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Factoring Polynomials over Finite Fields using Balance Test [article]

Chandan Saha
2008 arXiv   pre-print
We study the problem of factoring univariate polynomials over finite fields. Under the assumption of the Extended Riemann Hypothesis (ERH), (Gao, 2001) designed a polynomial time algorithm that fails to factor only if the input polynomial satisfies a strong symmetry property, namely square balance. In this paper, we propose an extension of Gao's algorithm that fails only under an even stronger symmetry property. We also show that our property can be used to improve the time complexity of best
more » ... terministic algorithms on most input polynomials. The property also yields a new randomized polynomial time algorithm.
arXiv:0802.2838v1 fatcat:2sadxgrjnrej5bfhfaog4p55qi

The Power of Depth 2 Circuits over Algebras [article]

Chandan Saha, Ramprasad Saptharishi, Nitin Saxena
2009 arXiv   pre-print
We study the problem of polynomial identity testing (PIT) for depth 2 arithmetic circuits over matrix algebra. We show that identity testing of depth 3 (Sigma-Pi-Sigma) arithmetic circuits over a field F is polynomial time equivalent to identity testing of depth 2 (Pi-Sigma) arithmetic circuits over U_2(F), the algebra of upper-triangular 2 x 2 matrices with entries from F. Such a connection is a bit surprising since we also show that, as computational models, Pi-Sigma circuits over U_2(F) are
more » ... trictly 'weaker' than Sigma-Pi-Sigma circuits over F. The equivalence further shows that PIT of depth 3 arithmetic circuits reduces to PIT of width-2 planar commutative Algebraic Branching Programs (ABP). Thus, identity testing for commutative ABPs is interesting even in the case of width-2. Further, we give a deterministic polynomial time identity testing algorithm for a Pi-Sigma circuit over any constant dimensional commutative algebra over F. While over commutative algebras of polynomial dimension, identity testing is at least as hard as that of Sigma-Pi-Sigma circuits over F.
arXiv:0904.2058v1 fatcat:up42y4zzb5fsrfevtuef2kk7iq

Quasi-polynomial Hitting-set for Set-depth-Delta Formulas [article]

Manindra Agrawal, Chandan Saha, Nitin Saxena
2012 arXiv   pre-print
We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition (X_1,...,X_d) of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑_i=1^k ∏_j=1^d f_i,j(x_X_j), where f_i,j is a sparse polynomial in F[x_X_j]. Extending this definition to any depth - we call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in C respects a (unknown) partition on the
more » ... if Delta is even then the product gates of the bottom-most Pi-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-Delta formulas (over any field) with running time polynomial in exp((Delta^2 log s)^Delta - 1), where s is the size bound on the input set-depth-Delta formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of Delta=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995). Our work settles this question, not only for depth-3 but, up to depth epsilon.log s / loglog s, for a fixed constant epsilon < 1. The technique is to investigate depth-Delta formulas via depth-(Delta-1) formulas over a Hadamard algebra, after applying a 'shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-Delta formulas.
arXiv:1209.2333v1 fatcat:52f2hvrvfbgh5g35c5yic7jsei

Randomized polynomial-time equivalence between determinant and trace-IMM equivalence tests [article]

Janaky Murthy, Vineet Nair, Chandan Saha
2020 arXiv   pre-print
[GGKS19] Ankit Garg, Nikhil Gupta, Neeraj Kayal, and Chandan Saha. Determinant equiva- lence test over finite fields and over Q.  ...  [KNS19] Neeraj Kayal, Vineet Nair, and Chandan Saha. Average-case linear matrix factorization and reconstruction of low width algebraic branching programs.  ... 
arXiv:2006.08272v1 fatcat:egtelgukxzhmledkfqosktxwtm

Superconductivity in doped Weyl semimetal Mo_0.9Ir_0.1Te_2 with broken inversion symmetry [article]

Manasi Mandal, Chandan Patra, Anshu Kataria, Suvodeep Paul, Surajit Saha, R. P. Singh
2021 arXiv   pre-print
This work presents the emergence of superconductivity in Ir - doped Weyl semimetal T_d - MoTe_2 with broken inversion symmetry. Chiral anomaly induced planar Hall effect and anisotropic magneto-resistance confirm the topological semimetallic nature of Mo_1-xIr_xTe_2. Observation of weak anisotropic, moderately coupled type-II superconductivity in T_d -Mo_1-xIr_xTe_2 makes it a promising candidate for topological superconductor.
arXiv:2108.10300v1 fatcat:6cybsqskrze4rjurxrsyflfncq

Multi-k-ic Depth Three Circuit Lower Bound

Neeraj Kayal, Chandan Saha
2016 Theory of Computing Systems  
A recent work by Kayal and Saha [11] uses the projected shifted partials measure to prove an exponential lower bound for depth three circuits with arbitrarily large formal degree but with somewhat low  ...  the constant 2 25 in the above theorem.) 2 The measure -evaluation dimension 8 'multiquadratic' sounds better here 9 The works of Grenet, Koiran, Portier, and Strozecki [4] and of Agrawal, Saha  ... 
doi:10.1007/s00224-016-9742-9 fatcat:2ll7reucvzcvpnraqdwthytvou

