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A Las Vegas randomized algorithm is given to compute the Hermite normal form of a nonsingular integer matrix $A$ of dimension $n$. The algorithm uses quadratic integer multiplication and cubic matrix multiplication and has running time…

Given a full column rank $M \in \Z^{\ell \times m}$ and an $F \in \Z^{n \times m}$ we present an algorithm to compute the $n \times n$ basis in Hermite form of the integer lattice comprised of all rows $p \in \Z^{1 \times n}$ such that $pF…

Given a square, nonsingular matrix of univariate polynomials $\mathbf{F} \in \mathbb{K}[x]^{n \times n}$ over a field $\mathbb{K}$, we give a fast, deterministic algorithm for finding the Hermite normal form of $\mathbf{F}$ with complexity…

We present a variation of the modular algorithm for computing the Hermite Normal Form of an $\OK$-module presented by Cohen, where $\OK$ is the ring of integers of a number field K. The modular strategy was conjectured to run in polynomial…

The Hermite Normal Form (HNF) is a canonical representation of matrices over any principal ideal domain. Over the integers, the distribution of the HNFs of randomly looking matrices is far from uniform. The aim of this article is to present…

We present a variation of the modular algorithm for computing the Hermite normal form of an $\mathcal O_K$-module presented by Cohen, where $\mathcal O_K$ is the ring of integers of a number field $K$. An approach presented in (Cohen 1996)…

We consider the hash function $h(x) = ((ax+b) \bmod p) \bmod n$ where $a,b$ are chosen uniformly at random from $\{0,1,\ldots,p-1\}$. We prove that when we use $h(x)$ in hashing with chaining to insert $n$ elements into a table of size $n$…

Integer-forcing (IF) linear receiver has been recently introduced for multiple-input multiple-output (MIMO) fading channels. The receiver has to compute an integer linear combination of the symbols as a part of the decoding process. In…

Let R=F[D;sigma,delta] be the ring of Ore polynomials over a field (or skew field) F, where sigma is a automorphism of F and delta is a sigma-derivation. Given a an m by n matrix A over R, we show how to compute the Hermite form H of A and…

Squares (fragments of the form $xx$, for some string $x$) are arguably the most natural type of repetition in strings. The basic algorithmic question concerning squares is to check if a given string of length $n$ is square-free, that is,…

We introduce HBLLM, a wavelet-enhanced high-fidelity $1$-bit post-training quantization method for Large Language Models (LLMs). By leveraging Haar wavelet transforms to enhance expressive capacity through frequency decomposition, HBLLM…

In this paper, a polynomial-time algorithm is given to compute the generalized Hermite normal form for a matrix F over Z[x], or equivalently, the reduced Groebner basis of the Z[x]-module generated by the column vectors of F. The algorithm…

A linear time approximate maximum likelihood decoding algorithm on tail-biting trellises is prsented, that requires exactly two rounds on the trellis. This is an adaptation of an algorithm proposed earlier with the advantage that it reduces…

A fundamental problem in computer science is to find all the common zeroes of $m$ quadratic polynomials in $n$ unknowns over $\mathbb{F}_2$. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity…

In the k-mappability problem, we are given a string x of length n and integers m and k, and we are asked to count, for each length-m factor y of x, the number of other factors of length m of x that are at Hamming distance at most k from y.…

We consider the Abelian longest common factor problem in two scenarios: when input strings are uncompressed and are of size $n$, and when the input strings are run-length encoded and their compressed representations have size at most $m$.…

Recent research, such as BitNet, is paving the way for a new era of 1-bit Large Language Models (LLMs). In this work, we introduce a 1-bit LLM variant, namely BitNet b1.58, in which every single parameter (or weight) of the LLM is ternary…

Palindromes are strings that read the same forward and backward. The computation of palindromic structures within strings is a fundamental problem in string algorithms, being motivated by potential applications in formal language theory and…

We revisit a fundamental problem in string matching: given a pattern of length m and a text of length n, both over an alphabet of size $\sigma$, compute the Hamming distance between the pattern and the text at every location. Several…

We study the classic Text-to-Pattern Hamming Distances problem: given a pattern $P$ of length $m$ and a text $T$ of length $n$, both over a polynomial-size alphabet, compute the Hamming distance between $P$ and $T[i\, .\, . \, i+m-1]$ for…