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2
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0answers
47 views

Bit complexity of factoring univariate polynomial over $\mathbb{Q}$ (rationals)

What is the bit complexity of finding all the irreducible factors $f_1, ..., f_r$ of a degree-$d$ polynomial $f(x) = \sum_{i=0}^d a_i\cdot x^i \in \mathbb{Q}[x]$ whose all coefficients are $B$-bit ...
3
votes
0answers
43 views

Equal degree factoring of homogeneous polynomials over $\Bbb Q[x_1,\dots,x_n]$?

Given $f(x_1,\dots,x_n)\in\Bbb Q[x_1,\dots,x_n]$ of form $\prod_{i=1}^df_i(x_1,\dots,x_n)$ where each of $f,f_i$ are homogeneous and each $f_i$ is irreducible what is the best technique to factor such ...
4
votes
0answers
111 views

Factorizing semiprime $n=pq$ with $p \mid q-1$

Could we find a fast integer factorization algorithm for any large semiprime $n=pq$, if we know that $p \mid q-1$?
7
votes
1answer
131 views

What's the complexity of factoring over a set of generators (say in $GL_2$)?

In particular, if I have some char-0 field $k$ (let's take $\mathbb C$ for now) and I consider $G = GL_2(k)$ with arbitrary nontrivial distinct $A, B \in G$. Then for some $C \in GL_2(k)$ do we know ...
1
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0answers
116 views

Some doubts on Hensel's Lifting Lemma for Bivariate polynomial factorization [closed]

In a Bivariate polynomial factorization we use Hensel's lifting lemma. Suppose we have a polynomial of two variables then we can approximate the roots of polynomial with the help of Hensel's lifting ...
5
votes
0answers
185 views

Circuit complexity class of polynomial factoring and Hensel lifting in Zassenhaus' algorithm?

Given a primitive polynomial (gcd of coefficients is $1$) in $\Bbb Z[x]$ we have a polynomial time factoring algorithm for this that runs in time polynomial in degree $d$ and number of bits in ...
23
votes
1answer
906 views

Is the 2016 implementation of Shor's algorithm really scalable?

In the 2016 Science paper "Realization of a scalable Shor algorithm" [1], the authors factor 15 with only 5 qubits, which is fewer than the 8 qubits "required" according to Table 1 of [2] and Table 5 ...
-3
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1answer
169 views

The factoring problem reduces to order finding or is it the other way around? [closed]

initially i was not at all equipped in theoretical computer science and knew only basics of number of theory. I started working from scratch on the age old problem of primality testing which led me to ...
3
votes
1answer
99 views

Factoring assuming smoothness of some numbers

I have came across a lot of factorization methods and most of them seem to assume smoothness of some numbers. For example When $p-1$ is smooth When $|E(\mathbb{F}_p)|$ is smooth. (Elliptic curve ...
17
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0answers
262 views

Does Factoring have a Statistical Zero Knowledge Proof?

The title should be pretty self-explanatory, but to be more precise, consider the decision version of factoring, which is given input $(x,k)$, where $x$ and $k$ are binary encodings of integers, to ...
2
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0answers
101 views

An NP-complete variant of factoring and relation to factoring [closed]

After reading this post An NP-complete variant of factoring. I come up with a question. To summerize the post, we have the factoring problem (F) which ask for a number $p$ that is prime and divides ...
5
votes
2answers
431 views

Alternatives to Diffie Hellman

Assume that Discrete logarithms can be solved in linear time over any group (hence factorization is also trivial by a result of Eric Bach), is there any other candidate public key exchange problem ...
12
votes
2answers
2k views

What is worst case complexity of number field sieve?

Given composite $N\in\Bbb N$ general number field sieve is best known factorization algorithm for integer factorization of $N$. It is a randomized algorithm and we get an expected complexity of $O\Big(...
-2
votes
1answer
130 views

A curious statement in an old blog

In http://blog.computationalcomplexity.org/2009/08/finding-primes.html, a statement is added which reads "Oddly enough we would usually prefer a probabilistic over the deterministic method to find ...
-3
votes
1answer
266 views

Consequences of polynomial time algorithm to variant of integer factorization

Given $N,U,V\in\Bbb N$ is there $n\in[U,V]\cap\Bbb N$ such that $n|N$ is $\mathsf{NP}$-complete modulo Cramer's conjecture on prime gaps is shown in An NP-complete variant of factoring. So supposing ...
1
vote
0answers
162 views

Efficient algorithm to find number of factors

Given an integer n, is there an efficient algorithm to find the number of factors of n?
2
votes
2answers
124 views