Jacobian hits circuits

Manindra Agrawal, Chandan Saha, Ramprasad Saptharishi, Nitin Saxena
2012 Proceedings of the 44th symposium on Theory of Computing - STOC '12  
We present a single common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT), that have been hitherto solved using diverse tools and techniques, over fields of zero or large characteristic. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models -depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas -can be
more » ... d using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely:
doi:10.1145/2213977.2214033 dblp:conf/stoc/AgrawalSSS12 fatcat:micc6dhlqrfztncc35ddkuv3qq

Give-and-take based peer-to-peer content distribution networks

2014 Sadhana (Bangalore)  
Content Distribution Networks (CDNs) are widely used to distribute data to large number of users. Traditionally, content is being replicated among a number of surrogate servers, leading to high operational costs. In this context, Peer-to-Peer (P2P) CDNs have emerged as a viable alternative. An issue of concern in P2P networks is that of free riders, i.e., selfish peers who download files and leave without uploading anything in return. Free riding must be discouraged. In this paper, we propose a
more » ... criterion, the Give-and-Take (G&T) criterion, that disallows free riders. Incorporating the G&T criterion in our model, we study a problem that arises naturally when a new peer enters the system: viz., the problem of downloading a 'universe' of segments, scattered among other peers, at low cost. We analyse this N P−hard problem, and characterize the optimal download cost under the G&T criterion. We propose an optimal algorithm, and provide a sub-optimal algorithm that is nearly optimal, but runs much more quickly; this provides an attractive balance between running time and performance. Finally, we compare the performance of our algorithms with that of a few existing P2P downloading strategies in use. We also study the computation time for prescribing the strategy for initial segment and peer selection for the newly arrived peer for various existing and proposed algorithms, and quantify cost-computation time trade-offs.
doi:10.1007/s12046-014-0266-1 fatcat:3c32pg6fe5cxrpgj3x327jw334

Fast Integer Multiplication Using Modular Arithmetic

Anindya De, Piyush P. Kurur, Chandan Saha, Ramprasad Saptharishi
2013 SIAM journal on computing (Print)  
We give an N · log N · 2 O(log * N ) time algorithm to multiply two N -bit integers that uses modular arithmetic for intermediate computations instead of arithmetic over complex numbers as in Fürer's algorithm, which also has the same and so far the best known complexity. The previous best algorithm using modular arithmetic (by Schönhage and Strassen) has complexity O(N · log N · log log N ). The advantage of using modular arithmetic as opposed to complex number arithmetic is that we can
more » ... ely evade the task of bounding the truncation error due to finite approximations of complex numbers, which makes the analysis relatively simple. Our algorithm is based upon Fürer's algorithm, but uses FFT over multivariate polynomials along with an estimate of the least prime in an arithmetic progression to achieve this improvement in the modular setting. It can also be viewed as a p-adic version of Fürer's algorithm.
doi:10.1137/100811167 fatcat:7jy5vrfat5gmjbuuhfeu3cm5xu

Lower Bounds for Depth-Three Arithmetic Circuits with small bottom fanin

Neeraj Kayal, Chandan Saha
2016 Computational Complexity  
reduction results of Agrawal and Vinay [1] and Koiran[14] and Tavenas[25] and using a complexity measure introduced inKayal [10], the work of Gupta, Kamath, Kayal and Saptharishi [8] and Kayal, Saha  ...  Subsequently, work by Kayal, Limaye, Saha and Srinivasan [12, 11] removed the restriction on the bottom fanin and obtained a n Ω( √ d) lower bound for homogeneous depth four circuits for a family of polynomials  ... 
doi:10.1007/s00037-016-0132-0 fatcat:s6mnfmxlizhzve5aq4xwit2aum

Continuous-time Zero-Sum Stochastic Game with Stopping and Control [article]

Chandan Pal, Subhamay Saha
2020 arXiv   pre-print
We consider a zero-sum stochastic game for continuous-time Markov chain with countable state space and unbounded transition and pay-off rates. The additional feature of the game is that the controllers together with taking actions are also allowed to stop the process. Under suitable hypothesis we show that the game has a value and it is the unique solution of certain dynamic programming inequalities with bilateral constraints. In the process we also prescribe a saddle point equilibrium.
arXiv:2006.01420v2 fatcat:w6mvc2zicfdwzk7dwy4fy4spji

