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Special-purpose algorithm for factoring integers
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
Pollard's_p_−_1_algorithm
Integer factorization algorithm
theory, Williams's p + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms. It was invented
Williams's_p_+_1_algorithm
Algorithm in computational number theory
and computational algebra, Pollard's kangaroo algorithm (also Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm
Pollard's_kangaroo_algorithm
Topics referred to by the same term
Several algorithms created by British mathematician John Pollard: Pollard's kangaroo algorithm Pollard's p − 1 algorithm Pollard's rho algorithm Pollard (coin)
Pollard
Integer factorization algorithm
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and its
Pollard's_rho_algorithm
Algorithm for public-key cryptography
for p and q is trivial. Furthermore, if either p − 1 or q − 1 has only small prime factors, n can be factored quickly by Pollard's p − 1 algorithm, and
RSA_cryptosystem
Topics referred to by the same term
single-stranded DNA as well as RNA Period 1 of the periodic table Pollard's p − 1 algorithm for integer factorization P-ONE - a proposed neutrino detector P1
P1
Dixon's algorithm Fermat's factorization method General number field sieve Lenstra elliptic curve factorization Pollard's p − 1 algorithm Pollard's rho algorithm
List_of_algorithms
Decomposition of a number into a product
Brent. Algebraic-group factorization algorithms, among which are Pollard's p − 1 algorithm, Williams' p + 1 algorithm, and Lenstra elliptic curve factorization
Integer_factorization
Type of computer science algorithm
In computer science, an in-place algorithm is an algorithm that operates directly on the input data structure without requiring extra space proportional
In-place_algorithm
Volunteer project using software to search for Mersenne prime numbers
to rapidly eliminate many Mersenne numbers with small factors. Pollard's p − 1 algorithm is also used to search for smooth factors. The variant of LL used
Great Internet Mersenne Prime Search
Great_Internet_Mersenne_Prime_Search
Freeware application to search for primes
factor. As of 2024, test candidates are mainly filtered using Pollard's p − 1 algorithm. Trial division is implemented, but Prime95 is rarely used for
Prime95
march algorithm developed by R. A. Jarvis 1973 – Hopcroft–Karp algorithm developed by John Hopcroft and Richard Karp 1974 – Pollard's p − 1 algorithm developed
Timeline_of_algorithms
British mathematician
John M. Pollard (born 1941) is a British mathematician who has invented algorithms for the factorization of large numbers and for the calculation of discrete
John_Pollard_(mathematician)
Mathematical algorithm
Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 to solve the discrete logarithm problem, analogous to Pollard's
Pollard's rho algorithm for logarithms
Pollard's_rho_algorithm_for_logarithms
Type of prime number
two strong primes. This makes the factorization of n = pq using Pollard's p − 1 algorithm computationally infeasible. For this reason, strong primes are
Strong_prime
Lucas–Lehmer test for Mersenne numbers AKS primality test Pollard's p − 1 algorithm Pollard's rho algorithm Lenstra elliptic curve factorization Quadratic sieve
List_of_number_theory_topics
Integer having only small prime factors
n-powersmooth numbers have applications in number theory, such as in Pollard's p − 1 algorithm and ECM. Such applications are often said to work with "smooth
Smooth_number
Prime pair of the form (p, 2p+1)
prevent the system being broken by some factorization algorithms such as Pollard's p − 1 algorithm. However, with the current factorization technology,
Safe and Sophie Germain primes
Safe_and_Sophie_Germain_primes
Algorithm for integer factorization
with n ≠ (1 or n), so when simplifying fails, a non-trivial divisor of n is found. Analogous to the two-stage variant of Pollard's p − 1 algorithm, Lenstra
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
Quantum search algorithm
In quantum computing, Grover's algorithm, also known as the quantum search algorithm, is a quantum algorithm for unstructured search that finds with high
Grover's_algorithm
Algorithm for computing greatest common divisors
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers
Euclidean_algorithm
On finding a repeating loop in a sequence
are possible. The classic example is Pollard's rho algorithm for integer factorization, which searches for a factor p of a given number n by looking for