Factoring semiprimes whose factors very close to a power of two

Are there any factorization algorithms that run well on numbers $N = pq$ where $p,q$ are prime and $p = 2^b - k_p, q = 2^b - k_q$ for very small $k_p,k_q$? What about $p = 2^b + k_p, q = 2^b + k_q$ ...
18
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1answer
763 views

Why does Odlyzko improvement of Shor's Algorithm reduces the number of trials to $O(1)$

In his 1995 paper Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, Peter W. Shor discusses an improvement on the order-finding part of his ...
2
votes
1answer
145 views

Randomness versus Quantumness in factorization

The best deterministic factorization algorithm that is currently known runs in $O(N^{\frac{1}4+\epsilon})$ arithmetic steps. Randomness and quantumness improves upon this. I believe Quadratic/Number ...
3
votes
1answer
119 views

Algorithms for factorization using the decision version

Are there any practical algorithms for integer factorization that work by solving the decision version of the problem $\log N$ times to isolate a factor? Or is the decision version of only theoretical ...
2
votes
1answer
361 views

Continued Fraction Algorithm in Shor's Algorithm

I am just trying to make the final link of Shor's algorithm clear. Here $r$ is the order of $x$ modulo $N$. We have a number $\psi$, which for a rational number $\dfrac{s}{r}$ satisfies \begin{...
8
votes
2answers
762 views

Is square removal easier than factoring?

It seems to me that the square removal task can be reduced to the factoring task, but that there is no way to reduce factoring to square removal. Is there a way to make this "feeling" more precise, i....
2
votes
1answer
349 views

Complexity lower bound of finding the factorial of a number

I was wondering about the complexity of the factorial of a number mostly because this problem is not referenced in the complexity books I have read. Two similar problems, Matrix Multiplication and ...
75
votes
2answers
8k views

Was the reduction in Shor's algorithm originally discovered by Shor?

This is a "historical question" more than it is a research question, but was the classical reduction to order-finding in Shor's algorithm for factorization initially discovered by Peter Shor, or was ...
8
votes
1answer
251 views

Factoring low-degree polynomials

What is the fastest algorithm known for factoring polynomials with $n$ variables and total degree $\leq d$? Here, $n$ is growing and $d$ is fixed. Most work seem to consider the case when $d$ is ...
3
votes
1answer
324 views

Factoring with LLL when the form of the factors is given

Given a degree $2k$ reducible polynomial $$f(x)=\sum_{i=0}^{2k}a_ix^i\in\Bbb Z[x]$$ with $$\text{gcd}(a_{2k},\dots,a_0)=1$$ that is known to be of the form $f_1(x)f_2(x)$ with $\text{deg}\big(...
3
votes
1answer
676 views

What NP-complete problems are most similar to integer factoring?

The exact complexity of factoring integers (the decision problem) is a major open question in TCS (with important implications, especially in cryptography because of the RSA algorithm), and is widely ...
6
votes
1answer
200 views

Oracles which put integer factorization in P

I'm compiling a list of as many problems (decision or function) as I can find such that, if I had an oracle that could solve the problem in P, then integer factorization would also be in P. Here is a ...
-2
votes
1answer
339 views

Computational complexities in factoring

[Note: n is a given integer (not the number of its digits)] I'd like to know how O(sqrt(n)/log(n)) would compare against the computational complexity of the best available algorithms (as well as the ...
15
votes
0answers
421 views

Semiprime factorization, Groebner bases and a Nullstellensatz certificate

Suppose we have $N=pq$, with $p$ and $q$ are unknown odd primes. We can encode factorization problem in the system of polynomial equations. For instance, $p= 1+ \sum_{k=1}^n 2^k x_k$, $q= 1+ \sum_{k=1}...
16
votes
1answer
526 views

$(2^n)! = \sum_{k=0}^{m-1} a_k b_k^{c_k} $?

While reading Dick Lipton's blog, I stumbled across the following fact near the end of his Bourne Factor post: If, for every $n$, there exists a relation of the form $$ (2^n)! = \sum_{k=0}^{m-1} ...
1
vote
0answers
193 views

Estimating the number of operations required to factorize a $b$-bit integer using GNFS

You may skip to question According to Wikipedia, the complexity of factorizing a number n using the general number field sieve is ("heuristically"): $ \exp\left( \left(\sqrt[3]{\frac{64}{9}} + o(1)\...
7
votes
1answer
373 views

Layman Interpretation: Quantum Factoring Algorithm

I must firstly express that I know only a little about quantum computing and my knowledge comes largely from popular science texts and the media. So, I'm hoping that somebody will be able to help me ...
4
votes
1answer
237 views

Is it possible to design an efficient approximation algorithm for one NP-complete problem based on Shor's algorithm?

