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Questions tagged [comp-number-theory]

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Algorithm to check whether a given set is Sidon

We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct. What algorithms do ...
Sayan Dutta's user avatar
1 vote
0 answers
59 views

Why does splitting $n$ bit integers into chunks of size $\log(n)$ specifically, help in multiplying them

In integer multiplication algorithms such as the Schonhage-Strassen algorithm (and the recently described Harvey and van der Hoeven algorithm), integers of size $n$ are reduced to polynomials with ...
Vighnesh Iyer's user avatar
3 votes
1 answer
262 views

Is modular square roots modulo primes in $NC$?

Assume modulus is prime. Is modular square roots then in $NC$? If not then, are there special primes or prime powers or numbers related to these (such as $2^k\pm i$ where $i\in\{-1,0,+1\}$) where it ...
Turbo's user avatar
  • 12.9k
0 votes
0 answers
45 views

Is this a proof that diophantine equation solutions can't be bounded by power towers?

From this 2017 paper on upper bounds for solutions to diophantine equations: Conjecture 1. If a system of equations S ⊆ Bn has exactly one solution in positive integers x1, . . . , xn , then x1, ....
ghosts_in_the_code's user avatar
1 vote
1 answer
236 views

Analytic Number theory in TCS [closed]

Are there any applications of analytic number theory in TCS?
nocitome's user avatar
7 votes
0 answers
123 views

Is finding square roots as hard as factoring, when coins are *public* and the square root oracle is adversarial?

Background: There is a well known argument (due to Rabin) that demonstrates that if one has access to an machine that computes square roots of elements in $\mathbb{Z}_n$, with $n = pq$, then $n$ can ...
Elle Najt's user avatar
  • 1,469
20 votes
1 answer
514 views

Is prime-counting function #P-complete?

Recall $\pi(n)$ the number of primes $\le n$ is the prime-counting function. By "PRIMES in P", computing $\pi(n)$ is in #P. Is the problem #P-complete? Or, perhaps, there is a complexity reason to ...
Igor Pak's user avatar
  • 812
14 votes
1 answer
1k views

Deciding whether an interval contains a prime number

What is the complexity of deciding whether an interval of the natural numbers contains a prime? A variant of the Sieve of Eratosthenes gives an $\tilde O(L)$ algorithm, where $L$ is the length of the ...
Elliot Gorokhovsky's user avatar
1 vote
0 answers
146 views

Fixed dimension Integer programming minus LLL in fixed parameter $NC$?

If you remove LLL part then is remaining part of a. Lenstra algorithm b. Barvinok algorithm in $O(f(n)(\log(mL))^c)$ time on $O(g(n)(mL)^c)$ processors with fixed $c>0$ in fixed $n$ dimension, $m$ ...
Turbo's user avatar
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3 votes
0 answers
79 views

How to charactorize computational complexity based on finding solution to algebraic equations? [closed]

The Matiyasevich/MRDP Theorem relates two notions:one from computability theory, the other from number theory, thus Turing Machine and algorithms finding integral solution to algebraic equations can ...
XL _At_Here_There's user avatar
1 vote
1 answer
337 views

Relation between transcendental numbers and computational complexity?

Regarding the relation, there is the Hartmanis-Stearns conjecture, but beyond Turing Machine of the realtime output, there is no further conjecture or theorem. Obviously irrational algebraic number ...
XL _At_Here_There's user avatar
6 votes
0 answers
322 views

Is there any algorithm outputing $e$ in real time?

The Hartmanis-Stearns Conjecture says that a number computed in real time by a Turing Machine is either rational or transcendental. We know that there is some transcendental (Liouville) number that ...
XL _At_Here_There's user avatar
4 votes
0 answers
195 views

What is the computational complexity of Hilbert Tenth problem over $\mathbb{Q_p}$

We know that Hilbert Tenth problem over $\mathbb{Q_p}$ is decidable,so what is the computational complexity of Hilbert Tenth problem over $\mathbb{Q_p}$? Is it equivalent to Tarski elimination ...
XL _At_Here_There's user avatar
10 votes
1 answer
352 views

Algorithm to compute distance between powers

Given coprime $a, b$, can you quickly compute $$ \min_{x, y > 0} |a^x - b^y| $$ Here $x, y$ are integers. Obviously taking $x = y = 0$ gives an uninteresting answer; in general how close can these ...
Gautam's user avatar
  • 261
6 votes
2 answers
290 views

Is there a fast algorithm to quickly evaluate $a^{b^c}$ mod $n$?

