arXiv subjects
Publications and source records attributed to Chandan Dubey.
Given a non-empty genus in $n$ dimensions with determinant $d$, we give a randomized algorithm that outputs a quadratic form from this genus. The time complexity of the algorithm is poly$(n,\log d)$; assuming Generalized Riemann Hypothesis (GRH).
An $n$-ary integral quadratic form is a formal expression $Q(x_1,...,x_n)=\sum_{1\leq i,j\leq n}a_{ij}x_ix_j$ in $n$-variables $x_1,...,x_n$, where $a_{ij}=a_{ji} \in \mathbb{Z}$. We present a poly$(n,k, \log p, \log t)$ randomized algorithm that given a quadratic form $Q(x_1,...,x_n)$, a prime $p$, a positive integer $k$ and an integer $t$, samples a uniform solution of $Q(x_1,...,x_n)\equiv t \bmod{p^k}$.
An $n$-ary integral quadratic form is a formal expression $Q(x_1,..,x_n)=\sum_{1\leq i,j\leq n}a_{ij}x_ix_j$ in $n$-variables $x_1,...,x_n$, where $a_{ij}=a_{ji} \in \mathbb{Z}$. We present a randomized polynomial time algorithm that given a quadratic form $Q(x_1,...,x_n)$, a prime $p$, and a positive integer $k$ outputs a $\mathtt{U} \in \text{GL}_n(\mathbb{Z}/p^k\mathbb{Z})$ such that $\mathtt{U}$ transforms $Q$ to its $p$-adic canonical form.
We give several improvements on the known hardness of the unique shortest vector problem. - We give a deterministic reduction from the shortest vector problem to the unique shortest vector problem. As a byproduct, we get deterministic NP-hardness for unique shortest vector problem in the $\ell_\infty$ norm. - We give a randomized reduction from SAT to uSVP_{1+1/poly(n)}. This shows that uSVP_{1+1/poly(n)} is NP-hard under randomized reductions. - We show that if GapSVP_γ\in coNP (or coAM) then uSVP_{\sqrtγ} \in coNP (coAM respectively). This simplifies previously known uSVP_{n^{1/4}} \in coAM proof by Cai \cite{Cai98} to uSVP_{(n/\log n)^{1/4}} \in coAM, and additionally generalizes it to uSVP_{n^{1/4}} \in coNP. - We give a deterministic reduction from search-uSVP_γto the decision-uSVP_{γ/2}. We also show that the decision-uSVP is {\bf NP}-hard for randomized reductions, which does not follow from Kumar-Sivakumar \cite{KS01}.
We give a polynomial time Turing reduction from the $γ^2\sqrt{n}$-approximate closest vector problem on a lattice of dimension $n$ to a $γ$-approximate oracle for the shortest vector problem. This is an improvement over a reduction by Kannan, which achieved $γ^2n^{3/2}$.