Is it possible to design an efficient approximation algorithm for an $\sf{NP\text{-}complete}$ problem based on reductions from Shor's algorithm? Are known any (classical) approximation algorithms ...
2
votes
2answers
403 views

Factorization Using Statistical Methods

We consider a number $N=AB$ where $A$ and $B$ are primes. Along the whole number-line from $1$ to $N$ we have two success points or target points: $A$ and $B$. If we had millions of target points the ...
39
votes
2answers
2k views

Are the problems PRIMES, FACTORING known to be P-hard?

Let PRIMES (a.k.a. primality testing) be the problem: Given a natural number $n$, is $n$ a prime number? Let FACTORING be the problem: Given natural numbers $n$, $m$ with $1 \leq m \leq n$, ...
30
votes
2answers
952 views

How hard is it to count the number of factors of an integer?

Given an integer $N$ of length $n$ bits, how hard is it to output the number of prime factors (or alternatively number of factors) of $N$? If we knew the prime factorization of $N$, then this would ...
22
votes
3answers
743 views

Adding integers represented by their factorization is as hard as factoring? Reference request

I'm looking for a reference for the following result: Adding two integers in the factored representation is as hard as factoring two integers in the usual binary representation. (I'm pretty sure ...
11
votes
1answer
346 views

Lower bounds on the period in integer factorization?

In 1975, Miller has shown how to reduce the factorization of integer $N$ to finding the period $r$ of a function $f(x)=a^x\;\bmod\;N$ such that $f(x+r)=f(x)$ with some randomly chosen $a<N$. It is ...
0
votes
2answers
281 views

How to calculate the cost of factoring a large integer?

I would like to know how much it would cost to factor a large integer. The cost can be given computer operations, time to process it or even monetary value. I know there are people that factored 200 ...
28
votes
5answers
3k views

Fast Reduction from RSA to SAT

Scott Aaronson's blog post today gave a list of interesting open problems/tasks in complexity. One in particular caught my attention: Build a public library of 3SAT instances, with as few variables ...
39
votes
3answers
3k views

Is the integer factorization problem harder than RSA factorization: $n = pq$?

This is a cross-post from math.stackexchange. Let FACT denote the integer factoring problem: given $n \in \mathbb{N},$ find primes $p_i \in \mathbb{N},$ and integers $e_i \in \mathbb{N},$ such that $...
13
votes
2answers
631 views

Reference for Levin's optimal factoring algorithm ?

In Manuel Blum's "Advice to a Beginning Graduate Student": LEONID LEVIN believes as I do that whatever the answer to the P=NP? problem, it won't be like anything you think it should be. And he has ...
34
votes
3answers
3k views

Consequences of Factoring being in P?

Factoring is not known to be NP-complete. This question asked for consequences of Factoring being NP-complete. Curiously, no one asked for consequences of Factoring being in P (maybe because such a ...
42
votes
3answers
3k views

An NP-complete variant of factoring.

Arora and Barak's book presents factoring as the following problem: $\text{FACTORING} = \{\langle L, U, N \rangle \;|\; (\exists \text{ a prime } p \in \{L, \ldots, U\})[p | N]\}$ They add, further ...
17
votes
5answers
913 views

P with integer factorization oracle

I just read the "Is integer factorization an NP-complete problem?" question ... so I decided to spend some of my reputation :-) asking another question $Q$ having $P(\text{Q is trivial}) \approx 1$: ...
14
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0answers
582 views

Reference request: a more complete “faster factorization into coprimes”

Some months ago, before the advent of "CS-Theory", I asked a question on MathOverflow about efficiently factoring an integer N into coprime factors n1 and n2, where n1 is a multiple of a given a ...
20
votes
2answers
1k views

Why isn't Montgomery modular exponentiation considered for use in quantum factoring?

It is well known that modular exponentiation (the main part of an RSA operation) is computationally expensive, and as far as I understand things the technique of Montgomery modular exponentiation is ...
7
votes
1answer
641 views

Permanents - Approximation and connection to integer factorization

Given a non-negative integer matrix of size $n \times n$. If the permanent can be calculated in $n^{2}$ arithmetic operations but each operation is on word size $n^{k}$ bits for some constant $k$, ...
24
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2answers
3k views