Tagged Questions

167 views

How to define the input of computable function or Turing machine over real numbers

Computation or computability over $\mathbb{N}$ can be extended to computation or computability over $\mathbb{R}$ or even computation or computability over $\mathbb{C}$.The following is a formal ...
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Computational complexity of solution of Pell equation and more

What is computational complexity for computing integral solution of Pell equation .It seems to be in P ,and could any one give an algorithm and reference for proof of it's complexity? And more,could ...
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Complexity of Membership-Testing for finite abelian groups

Consider the following abelian-subgroup membership-testing problem. Inputs: A finite abelian group $G=\mathbb{Z}_{d_1}\times\mathbb{Z}_{d_1}\ldots\times\mathbb{Z}_{d_m}$ with ...
278 views

Hermit H-machines

I call an H-machine a machine that can be connected to turing machines and that takes as input a natural integer n and instantly returns the n'th digit of the mathematical constant H. Is there a ...
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Finding colinear points in F_q^n

Forgive me if this is well known, it's not really my field, but it's a problem I've run across and thought about a bit. Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $n\ge2$, and let ...
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Reducing factoring prime products to factoring integer products (in average-case)

My question is about the equivalence of the security of various candidate one-way functions that can be constructed based on the hardness of factoring. (This question has been asked also in the CS ...
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Efficient computation of the least fraction with square denominator greater than the square root of 2.

The least rational number greater than $\sqrt{2}$ that can be written as a ratio of integers $x/y$ with $y\le10^{100}$ can be found in a moment using a little Python program. Can anyone write a ...
281 views

Computation for composition of polynomials

Let $R$ be a ring, $f(X)$ be a polynomial with coefficients in $R$ of degree $n$. It's known that for any $\alpha \in R$, one can evaluate $f$ at $\alpha$, i.e compute $f( \alpha)$ in $O(n)$ ...
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Walsh Fourier Transform of the Möbius function

This question is related to this previous question where I asked about ordinary Fourier coefficients. Special case: is MÃ¶bius nearly Orthogonal to Morse ! Harold Calvin Marston Morse (24 March ...
Suppose we have an odd composite $N$ and want to find numbers $a_1,\ldots,a_k$ such that each $a_i^2$, reduced mod $N$, is $b$-smooth. Of course we can use the quadratic sieve algorithm (minus the ...
It is not difficult to see that any reduced fraction $\frac{p}{q}$ where $0 < p < q$ and both $p$ and $q$ have at most $N$ digits (where $N$ is a fixed integer) can be reconstructed from its ...