Questions tagged [modular-arithmetic]
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Understanding Polynomial Rolling Hash Function by Modular Arithmetic
I was learning the Polynomial Hash function in python, the one used in Rabin Karp Algorithm
This is the implementation I was taught:
...
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Finding solution to Mv=v over $\mathbb{Z}$={0,1} for matrix M given a set linearly independent v
Under mod 2 arithmetic ($\mathbb{Z}$={0,1}), given a set $V$ of $n$x$1$ linearly independent vectors $\{x_1,...,x_n\}$ I'd like to find a $n^2$ binary matrix $M$ such that $Mv=v$ where $v \in V$ and $...
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Name of graph family defined by modular sum
In the context of finite, simple, undirected graphs, associate with each node $v\in V$ an integer $n(v)$ (you can limit this to positive integers without loss of generality). Create the set of edges ...
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$n \pmod k$ circuit
I wondered if, for a fixed integer $k ≥ 3$, how can I construct a circuit for each $n \in \mathbb{N}$, that takes as input an n-bit integer $x$ and outputs whether 3 divides $k$?
Considering an n-bit ...
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Algorithm for checking whether a set of hyperplanes covers $\mathbb{Z}_r^n$
In what follows, $r \in \mathbb{N}$ is not necessarily prime. $\mathbb{Z}_r$ is shorthand for $\mathbb{Z}/r\mathbb{Z}$.
Given a set of $h$ hyperplanes $A \vec x = b \mod r$, we can check whether the ...
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Floating-point modular multiplication algorithm
Is there a well-known algorithm for modular multiplication of floating-point numbers?
I would like to multiply some large angle in single precision (6-7 significant digits) and wrap it back to 360 ...
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Within the set of signed integers representable by a bit string of length n, are any two elements equivalent to each other mod 2^n?
Donald Knuth's The Art of Computer Programming, Volume 1 Fascicle 1 contains the following exercise:
If $\alpha$ is any string of 0s and 1s, let $\operatorname{s}(\alpha)$ and $\operatorname{u}(\...
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Are there any mathematical properties of consecutive integer power modulo operations that could be exploited for algorithmic speed gain?
I'm attempting to search through all the integers between 10^15 and 10^16 to check if they are in the oeis sequence A277274, and the entirety of the program can be summarized as mostly equivalent to :
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Understanding Rabin-Karp's rolling hash computation
Possibly related to this. Let $T$ be the text and $n$ be the length of the pattern. I understand that if substrings of $T$ are interpreted as base-$d$ numbers where $d$ is the alphabet's size, then ...
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Universal class $\mathcal{H}_{p, m}$ of hash functions has $p(p-1)$ members
In CLRS it is stated that the class $\mathcal{H}_{p, m} = \{ h_{ab}:\mathbf{Z}_p \to \mathbf{Z}_m \mid a \in \mathbf{Z}_p^*, b \in \mathbf{Z}_p\}$, $h_{ab}(x) = (ax+b) \mod p \mod m$, $m < p$ prime ...
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Fastest algorithm for polynomial multiplication in 256-bit finite fields
I am looking for the fastest algorithm (in practice) to multiply two polynomials $f(X)\cdot h(X)$ in $\mathbb{F}_p[X]$. The prime $p$ is roughly $256$ bits but the integer $p-1$ might not have any ...
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Efficient algorithm to "lift" a number in CRT representation mod r to mod $r^2$
Integers between 0 and a square-free number $r$ minus one can be represented by their value modulo each of $r$'s prime factors, according the Chinese remainder theorem.
Given a number represented like ...
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Does the reliability of polynomial hashing depend on whether the modulus is prime, for coprime base and modulus?
A polynomial hash of a string $s$ with base $b$ and modulus $M$ is defined as
$$
H(s) = (s_0 + s_1 b + s_2 b^2 + \dots + s_{n-1} b^{n-1}) \mod M.
$$
I have proven (and this is quite obvious) that ...
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upper bound on the smallest modulus for perfect hashing of a Huffman tree
Given a full binary tree with 256 leaves and depth <= 64,
let H be the set of Huffman codes described by the tree (using 0 to go left, and 1 to go right, where ...
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Find array of coprime integers whose average is maximized
I am creating a class to store large integers in a residue number system. I want each "integer" to be 4-12GB in size and be comprised of 64-bit moduli. These moduli must be pairwise coprime ...
