebner Basis. In case of HF-hash, the compression function consists of 32 polynomials with 64 variables which were taken from the first 32 polynomials of hidden field equations challenge-1 by forcing last 16 variables as 0. In GB-hash we have designed the compression function in such way that these 32 polynomials with 64 variables form a minimal Groebner basis of the ideal generated by them with respect to graded lexicographical (grlex) ordering as well as with respect to graded reverse lexicographical (grevlex) ordering. In this paper we will prove that GB-hash is more secure than HF-hash as well as more secure than SHA-256. We have also compared the efficiency of our GB-hash with SHA-256 and HF-hash.

cs.CR↗

arXiv2011

Minimal Graded Free Resolutions for Monomial Curves Defined by Arithmetic Sequences

Let $\mm=(m_0,...,m_n)$ be an arithmetic sequence, i.e., a sequence of integers $m_0<...<m_n$ with no common factor that minimally generate the numerical semigroup $\sum_{i=0}^{n}m_i\N$ and such that $m_i-m_{i-1}=m_{i+1}-m_i$ for all $i\in\{1,...,n-1\}$. The homogeneous coordinate ring $Γ_\mm$ of the affine monomial curve parametrically defined by $X_0=t^{m_0},...,X_n=t^{m_n}$ is a graded $R$-module where $R$ is the polynomial ring $k[X_0,...,X_n]$ with the grading obtained by setting $°{X_i}:=m_i$. In this paper, we construct an explicit minimal graded free resolution for $Γ_\mm$ and show that its Betti numbers depend only on the value of $m_0$ modulo $n$. As a consequence, we prove a conjecture of Herzog and Srinivasan on the eventual periodicity of the Betti numbers of semigroup rings under translation for the monomial curves defined by an arithmetic sequence.

math.AC↗

arXiv2011

The Rees Algebra for Certain Monomial Curves

In this article, we find the equations defining the Rees algebra for certain Monomial Curves explicitly and use them to prove that the blowup scheme is not smooth. This proves a conjecture of Francia in affirmative, which says that a dimension one prime in a regular local ring is a complete intersection if it has a smooth blowup.

math.AC↗

arXiv2010

Minimal free resolutions for certain affine monomial curve

Given an arbitrary field k and an arithmetic sequence of positive integers m_0<...<m_n, we consider the affine monomial curve parameterized by X_0=t^{m_0},...,X_n=t^{m_n}. In this paper, we conjecture that the Betti numbers of its coordinate ring are completely determined by n and the value of m_0 modulo n. We first show that the defining ideal of the monomial curve can be written as a sum of two determinantal ideals. Using this fact, we describe the minimal free resolution of the coordinate ring in the following three cases: when m_0 is 1 modulo n (determinantal), when m_0 is n modulo n (almost determinantal), and when m_0 is 2 modulo n and n=4 (Gorenstein of codimension 4).

math.AC↗

arXiv2009

HF-hash : Hash Functions Using Restricted HFE Challenge-1

Vulnerability of dedicated hash functions to various attacks has made the task of designing hash function much more challenging. This provides us a strong motivation to design a new cryptographic hash function viz. HF-hash. This is a hash function, whose compression function is designed by using first 32 polynomials of HFE Challenge-1 with 64 variables by forcing remaining 16 variables as zero. HF-hash gives 256 bits message digest and is as efficient as SHA-256. It is secure against the differential attack proposed by Chabaud and Joux as well as by Wang et. al. applied to SHA-0 and SHA-1.

cs.CR↗