# Open problems in number theoretic complexity, II

@inproceedings{Adleman1994OpenPI,
title={Open problems in number theoretic complexity, II},
author={Leonard M. Adleman and Kevin S. McCurley},
booktitle={ANTS},
year={1994}
}
• Published in ANTS 6 May 1994
• Mathematics
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…
64 Citations

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## References

SHOWING 1-10 OF 161 REFERENCES

### 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

• Computer Science
• 1993
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

• Mathematics
J. Algorithms
• 1995
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

• Mathematics, Computer Science
STOC '86
• 1986
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

• Mathematics
23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
• 1982
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