John Pollard's p − 1 algorithm is also used to search for larger factors.

** Pollard's p − 1 algorithm

The factors p and q of n should both be very large, around the same order of magnitude as the square root of n ; this makes trial division and Pollard's rho algorithm impractical.

The number may also be chosen so that none of p − 1, p + 1, q − 1, or q + 1 are smooth numbers, protecting against Pollard's p − 1 algorithm or Williams ' p + 1 algorithm.

* Pollard's p − 1 Method

B-smooth and B-powersmooth numbers have applications in number theory, such as in Pollard's p − 1 algorithm.

# redirect Pollard's p − 1 algorithm

* Pollard's p − 1 algorithm for integer factorization

It is analogous to Pollard's p − 1 algorithm.

If, this algorithm degenerates into a slow version of Pollard's p − 1 algorithm.

Based on Pollard's p-1 and Williams ' p + 1 factoring algorithms, Eric Bach and Jeffrey Shallit developed techniques to factor n efficiently provided that is has a prime factor p such that any k < sup > th </ sup > cyclotomic polynomial Φ < sub > k </ sub >( p ) is smooth.

** Pollard's kangaroo algorithm

** Pollard's rho algorithm

Similarly, there are simple randomized in-place algorithms for primality testing such as the Miller-Rabin primality test, and there are also simple in-place randomized factoring algorithms such as Pollard's rho algorithm.

All generic attacks on the discrete logarithm problem in finite abelian groups such as the Pohlig-Hellman algorithm and Pollard's rho method can be used to attack the DLP in the Jacobian of hyperelliptic curves.

Pollard's rho algorithm is a special-purpose integer factorization algorithm.

* ( this section discusses only Pollard's rho algorithm ).

# REDIRECT Pollard's rho algorithm

Pollard's rho algorithm for logarithms is an algorithm for solving the discrete logarithm problem analogous to Pollard's rho algorithm for solving the Integer factorization problem.

# REDIRECT Pollard's rho algorithm

( Listen here ) The record, which contains sound bites from Robert Kennedy and Chicago mayor Richard J. Daley, extolled Pollard's qualifications for the Oval Office: " Those who saw him as C. W. Moss / Know this hippie is really boss!

If the hyperelliptic curve is chosen with care, then Pollard's rho method is the most efficient way to solve DLP.

As late as 1933, just north of Pollard's Gulch on what is now Interstate 5, one could see a few old tumbledown buildings, one ancient log cabin, and some veteran apple trees — all that remained of Portuguese Flat.

Another significant early text is Peggy Pollard's 1941 play Beunans Alysaryn, modelled on the 16th-century saints ' plays.

This interpretation was enhanced by the High and Late Victorian historians, James Anthony Froude and A. F. Pollard, who saw Somerset as a champion of political liberty whose desire " to do good " was thwarted by, in Pollard's phrase, " the subtlest intriguer in English History ".

Sprout and Pollard marked the occasion by releasing simultaneous solo albums on the same day in 1996: Sprout's Carnival Boy and Pollard's Not in My Airforce, with each making a guest appearance on the other's album.

However, after another year of rigorous touring, the " Guided by Verde " lineup split in late 1997 following Pollard's announcement in an interview that he intended to work with other musicians on the next Guided by Voices project.

Later Pollard albums have been released on Merge and Pollard's own label, Guided by Voices Inc.

With Pollard's chest crushed by the weight of the fallen horse, and his ribs and arm broken, Howard had to find a new jockey.

Pollard's arguments are nowadays generally discredited by the revisionist school of history led by academics such as Christopher Haigh.

Sam Pollard's A-Hmao was adapted to write Lipo, another Lisoish language ( sometimes called Eastern Lisu ) spoken by the Lisu people.

* — Written by Samuel Pollard's son, a well-known professor of physics and biophysics.

* John Pollard's web site

Pollard's son Fritz Pollard, Jr. won the bronze medal for 110 m hurdles at the 1936 Summer Olympics in Berlin.

Pollard, still convalescing, and Seabiscuit recovered together at Howard's ranch, with Pollard's new wife Agnes, who had nursed him through his initial recovery.

In honor of the Pollard's, it was named the Pollard Activities Building in 1970.

" Sermon " 17, The Story of Sam Pollard, written for Atheists offers a very personal look at Sam Pollard's life and motivation

#* For example, William G. Pollard's studies in Physicist and Christian: A dialogue between the communities

In Pollard's appeal to United States Court of Appeals for the D. C.

Sprout, who was briefly featured in an early -' 80s version of the band, had re-joined circa Propeller and soon became Pollard's primary musical foil, in addition to contributing several of his own songs to the band's catalog.

The band surprised early audiences accustomed to the generally shambling, lo-fi and collage-like quality of the records with their energetic live show, featuring Pollard's homegrown rock theatrics ( consisting of karate-kicks, leaps, and Roger Daltrey-inspired mic-twirling ), Mitch Mitchell's windmilling and chain smoking, sometime bassist Greg Demos ' striped pants, a never-ending barrage of tunes that all seemed to clock in under 90 seconds, and prodigious alcohol consumption all around.

2000 was capped with the release of the massive Suitcase, a four-disc, 100-song trawl through three decades worth of Pollard's enormous reserve of unreleased material.

The 1998 – 1999 lineup of the band reunited for a few songs, for Pollard's 50th birthday in 2007.

A local doctor broke and reset Pollard's leg to aid his recovery, and slowly Red regained the confidence to sit on a horse.

Seymour Hersh wrote an article entitled " The Traitor " arguing against Pollard's release.