I need to quickly evaluate $a^{b^c} \mod n$ where $c$ is pretty big. Using the usual repeated squaring trick, this can be performed in $O(\log(b^c)) = O(c)$ time. In my problem, $c$ is huge, (say, $&...
Gautam's user avatar
  • 261
9 votes
2 answers
386 views

Any computable transcendental number that is computable in P time but not $O(n)$

Is there any known computable transcendental number such that its $n$th digit is computable in polynomial time, but not in $O(n)$?
XL _At_Here_There's user avatar
2 votes
0 answers
119 views

What is the complexity of computation of zero point by Rieman zeta function?

What is the complexity of computation of zero point by Rieman zeta function? That is, given s, whether ζ(s)=0? Is it in P? Any reference is appreciated.
XL _At_Here_There's user avatar
-2 votes
1 answer
707 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 ...
Muhammad Usman Qureshi's user avatar
3 votes
0 answers
82 views

Finding degree two subfield

Let $K=\frac{\mathbb{Q}[x]}{<f(x)>}$ where $f(x)$ is irreducible over $\mathbb{Q}$ and has even degree. I want to find $K_2$ such that $ \mathbb{Q} \subseteq K_2\subseteq K$ and $[K_2:\mathbb{Q}]...
xyz's user avatar
  • 243
10 votes
0 answers
230 views

Does a polynomial-time algorithm for factoring product of two primes imply a polynomial-time algorithm for factoring in general?

Is it known if the existence of a polynomial-time algorithm for the promise problem of factoring of numbers with two prime factors implies that factoring in general has a polynomial-time algorithm?
eof's user avatar
  • 101
7 votes
0 answers
173 views

Simplified lattices

Consider the following question: Let $N$ be some large prime number, and suppose we are given $n$ uniformly independent samples $g_i$ from $0...,N-1$. Think of $N$ as being exponentially large in $n$...
Lior Eldar's user avatar
  • 1,224
5 votes
1 answer
1k views

Computational complexity of modular power towers (tetration)

The complexity of modular addition is known: $g + p \mod N$ (for $|p| \approx |g| \approx |N|$) can be computed in $O(n = |N|)$. The complexity of modular multiplication is open though some results ...
Ross Snider's user avatar
  • 2,045
6 votes
1 answer
296 views

Integer factorization using polynomial whose roots are prime factors

Let $n$ be a square-free positive integer, let $n=p_{1}p_{2}\ldots p_{k}$ be the prime factorization of $n$ into $k$ distinct primes $p_{i}$. For such $n$, define $F_{n}(x)\triangleq\prod_{i=1}^{k}(x-...
Gorav Jindal's user avatar
8 votes
1 answer
855 views

Downward self-reducibility of factorization

Is integer factorization downward self-reducible? Is anything known about this?
Gorav Jindal's user avatar
4 votes
1 answer
144 views

Solving a system of sums-of-powers polynomials

What is the complexity of calculating the values of the integers $x_i$, where $0 \leq x_1 < x_2 < \dots < x_k < n$, given only the values $s_m = \sum_{i=1}^k x_i^m$? for $1 \leq m \leq k$? ...
jbapple's user avatar
  • 11.2k
1 vote
1 answer
119 views

What is necessary and/or sufficient requirement for a subring of a field to be computable? [closed]

As title asks, what is necessary and/or sufficient requirement for a subring of a field to be a computable ring? Conditions on either field or subring are fine.
Korpam's user avatar
  • 11
2 votes
0 answers
269 views

Sieve Methods for Twin Primes - How to extract algorithm from formula

I am reading Cojocaru and Murty's Introduction to Sieve Methods and their Applications. They wait until Chapter 5 to discuss the Sieve of Eratosthenes for finding primes - and their version of it is ...
john mangual's user avatar
2 votes
1 answer
290 views

n irrational number whose digits are pseudo-random: conceptual mismatch?

Are there irrational numbers whose digits are considered pseudo-random? Both concepts seem to be mismatched as a pseudo-random number generator typically is periodic and therefore generates a ...
mathersjj1's user avatar
13 votes
1 answer
881 views

Complexity class of this problem?

I am trying to understand to which complexity class the following problem belongs: Exponentiating Polynomial Root Problem (EPRP) Let $p(x)$ be a polynomial with $\deg(p) \geq 0$ with coefficients ...
Massimo Cafaro's user avatar
12 votes
3 answers
3k views

Using the de Bruijn sequence to find the $\lceil\log_2 v \rceil$ of an integer $v$

Sean Anderson published bit twiddling hacks containing the Eric Cole's algorithm to find the $\lceil\log_2 v \rceil$ of an $N$-bit integer $v$ in $O(\lg(N))$ operations with multiply and lookup. The ...
Yury Bayda's user avatar
6 votes
1 answer
279 views

Is the primality problem with unary input NLOGSPACE-Hard?