Sulcus Vocalis: Our Experience

Soumitra Ghosh, Baisakhi Bakat, Abhishek Gupta, Sukamal Das, Chandan Saha, Barin K Roychaudhuri
2018 International Journal of Phonosurgery & Laryngology  
Aims and objectives: • Evaluate the incidence of sulcus vocalis. • To document and analyze the outcome of treatment of sulcus vocalis.
doi:10.5005/jp-journals-10023-1156 fatcat:7kiwgk2sybhbpf2g3ow67gbg6m

Square root Bound on the Least Power Non-residue using a Sylvester-Vandermonde Determinant [article]

Michael Forbes, Neeraj Kayal, Rajat Mittal, Chandan Saha
2011 arXiv   pre-print
We give a new elementary proof of the fact that the value of the least k^th power non-residue in an arithmetic progression {bn+c}_n=0,1..., over a prime field _p, is bounded by 7/√(5)· b ·√(p/k) + 4b + c. Our proof is inspired by the so called Stepanov method, which involves bounding the size of the solution set of a system of equations by constructing a non-zero low degree auxiliary polynomial that vanishes with high multiplicity on the solution set. The proof uses basic algebra and number
more » ... ry along with a determinant identity that generalizes both the Sylvester and the Vandermonde determinant.
arXiv:1104.4557v1 fatcat:vzusint3kzg6he5nhs7pprmhpy

A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

Chandan Saha, Ramprasad Saptharishi, Nitin Saxena
2012 Computational Complexity  
Polynomial identity testing (PIT) problem is known to be challenging even for constant depth arithmetic circuits. In this work, we study the complexity of two special but natural cases of identity testing -first is a case of depth-3 PIT, the other of depth-4 PIT. Our first problem is a vast generalization of: Verify whether a bounded top fanin depth-3 circuit equals a sparse polynomial (given as a sum of monomial terms). Formally, given a depth-3 circuit C, having constant many general product
more » ... ates and arbitrarily many semidiagonal product gates, test if the output of C is identically zero. A semidiagonal product gate in C computes a product of the form m · b i=1 e i i , where m is a monomial, i is an affine linear polynomial and b is a constant. We give a deterministic polynomial time test, along with the computation of leading monomials of semidiagonal circuits over local rings. The second problem is on verifying a given sparse polynomial factorization, which is a classical question (von zur Gathen, FOCS 1983): Given multivariate sparse polynomials f, g 1 , . . . , g t explicitly, check if f = t i=1 g i . For the special case when every g i is a sum of univariate polynomials, we give a deterministic polynomial time test. We characterize the factors of such g i 's and even show how to test the divisibility of f by the powers of such polynomials. The common tools used are Chinese remaindering and dual representation. The dual representation of polynomials (Saxena, ICALP 2008) is a technique to express a product-of-sums of univariates as a sum-ofproducts of univariates. We generalize this technique by combining it with a generalized Chinese remaindering to solve these two problems (over any field).
doi:10.1007/s00037-012-0054-4 fatcat:fj53gkc24fckpapkxbcl2obs6a

Evaluation of oxidative stress and the microenvironment in oral submucous fibrosis

Vertika Rai, Surajit Bose, Satadal Saha, Chandan Chakraborty
2019 Heliyon  
Oral Submucous fibrosis (OSF) is a chronic inflammatory mucosal disease of unknown etiology. Statistics show cases of OSF which has a high rate of overall prevalence and increase the chance of malignant transformation. As we know malignant cells is situated in a very complex microenvironment with altered metabolic pathway including intermediates which participate in oxidative stress process which enhances metabolic rewiring and promotes tumor progression. This study aims to evaluate the tumor
more » ... croenvironment and their role in metabolic reprogramming. This study was conducted on the serum sample of OSF (n = 20) compared to the healthy group (n = 20) using ELISA. The serum levels of intermediate by-products of metabolic pathway and oxidative stress induced biomolecular damage products were determined. The sensitivity of results was analyzed by correlating it with markers of metabolic status (Glucose, Total cholesterol, Total protein). Metabolic pathway intermediates molecules like Fatty Acids (FAA), Ascorbic acid, Citrate, Oxaloacetate (OAA), levels were significantly high in the serum of OSF cases. This indicated that intermediates act as a metabolic switch that drives cells to adapt malignant transformation pathway. Markers related to oxidative DNA damage (8-hydroxy-2' -deoxyguanosine), Oxidative lipid peroxidation (8-epi-Prostaglandin F2α), and Protein carbonyl were significantly up-regulated. This significant increase in oxidative stress marker revealed the reprogramming of the metabolic pathway for fulfilling the nutritional requirement of cancer cells. A further significant correlation was observed with metabolic products confirmed altered metabolic status. Our findings could identify the differentiating intermediate pathway metabolites and oxidative damage to biomolecules that are leading to rewiring of metabolism in the OSF group. Findings described in the study can be helpful to explain further the molecular aspects that lead to the progression of OSF towards carcinogenesis.
doi:10.1016/j.heliyon.2019.e01502 pmid:31011652 pmcid:PMC6462775 fatcat:qompr5fxxbbifmggva4iftwpoa
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