# Integer Factoring

@article{Lenstra2000IntegerF, title={Integer Factoring}, author={Arjen K. Lenstra}, journal={Designs, Codes and Cryptography}, year={2000}, volume={19}, pages={101-128} }

Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.

## 69 Citations

### A log-log speedup for exponent one-fifth deterministic integer factorisation

- MathematicsMath. Comput.
- 2022

Building on techniques recently introduced by the second author, and further developed by the first author, we show that a positive integer N may be rigorously and deterministically factored into…

### A Survey of Techniques Used in Algebraic and Number Theoretic Algorithms

- Computer Science, Mathematics
- 2005

This work surveys some of the important tools and techniques used in designing algorithms for problems in algebra and number theory and shows how they are applied to design algorithms for several basic algebraic and number theoretic operations.

### Computational methods in public key cryptology

- Computer Science, Mathematics
- 2002

These notes informally review the most common methods from computational number theory that have applications in public key cryptology.

### Zero-knowledge proofs technique using integer factorization for analyzing robustness in cryptography

- Mathematics, Computer Science2016 3rd International Conference on Computing for Sustainable Global Development (INDIACom)
- 2016

We have proved that zero-knowledge proofs technique using integer factorization problem has big-oh O(τ1/4)for factoring integers algorithm given by Pollard's rho in comparison with Henry for discrete…

### Quantum Algorithms for Integer Factorization

- Computer Science, Mathematics
- 2015

If IFP can be solved in polynomial-time, then RSA and many other cryptographic systems can be broken completely and efficiently.

### A time-space tradeoff for Lehman's deterministic integer factorization method

- Mathematics, Computer ScienceMath. Comput.
- 2021

This paper constructs a time-space tradeoff for Lawrence's generalization and applies it together with Lehman's result to obtain a deterministic integer factorization algorithm with runtime complexity $O(N^{2/9+o(1)})$.

### Factoring Based Cryptography

- Mathematics, Computer ScienceCybercryptography: Applicable Cryptography for Cyberspace Security
- 2018

This chapter discusses various factoring based cryptographic systems and protocols in the context of public-key cryptography.

### The Sum of Binomial Coefficients and Integer Factorization

- MathematicsIntegers
- 2016

It is proved that for any a coprime to n there exists a modulus r such that the combinatorial sum has a nontrivial greatest common divisor with n.

### A babystep-giantstep method for faster deterministic integer factorization

- Computer ScienceMath. Comput.
- 2018

This paper combines Strassen's approach with a babystep-giantstep method to improve the currently best known bound by a superpolynomial factor.

### A special purpose integer factorization algorithm

- Mathematics, Computer ScienceCCSEIT
- 2012

A special purpose factorization algorithm is proposed to find the factors of a composite number which is the product of two primes and the efficiency of the scheme is proved theoretically.

## References

SHOWING 1-10 OF 83 REFERENCES

### An Implementation of the Elliptic Curve Integer Factorization Method

- Mathematics, Computer Science
- 1995

This paper describes the second author’s implementation of the elliptic curve method for the factorization of integers as it is currently available in the computational algebra package Magma, which…

### A monte carlo method for factorization

- Computer Science
- 1975

A novel factorization method involving probabilistic ideas is described briefly, and it is suggested that this method should be considered as a viable alternative to traditional factorization methods.

### A method of factoring and the factorization of

- Mathematics
- 1975

The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is…

### The multiple polynomial quadratic sieve

- Computer Science
- 1987

A modification, due to Peter Montgomery, of Pomerance's Quadratic Sieve for factoring large integers is discussed along with its implementation, which enables one to factor numbers in the 60-digit range in about a day, using a large minicomputer.

### Reduction of Huge, Sparse Matrices over Finite Fields Via Created Catastrophes

- Computer ScienceExp. Math.
- 1992

We present a heuristic method for the reduction modulo 2 of a large, sparse bit matrix to a smaller, dense bit matrix that can then be solved by conventional means, such as Gaussian elimination. This…

### An Implementation of the Number Field Sieve

- Mathematics, Computer ScienceExp. Math.
- 1996

An implementation of the NFS is described, including the choice of two quadratic polynomials, both classical sieving and a special form of lattice sieving (line sieving), the block Lanczos method and a new square root algorithm.

### Solving Large Sparse Linear Systems over Finite Fields

- Computer ScienceCRYPTO
- 1990

It is shown that very large sparse systems can be solved efficiently by using combinations of structured Gaussian elimination and the conjugate gradient, Lanczos, and Wiedemann methods.

### Theorems on factorization and primality testing

- Computer ScienceMathematical Proceedings of the Cambridge Philosophical Society
- 1974

This paper is concerned with the problem of obtaining theoretical estimates for the number of arithmetical operations required to factorize a large integer n or test it for primality and uses a multi-tape Turing machine for this purpose.

### The lattice sieve

- Computer Science
- 1993

A possible improvement to the Number Field Sieve is described, which can reduce the time for the sieve stage by a factor comparable with log(B1), where much factoring takes place.

### A $p+1$ method of factoring

- Mathematics
- 1982

Let N have a prime divisor p such that p + 1 has only small prime divisors. A method is described which will allow for the determination of p, given N. This method is analogous to the p — 1 method of…