Consider the language $L=\{1^n : n \text{ is prime}\}$. Is this language NLOGSPACE-Hard? The motivation for this question is that $L$ is a good candidate for reducing to other languages related to my ...
Jonathan Mosheiff's user avatar
3 votes
1 answer
114 views

Complexity of higher order residues

Let $P$ be a prime. Given a number $a$, what is the computational complexity in establishing if $a$ is a cubic or higher order residue modulo $P$? Are there any good algorithms?
Turbo's user avatar
  • 12.9k
1 vote
1 answer
222 views

What algorithms do you know for beltway reconstruction? [closed]

I've faced the beltway reconstruction problem and I've developed a simple backtrack algorithm, what algorithms do you know for this problem? Beltway Reconstruction Problem: Assume there is a set of ...
Mahdi Khosravi's user avatar
6 votes
2 answers
195 views

Difference Sets

Suppose we have a set $$P=\{p_1,p_2,...,p_K\}$$ where $$1\leq p_k\leq N , k=1,...,K \qquad \& \quad p_k \in \mathbb{N} $$ and $p_k$'s are distinct. We calculate the differences as: $$d=p_i-p_j\mod ...
Mahdi Khosravi's user avatar
19 votes
4 answers
711 views

How to obtain the unknown values $a_i,b_j$ given an unordered list of $a_i-b_j\mod N$?

Can anyone help me with the following problem? I want to find some values $a_i,b_j$ (mod $N$) where $i=1,2,…,K, j=1,2,…,K $ (for example $K=6$), given a list of $K^2$ values that correspond to the ...
a guest's user avatar
  • 193
5 votes
0 answers
93 views

How short can reversible representations of the n-bit primes be?

For $\: 1 < n \:$ and $\: n^{o(1)} < \sigma \leq n \:$, $\:$ how small can $L$ be for there to be for there to be an efficiently computable (deterministic) function $\;\; f \: : \: \{0\hspace{....
user avatar
15 votes
1 answer
873 views

Is there a quantum NC algorithm for computing GCD?

From the comments on one of my questions on MathOverflow I get the feeling that the question regarding GCD being in $\mathsf{NC}$ vs. $\mathsf{P}$ is akin to the question regarding Integer ...
Turbo's user avatar
  • 12.9k
27 votes
2 answers
1k views

Complexity of factoring in number fields

What is known about the computational complexity of factoring integers in general number fields? More specifically: Over the integers we represent integers via their binary expansions. What is the ...
Gil Kalai's user avatar
  • 6,053
7 votes
1 answer
1k views

Discrete log in GL(2,p)

Let $p$ be a large prime. Let $A$ be a $2\times 2$ matrix with coefficients in $GF(p)$ (i.e., coefficients taken modulo $p$). Let $B=A^k$, where $k$ is an integer not given to us. Given $p$, $A$, ...
D.W.'s user avatar
  • 12.1k
16 votes
1 answer
737 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} ...
user834's user avatar
  • 2,806
19 votes
3 answers
2k views

Determinant modulo m

What are the known efficient algorithms for computing a determinant of an integer matrix with coefficients in $\mathbb{Z}_m$, the ring of residues modulo $m$. The number $m$ may not be prime but ...
Valeriy Sokolov's user avatar
13 votes
4 answers
3k views

Computing the Mobius function

The Mobius function $\mu(n)$ is defined as $\mu(1)=1$, $\mu(n)=0$ if $n$ has a squared prime factor, and $\mu(p_1 \dots p_k)= (-1)^k$ if all the primes $p_1,\dots,p_k$ are different. Is it possible to ...
Craig Feinstein's user avatar
7 votes
2 answers
294 views

Complexity class of phase information in Gauss sum

Have number theoretic functions such as Gauss sums been studied from a complexity view point? Where can I get a good introduction into complexity of Gauss sum estimation (beyond the quadratic case)? ...
v s's user avatar
  • 2,208
12 votes
0 answers
278 views

Conditional density of primes

We have some theorems about the density of prime numbers, the most famous one is probably the prime number theorem. My question is about the density of primes when we choose random numbers from a ...
Kaveh's user avatar
  • 21.7k