Open problems in number theoretic complexity, II

  title={Open problems in number theoretic complexity, II},
  author={Leonard M. Adleman and Kevin S. McCurley},
Publisher Summary In the past decade, there has been a resurgence of interest in computational problems of a number theoretic nature. This period has been characterized by a growing awareness of the practical aspects of number theoretic computations and at the same time by an increased understanding of the relevance of deep theory to the problems that arise. This chapter presents a collection of 36 open problems in number theoretic complexity. Questions about the integers have natural… 

Algorithmic number theory-the complexity contribution

  • L. Adleman
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
A brief history of the symbiotic relationship between number theory and complexity theory will be presented and some of the technical aspects underlying 'modern' methods of primality testing and factoring will be described.

Detection of Primes in the Set of Residues of Divisors of a Given Number

An algorithm is presented which solves the problem of finding efficiently a small set of residues containing all residues coming from prime factors satisfying some natural conditions and it is shown that there are plenty of such numbers.

Algorithmic Number Theory and Its Relationship to Computational Complexity

A brief history of the symbiotic relationship between number theory and complexity theory will be presented, and some of the technical aspects underlying ‘modern’ methods of primality testing and factoring will be described.

A note on quadratic residuosity and UP

How to Recognize Whether a Natural Number is a Prime

The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.

Approximating the Number of Prime Factors Given an Oracle to Euler's Totient Function

In this work we devise the first efficient deterministic algorithm for approximating ω ( N ) – the number of prime factors of an integer N ∈ N , given in addition oracle access to Euler’s Totient

Deterministic factoring with oracles

This work revisits the problem of integer factorization with number-theoretic oracles, and shows that if N is a squarefree integer with a prime factor p, then it can be recovered in deterministic polynomial time given $ϕ(N)$.

A deterministic version of Pollard's p-1 algorithm

  • B. Zralek
  • Mathematics, Computer Science
    Math. Comput.
  • 2010
Applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms are presented and it is proved that oracle queries for values of $\phi$ are sufficient to completely factor any integer $n$ in less than deterministic polynomial time.

The Least Witness of a Composite Number

This paper is interested in obtaining upper bounds for w(n) without assuming the GRH, the Generalized Riemann Hypothesis, to be the least witness for a composite n.

Improved incremental prime number sieves

An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the first prime number sieve that is simultaneously sublinear, additive, and smoothly incremental: it



Explicit bounds for primality testing and related problems

Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits

Probabilistic Algorithms in Finite Fields

  • M. Rabin
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 1980
We present probabilistic algorithms for the problems of finding an irreducible polynomial of degree n over a finite field, finding roots of a polynomial, and factoring a polynomial into its

Algorithms in algebraic number theory

It is shown that the study of algorithms not only increases the understanding of algebraic number fields but also stimulates the curiosity about them.

Factoring integers with elliptic curves

This paper is devoted to the description and analysis of a new algorithm to factor positive integers that depends on the use of elliptic curves and it is conjectured that the algorithm determines a non-trivial divisor of a composite number n in expected time at most K( p)(log n)2.

Factorization of polynomials over finite fields.

Dickson [1, Ch. V, Th. 38] has given an interesting necessary condition for a polynomial over a finite field of odd characteristic to be irreducible. In Theorem 1 below, I will give a generalization

The Development of the Number Field Sieve

The number field sieve is an algorithm to factor integers of the form $r^e-s$ for small positive $r$ and $s$. The authors present a report on work in progress on this algorithm. They informally

Counting the Integers Factorable via Cyclotomic Methods

It is shown that the p − 1 and p + 1 integer factoring algorithms based on the first two cyclotomic polynomials each factor a positive proportion more integers in t steps than trial division but far fewer than the elliptic curve method.

Finding irreducible polynomials over finite fields

Irreducible polynomials in Fp[X] are used to carry out the arithmetic in field extension of Fp to solve the random polynomial time problem of finding irreducibles of any degree over Fp.

An application of higher reciprocity to computational number theory

The Higher Reciprocity Laws are considered to be among the deepest and most fundamental results in number theory. Yet, they have until recently played no part in number theoretic algorithms. In this