Cycle_detection
Best results achieved to date
Signature Algorithm, and the elliptic curve cryptography analogues of these. Common choices for G {\displaystyle G} used in these algorithms include the
Discrete_logarithm_records
cryptosystem. This article covers algorithms to count points on elliptic curves over fields of large characteristic, in particular p > 3. For curves over fields
Counting points on elliptic curves
Counting_points_on_elliptic_curves
Problem of inverting exponentiation in groups
calculus algorithm Number field sieve Pohlig–Hellman algorithm Pollard's rho algorithm for logarithms Pollard's kangaroo algorithm (aka Pollard's lambda
Discrete_logarithm
Australian mathematician and computer scientist
of the Pollard rho algorithm. He later factored the tenth and eleventh Fermat numbers using Lenstra's elliptic curve factorisation algorithm. In 2002
Richard_P._Brent
Algorithm used by Google Search to rank web pages
PageRank (PR) is an algorithm used by Google Search to rank web pages in their search engine results. It is named after both the term "web page" and co-founder
PageRank
Approach to public-key cryptography
_{q}} . Because all the fastest known algorithms that allow one to solve the ECDLP (baby-step giant-step, Pollard's rho, etc.), need O ( n ) {\displaystyle
Elliptic-curve_cryptography
Digital signature scheme
In public-key cryptography, Edwards-curve Digital Signature Algorithm (EdDSA) is a digital signature scheme using a variant of Schnorr signature based
EdDSA
Probabilistic algorithm for computing discrete logarithms
In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms. Dedicated to the discrete
Index_calculus_algorithm
Attribute of machine learning models
The sample complexity of a machine learning algorithm represents the number of training-samples that it needs in order to successfully learn a target
Sample_complexity
Mathematical operation on points on an elliptic curve
exponentiation. The algorithm works as follows: To compute sP, start with the binary representation for s: s = s 0 + 2 s 1 + 2 2 s 2 + ⋯ + 2 n − 1 s n − 1 {\displaystyle
Elliptic curve point multiplication
Elliptic_curve_point_multiplication
Integer factorization algorithm
In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field
Rational_sieve
Scientific area at the interface between computer science and mathematics
division algorithm: for polynomials in several indeterminates Pollard's kangaroo algorithm (also known as Pollard's lambda algorithm): an algorithm for solving
Computer_algebra
HTTP/2 library in C
tools to decompress and compress using the HPACK header compression algorithm. nghttp3 is an implementation of HTTP/3 in C and authored by Tsujikawa
Nghttp2
Integer factorization algorithm
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field
Quadratic_sieve
developing the algorithm was to provide a test that primes would always pass and composites would pass with a probability of less than 1/7710. The test
Quadratic_Frobenius_test
Quality measure in cluster analysis
2004.10073. Van der Laan, Mark; Pollard, Katherine; Bryan, Jennifer (2003). "A new partitioning around medoids algorithm". Journal of Statistical Computation
Silhouette_(clustering)
Number divisible only by 1 and itself
factorization algorithms are known, they are slower than the fastest primality testing methods. Trial division and Pollard's rho algorithm can be used to
Prime_number
Type of cryptographic attack
contract, not just the fraudulent one. Pollard's rho algorithm for logarithms is an example for an algorithm using a birthday attack for the computation
Birthday_attack
Key agreement protocol
{\displaystyle x_{0}(tQ)} requires about O ( p 1 / 2 ) {\displaystyle O(p^{1/2})} time using the Pollards rho algorithm. The most famous example of Montgomery
Elliptic-curve_Diffie–Hellman
Graph with at most one cycle per component
applications in cryptography and computational number theory, as part of Pollard's rho algorithm for integer factorization and as a method for finding collisions
Pseudoforest
Central nervous system stimulant
[supplementation] with zinc is not integrated in any ADHD treatment algorithm. Scholze P, Nørregaard L, Singer EA, Freissmuth M, Gether U, Sitte HH (June
Amphetamine
In cryptography, XTR is an algorithm for public-key encryption. XTR stands for 'ECSTR', which is an abbreviation for Efficient and Compact Subgroup Trace
XTR
UK parliamentary by-election
