# Trading Time for Space in Prime Number Sieves

@inproceedings{Sorenson1998TradingTF, title={Trading Time for Space in Prime Number Sieves}, author={J. Sorenson}, booktitle={ANTS}, year={1998} }

A prime number sieve is an algorithm that finds the primes up to a bound n. We present four new prime number sieves. Each of these sieves gives new space complexity bounds for certain ranges of running times. In particular, we give a linear time sieve that uses only O(√n/(log log n)2) bits of space, an O l(n/ log log n) time sieve that uses O(n/((log n)l log log n)) bits of space, where l>1 is constant, and two super-linear time sieves that use very little space.

## 10 Citations

The Pseudosquares Prime Sieve

- Computer Science, MathematicsANTS
- 2006

The pseudosquares prime sieve is presented, which finds all primes up to n in sublinear time using very little space and the primes generated by the algorithm are proven prime unconditionally.

An improved sieve of Eratosthenes

- MathematicsMath. Comput.
- 2020

The sieve of Eratosthenes will be able to use it to factor integers, and not just to produce lists of consecutive primes, and also has close ties to Voronoi's work on the Dirichlet divisor problem.

A Fast Algorithm for Appoximately Counting Smooth Numbers

- MathematicsANTS
- 2000

An algorithm for estimating the value of Ψ(x,y) with a running time roughly proportional to \(\sqrt{y}\).

Dissecting a Sieve to Cut Its Need for Space

- Computer Science, MathematicsANTS
- 2000

A “dissected” sieving algorithm which enumerates primes in the interval [x 1, x 2], using \(O(x_{2}^{1/3})\) bits of memory and using arithmetic operations on numbers of \(O(\rm ln \it x_{2}\) bits.

Sieve algorithms for the discrete logarithm in medium characteristic finite fields. (Algorithmes de crible pour le logarithme discret dans les corps finis de moyenne caractéristique)

- Computer Science, Mathematics
- 2017

This thesis proposes and study two new sieve algorithms allowing us to treat any dimensions, with an emphasis on the three-dimensional case, and provides a complete implementation of the relation collection for some variants of the NFS in three dimensions.

Computational Number Theory and Applications to Cryptography Wyoming Summer School, June 19–July 7, 2006

- Computer Science, Mathematics
- 2006

The Greatest common divisor (GCD) algorithms are discussed, beginning with Euclid’s algorithm, and the extended Euclidean algorithm, which are followed by variations and improvements such as Lehmer's algorithm and FFT-based methods.

Generating Primes Using Partitions

- MathematicsArXiv
- 2015

This paper presents a new technique of generating large prime numbers using a smaller one by employing Goldbach partitions and shows how this method produces candidate prime numbers that are subsequently tested using either Miller Rabin or AKS primality tests.

Two Compact Incremental Prime Sieves

- Mathematics, Computer ScienceLMS Journal of Computation and Mathematics
- 2015

The rolling sieve is described, a practical, incremental prime sieve that takes $O(n\log\log n)$ time and $O(\sqrt{n}$ bits of space, and how to modify the sieve of Atkin and Bernstein (2004) to obtain a sieves that is simultaneously sublinear, compact, and incremental is shown.

Modern computer algebra

- Computer ScienceSIGA
- 2002

This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems.

## References

SHOWING 1-10 OF 14 REFERENCES

A Space-E cient Fast Prime Number Sieve

- Computer Science, Mathematics
- 1996

A new algorithm is presented that matches the running time of the best previous prime number sieve, but uses less space by a factor of (log n).

Two Fast Parallel Prime Number Sieves

- Computer Science, MathematicsInf. Comput.
- 1994

Two parallel prime number sieves for an algebraic EREW PRAM model of computation are presented and analyzed and it is found that the second sieve is more work-efficient when communication latency is significant.

A sublinear additive sieve for finding prime number

- Computer ScienceCACM
- 1981

A new algorithm is presented for the problem of finding all primes between 2 and N that improves on Mairson's sieve algorithm by using a dynamic sieve technique that avoids most of the nonprimes in the range 2 to N, and byUsing a tabulation method to simulate multiplications.

Fast Compact Prime Number Sieves (among Others)

- Mathematics, Computer ScienceJ. Algorithms
- 1983

The segmented sieve of eratosthenes and primes in arithmetic progressions to 1012

- Mathematics
- 1977

The sieve of Eratosthenes, a well known tool for finding primes, is presented in several algorithmic forms. The algorithms are analyzed, with theoretical and actual computation times given. The…

Analytic methods in the analysis and design of number-theoretic algorithms

- Computer Science, Mathematics
- 1985

This book makes a substantial contribution to the understanding of a murky area of number theory that is important to computer science, an area relevant to the design and analysis of number-theoretic…

Mathematics for the Analysis of Algorithms

- Mathematics
- 1999

Preface Binomial Identities.- Summary of Useful Identities.- Deriving the Identities.- Inverse Relations.- Operator Calculus.- Hypergeometric Series.- Identities with the Harmonic Numbers Recurrence…