YouTube video by Tom Powell Jr

**Authors:** Daqing Wan We prove that exact Euclidean SVP is NP-hard under deterministic polynomial-time many-one reductions for full-rank cyclic integer lattices, equivalently full-rank ideals of $R_N:=\mathbb{Z}[X]/(X^N-1)$ in the coefficient norm. Hardness holds with $N=q-1$ for a varying odd prime $q$. As an application, we prove the same hardness for the algebraic class of NTRU-form lattices $\\{(x,z)\in R_N^2:Hx\equiv z\pmod{QR_N}\\}$, where $H,Q$ are unrestricted inputs. The decision problems are NP-complete, and the exact search problems are NP-hard under polynomial-time Turing reductions. No hardness claim is made for cryptographic NTRU parameter subclasses or key-generation distributions.

We introduce a new method for studying mod-$\ell$ congruences between eigenforms through the modular representation theory of $\mathrm{PGL}_2(\mathbb{F}_p)$. When $p\equiv \pm 1 \pmod{\ell}$, we use t...

We present a formal verification of Wolstenholme's theorem -- $\binom{2p}{p} \equiv 2 \pmod{p^3}$ for prime $p \geq 5$ -- in Lean~4 with Mathlib. The proof proceeds by expanding the shifted factorial ...

A Matching Vector ($\mathbf{MV}$) family modulo a positive integer $m\ge 2$ is a pair of ordered lists $U=(u_1,\ldots,u_K)$ and $V=(v_1,\ldots,v_K)$ with $u_i,v_j\in \Z_m^n$ such that $\langle u_i,v_i\rangle=0 \pmod m$ for every $i\in[K]$, while $\langle u_i,v_j\rangle\ne 0 \pmod m$ for every $i\ne j$. It is called $r$-restricted if the set of all inner products $\langle u_i,v_j\rangle$ has size at most $r$, including the diagonal value $0$. Restricted matching vector families are central to the matching-vector construction of constant-query subexponential locally decodable codes: an $r$-restricted family of size $K$ in $\Z_m^n$ gives an $r$-query matching-vector code with message length $K$ and codeword length $N=m^n$. The best previous bound tailored to the $3$-restricted case was $MV(m,n,3)\le \exp\bigl(O_m(n/\log n)\bigr)$ for fixed $m$, due to Bhowmick, Dvir and Lovett~\cite{bdl13}, who showed this bound under the polynomial Freiman Ruzsa conjecture, which was proved by Gowers, Green, Manners, and Tao~\cite{PFR24}. In this work we prove the subexponential upper bound \[ MV(m,n,3) \le \exp\bigl(O_m(\sqrt{n\log n})\bigr) \] for every fixed modulus $m$. Consequently, every $3$-query matching-vector code over a fixed modulus has codeword length \[ N \ge \exp\left(\Omega_m\left(\frac{(\log K)^2}{\log\log K}\right)\right). \] Our proof answers an explicit technical question left by the work of Aggarwal, Dutta, Li, Obremski, and Saraogi~\cite{ADLOS}: the entropy-growth method for sums of matching vectors can be pushed far beyond the initial constant-sum regime. We show that, for $3$-restricted families, entropy continues to grow for sums of length $L$ as large as $\Theta_m(\sqrt{n/\log n})$. The new ingredient is an exact-count rank bootstrap: a long-sum collision changes the exact number of one residue class seen from a suitable pivot, and exact-weight polynomials convert a disjoint collision family into a low-rank identity matrix.

We study pseudorandom bit generators for Boolean linear sums modulo a fixed integer $M$. A distribution $X\in\{0,1\}^n$ $\varepsilon$-fools these tests if, for every $a\in\mathbb{Z}_M^n$, the distribution of \[ \sum_{i=1}^n a_iX_i \pmod M \] is $\varepsilon$-close in statistical distance to the corresponding distribution under independent uniform bits. Lovett, Reingold, Trevisan, and Vadhan constructed generators with optimal seed length for every fixed prime-power modulus. We extend this result to every fixed modulus of the form \[ M=2^{t}\cdot p^{k}, \] where $t,k\ge 0$ and $p$ an odd prime. Specifically, we give an explicit generator with seed length \[ O_M\!\left(\log n+\log\frac{1}{\varepsilon}\right), \] which is optimal up to the constant depending on $M$. Our main contribution is a modulus-doubling reduction that transforms any generator fooling Boolean linear sums modulo $m$ into one fooling such sums modulo $2m$, while preserving the optimal asymptotic seed length.

YouTube video by Dr. Arsalan Aspires