May 2026. Rentoul, John (26 May 2026). "Reform vs Restore – how the 'algorithm election' could split the hard right for good". The Independent. Archived
2026_Makerfield_by-election
Password cracking dataset
and nine character NTLM passwords. A5/1 Brute-force attack DistrRTgen Pollard's kangaroo algorithm Oechslin, P. (2003). "Making a Faster Cryptanalytic
Rainbow_table
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
Hyperelliptic curve cryptography
Hyperelliptic_curve_cryptography
Function of four real variables that defines how light is reflected at an opaque surface
the optics of real-world light, in computer graphics algorithms, and in computer vision algorithms. The function takes an incoming light direction, ω i
Bidirectional reflectance distribution function
Bidirectional_reflectance_distribution_function
Peak finding software
Sean; Holloway, Alisha K.; Pollard, Katherine S. (2017-05-01). "Features that define the best ChIP-seq peak calling algorithms". Briefings in Bioinformatics
MACS_(software)
algorithm for fluids of fully anisotropic molecules" (PDF). The Journal of Chemical Physics. 103 (1): 481. Bibcode:1995JChPh.103..481L. doi:10.1063/1
Jaguar_(software)
American academic
who helped to develop the Smith-Waterman algorithm with Michael Waterman in 1981. The Smith-Waterman algorithm serves as the basis for multi sequence comparisons
Temple_F._Smith
Type of elliptic curve
have a small D. The following algorithm represents a doubling of a point P 1 = ( X 1 : Z 1 ) {\displaystyle P_{1}=(X_{1}:Z_{1})} on an elliptic curve in
Montgomery_curve
Calendar year with a day (or month) added
calendar, makes a further adjustment for the small error in the Julian algorithm; this extra leap day occurs in each year that is a multiple of 4, except
Leap_year
List of scientists who are Christians
for Scientists and Engineers. He specializes in sketching and streaming algorithms. Rosalind Picard (born 1962): professor of Media Arts and Sciences at
List of Christians in science and technology
List_of_Christians_in_science_and_technology
Computer file system architecture design
the same directory position. The checksum is calculated using the algorithm below. (pFCBName is a pointer to the name as it appears in a regular directory
Design_of_the_FAT_file_system
Method of deriving conclusions
§ 1. Introduction Grandy 1979, p. 122 Linnebo 2014, p. 123 Pollard 2015, p. 98 Moschovakis 2024, Lead section, § 1. Rejection of Tertium Non Datur Sider
Rule_of_inference
Card trick and probabilistic concept
Ergodic theory Geometric distribution Overlapping instructions Pollard's kangaroo algorithm Random walk Self-synchronizing code According to Diaconis & Graham
Kruskal_count
Applications of category theory
power of abstraction in order to prove new results or to develop new algorithms about the field. Samson Abramsky John C. Baez Bob Coecke Joachim Lambek
Applied_category_theory
Notion in supervised machine learning
the size of ε-nets, which determines the complexity of approximation algorithms based on them; range sets without finite VC dimension may not have finite
Vapnik–Chervonenkis_dimension
Writing systems for indigenous North American languages
asymmetric in form, it is derived from p- and therefore rotates 90° as p- does. Here is where the two algorithms to derive vowel orientations, which are
Canadian_Aboriginal_syllabics
American conservative non-profit media organization
gain "easy access to white supremacist logic." She also demonstrated an algorithmic connection on YouTube between PragerU, Fox News, and alt-right personalities
PragerU
Windows clients and Unix file server systems, and co-inventor of the rsync algorithm Jeff Waugh – software developer Aden Ekanayake – Australian rugby union
List of alumni of Barker College
List_of_alumni_of_Barker_College
Governor of Virginia from 2018 to 2022
policy, inmates were required to have a place to stay upon release. An algorithm-calculated recidivism rate is also used to determine whether an inmate
Ralph_Northam
Full-size luxury car manufactured by Cadillac
are mated to a 10-speed automatic transmission, which uses Performance Algorithm Shift programming to achieve the ideal shifting patterns. The CT6-V also
Cadillac_CT6
Unicode code point property names and their uses
interpreted by the algorithm: v t e Bidirectional character type (Bidi_Class Unicode character property)[1] In normal situations, the algorithm can determine
Unicode_character_property
Human protein-coding gene
2017-03-02. Brendel, V.; Bucher, P.; Nourbakhsh, I. R.; Blaisdell, B. E.; Karlin, S. (1992-03-15). "Methods and algorithms for statistical analysis of protein
