A 𝑝+1 method of factoring

Williams, H. C.

doi:10.1090/s0025-5718-1982-0658227-7

Cited by 35 publications

(6 citation statements)

References 7 publications

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“…This is especially useful when s = 2, as highlighted in Section 7. The algorithm is similar in spirit to Williams' p + 1 factoring method [66], where the arithmetic of the norm 1 subgroup of k * is performed using Lucas sequences on a subfield of index 2 of k .…”

Section: Appendices a Rain's Conic Algorithmmentioning

confidence: 99%

Computing isomorphisms and embeddings of finite fields

Brieulle

Feo

Doliskani

et al. 2019

ACM Commun. Comput. Algebra

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Let F q be a finite field. Given two irreducible polynomials f, g over F q , with deg f dividing deg g, the finite field embedding problem asks to compute an explicit description of a field embedding of. When deg f = deg g, this is also known as the isomorphism problem.This problem, a special instance of polynomial factorization, plays a central role in computer algebra software. We review previous algorithms, due to Lenstra, Allombert, Rains, and Narayanan, and propose improvements and generalizations. Our detailed complexity analysis shows that our newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better.We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as open source. Our experiments show that our new variants consistently outperform available software.

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Section: Appendices a Rain's Conic Algorithmmentioning

confidence: 99%

Computing isomorphisms and embeddings of finite fields

Brieulle

Feo

Doliskani

et al. 2019

ACM Commun. Comput. Algebra

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“…The prime numbers p and q in RSA should also to be chosen with the properties that p±1 and q ±1 have at least one prime factor greater than 10 20 [321], otherwise, p could be found efficiently by using Pollard's "p − 1" factoring algorithm [242] and Williams' "p + 1" factoring algorithm [329].…”

Section: The "P ± 1" and Ecm Attacksmentioning

confidence: 99%

“…Note that there is a similar method to "p − 1", called "p + 1", proposed by H. C. Williams [329] in 1982. It is suitable for the case where N has a prime factor p for which p + 1 has no large prime factors.…”

Section: The "P ± 1" and Ecm Attacksmentioning

confidence: 99%

Cryptanalytic Attacks on RSA

Yan¹

2008

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“…His most spectacular example is the divisor p = 174463386657191516033932614401 0£2 740 + 1. He succeeded using the p -1 method because There is also a p + 1 algorithm (see [37]) which discovers factors p of N when all prime factors of of p + 1 are small. It uses Lucas sequences in place of powers of 2.…”

Section: Js Calls a Subroutine Rmul Is The One Which Multiplies Non-mentioning

confidence: 99%

Methods of factoring large integers

Wagstaff

Smith

1987

Number Theory

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o. Introduction. Several papers on factoring and primality testing justify these endeavors by quoting a pass age from Disquisitiones Arithmeticae in which Gauss says that they are important and useful. We agree with Gauss and explain in Section 1 our views on the importance and usefulness of factoring. Sections 2 and 3 describe two of the fastest general integer factoring algorithms, the continued fraction method and the quadratic sieve method. These are the two algorithms we considered in 1981 when we made our plans to build the first special processor for factoring large integers. In Section 4 we present this processor, which uses the continued fraction algorithm, and mention another processor now being designed to factor numbers via the quadratic sieve algorithm. Section 6 describes Lenstra's elliptic curve "algorithm. As a prelude to his method, in Section 5 we discuss briefly Pollard's p-1 algorithm, which is older but closely related to Lenstra's method. Section 7 summarizes the running times of the four factoring algorithms. Section 8 describes a special computer for the elliptic curve method. Good general references for the factoring methods we discuss-and some we omit-are [14], [29] and [38]. '" Work partially supported by NSF grants.

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A 𝑝+1 method of factoring

Cited by 35 publications

References 7 publications

Computing isomorphisms and embeddings of finite fields

Computing isomorphisms and embeddings of finite fields

Cryptanalytic Attacks on RSA

Methods of factoring large integers

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