Pollard's p − 1 algorithm
Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm. The factors it finds are ones for which the number preceding the factor, p − 1, is powersmooth; the essential observation is that, by working in the multiplicative group modulo a composite number N, we are also working in the multiplicative groups modulo all of N's factors.
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- enPollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm. The factors it finds are ones for which the number preceding the factor, p − 1, is powersmooth; the essential observation is that, by working in the multiplicative group modulo a composite number N, we are also working in the multiplicative groups modulo all of N's factors.
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- enPollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm. The factors it finds are ones for which the number preceding the factor, p − 1, is powersmooth; the essential observation is that, by working in the multiplicative group modulo a composite number N, we are also working in the multiplicative groups modulo all of N's factors. The existence of this algorithm leads to the concept of safe primes, being primes for which p − 1 is two times a Sophie Germain prime q and thus minimally smooth. These primes are sometimes construed as "safe for cryptographic purposes", but they might be unsafe — in current recommendations for cryptographic strong primes (e.g. ), it is necessary but not sufficient that p − 1 has at least one large prime factor. Most sufficiently large primes are strong; if a prime used for cryptographic purposes turns out to be non-strong, it is much more likely to be through malice than through an accident of random number generation. This terminology is considered obsolete by the cryptography industry: ECM makes safe primes just as easy to factor as non-safe primes, so size is the important factor.
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- Algorithm
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- Pollard's p − 1 algorithm
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- enPollard's p − 1 algorithm
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- gitlab.inria.fr/zimmerma/ecm
- www.ams.org/bookpages/stml-68
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- Algebraic-group factorisation algorithm
- Algorithm
- ANSI X9.31
- Category:Integer factorization algorithms
- Fermat's little theorem
- Greatest common divisor
- Great Internet Mersenne Prime Search
- Integer
- Integer factorization
- John Pollard (mathematician)
- Lenstra elliptic-curve factorization
- Modular arithmetic
- MPrime
- Natural logarithm
- Necessary but not sufficient
- Number theory
- Obsolete
- Prime95
- Random number generation
- Safe prime
- Smooth number
- Sophie Germain prime
- Strong prime
- Williams's p + 1 algorithm
- SameAs
- Algorithme p-1 de Pollard
- Algoritmo p − 1 de Pollard
- m.02rzqd
- P−1-метод Полларда
- Pollardova p-1 metoda
- Pollard-p − 1-Methode
- Pollards p–1-methode
- Q1937853
- rRHf
- الگوریتم پی-۱ پولارد
- خوارزمية p - 1 لبولارد
- ขั้นตอนวิธีพีลบหนึ่งของพอลลาร์ด
- 폴라드의 P-1 알고리즘
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- Category:Integer factorization algorithms
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- Pollard's p − 1 algorithm?oldid=1113305485&ns=0
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- Wikipage page ID
- 578753
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- 1113305485
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