C19orf67
Act of making research datasets available
proposed, including privacy protection algorithms, data ”masking” methods, and regional privacy level calculation algorithm. A large number of journals and publishers
Data_publishing
American computational biologist (born 1972)
resources". Bioinformatics. 17 (1): 83–94. doi:10.1093/bioinformatics/17.1.83. PMID 11222265. Siepel, A. C. (2003). "An algorithm to enumerate sorting reversals
Adam_C._Siepel
Unemployment caused by technological change
well-defined procedures that can easily be performed by sophisticated algorithms" are at risk of displacement. The study, published in 2013, shows that
Technological_unemployment
Application of computational algorithms, methods and programs to phylogenetic analyses
or phylogenetic inference focuses on computational and optimization algorithms, heuristics, and approaches involved in phylogenetic analyses. The goal
Computational_phylogenetics
Discrete probability distribution
Berkopec, Aleš (2007). "HyperQuick algorithm for discrete hypergeometric distribution". Journal of Discrete Algorithms. 5 (2): 341–347. doi:10.1016/j.jda
Hypergeometric_distribution
Quality of a musical note or sound or tone
playing pairs of sounds to listeners, then using a multidimensional scaling algorithm to aggregate their dissimilarity judgments into a timbre space. The most
Timbre
American computational biologist
Computer Science and Artificial Intelligence Laboratory. Her research spans algorithms, bioinformatics, genomics, structural biology, and genomic privacy; landmark
Bonnie_Berger
on its photos of the busy streets of Manhattan. The technology uses an algorithm to search Google's image database for faces and blurs them, according
Google_Street_View_coverage
currently best known discrete logarithm attack is the generic Pollard's rho algorithm, requiring about 2 122.5 {\displaystyle 2^{122.5}} group operations
FourQ
Inflammation of the brain
Sejvar J, Bitnun A, et al. (October 2013). "Case definitions, diagnostic algorithms, and priorities in encephalitis: consensus statement of the international
Encephalitis
American businessman and software engineer (born 1955)
keeping me young," he declared." "Google Form S-1 Registration Statement". EDGAR. August 18, 2004. p. 29. "Google Management: Eric Schmidt, Executive"
Eric_Schmidt
Distribution of variables which satisfies a stability property under linear combinations
algorithms improved by John P. Nolan. Mandelbrot, B. (1960). "The Pareto–Lévy Law and the Distribution of Income". International Economic Review. 1 (2):
Stable_distribution
Human retrovirus, cause of AIDS
human immunodeficiency virus type 1 western blots: seroconversion risk, specificity of supplemental tests, and an algorithm for evaluation". The Journal of
HIV
Swiss-American computer scientist
Experimental Algorithmics, and he remained editor in chief of the journal until 2003. In 2001, Moret founded the Workshop in Algorithms for Bioinformatics
Bernard_Moret
being, these were quickly applied to digital signal processing using algorithms for improving radar performance. Other advances in radar systems and applications
History_of_radar
Electronic device that alters audio
vibrations in a spring. Digital reverb effects use various signal processing algorithms to create the reverb effect, often by using multiple feedback delay circuits
Effects_unit
Method used to analyze protein interactions with DNA
Thomas R, Thomas S, Holloway AK, Pollard KS (2017). "Features that define the best ChIP-seq peak calling algorithms". Brief Bioinform. 18 (3): 441–450
ChIP_sequencing
Interdisciplinary branch of design and fine arts
future of graphic design. Easily-accessible computer software using AI algorithms will complete many practical tasks performed by graphic designers, allowing
Graphic_design
Subset of characters in Unicode
supported through Unicode’s flexible scripts, combining marks and collation algorithms. Writing system is sometimes treated as a synonym for "script". However
Script_(Unicode)
Continuous probability distribution, named after Benjamin Gompertz
H.; Dohi, T. (2009). "Gompertz software reliability model: estimation algorithm and empirical validation". Journal of Systems and Software. 82 (3): 535–543
Gompertz_distribution
British statute adopting the Gregorian calendar
p. 10. Poole (1921), pp. 13–16. Pollard (1940), p. 179. Journals of the House of Lords. Vol. 1. 1509–1577. Journals of the House of Commons. Vol. 1.
Calendar_(New_Style)_Act_1750
Theorem in physics
J.; Spekkens, Robert W. (2015-03-03). "The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations
Bell's_theorem
Film industry honor
for motion picture cinematographers, and to William B. Pollard for contributing new algorithms on which the programs are based Photography Hal Landaker
Academy Award for Technical Achievement
Academy_Award_for_Technical_Achievement
Extrapolation method to detect common ancestors
tree. P ( D | S , θ ) {\displaystyle P(D|S,\theta )} is the likelihood of the observed data which can be computed by Felsenstein's pruning algorithm as given
Ancestral_reconstruction
German computational biologist
Families of Structurally Similar Proteins (FSSP) database and the DSSP algorithm for assigning secondary structure to the amino acids of a protein, given
Chris_Sander_(scientist)
American computational biologist
genomics, bioinformatics and their interfaces with machine learning and algorithms. Singh was awarded a Presidential Early Career Award for Scientists and
Mona_Singh_(scientist)
Orbits where planets may have liquid surface water
How Homo sapiens will Conquer Space. Springer. p. 110. ISBN 978-3-319-17004-6. Williams, D.; Pollard, D. (2002). "Earth-like worlds on eccentric orbits:
Habitable_zone
Computational biologist
into X-ray crystallography data, facilitating phase replacement. The algorithm has been extensively used by other researchers to predict and gain insights
Debora_Marks
Computer filing system
blocks of media, usually separating key blocks and data blocks. Efficient algorithms can be developed with pyramid structures for locating records. Typically
File_system
(Ph.D. 1973) – computational linguist who developed the first stemming algorithm for word matching Antonio Martino (Ph.D. 1968) – professor of Economics
List of University of Chicago alumni
List_of_University_of_Chicago_alumni
Annual award
2009 Interview with N. Koblitz 2013 Award ceremony: J.-J. Quisquater and C. P. Schnorr: 2014 Award ceremony: B. Preneel: 2015 Award ceremony: I.B. Damgård
RSA Award for Excellence in Mathematics
RSA_Award_for_Excellence_in_Mathematics
American politician (born 1972)
social media platforms and prevent social media companies from feeding "algorithmically targeted" content to users under 17. Schatz said: "There is no good
Brian_Schatz
Obsolete device purported to detect lies
captures less physiological information than a polygraph, and uses an algorithm, not the judgment of a polygraph examiner, to render a decision whether
Polygraph
POLLARDS P-1-ALGORITHM
POLLARDS P-1-ALGORITHM
Surname or Lastname
North German form of Fries 1.Dutch
North German form of Fries 1.Dutch : variant of Frese.English : metonymic occupational name for a weaver of frieze, a coarse woolen cloth with a thick nap, Old French frise.
Surname or Lastname
English and Irish
English and Irish : according to MacLysaght, this is a surname of Dutch origin which was taken to Ireland early in the 18th century.French : from a personal name composed of the Germanic elements boll ‘friend’, ‘brother’ + hard ‘hardy’, ‘strong’.
Surname or Lastname
English
English : variant of Holland 1.Dutch : variant of Holland 2.Dutch : habitational name from places called Holland in northern France, named with Middle Dutch onland(e) ‘marsh’.
Surname or Lastname
Ukrainian, Jewish (from Ukraine), Polish, Serbian, and Hungarian (Cáp)
Ukrainian, Jewish (from Ukraine), Polish, Serbian, and Hungarian (Cáp) : from Ukrainian tsap ‘billy goat’, Polish cap, and so probably a nickname for someone thought to resemble the animal in some way or perhaps a metonymic occupational name for a goat herd.Czech (Čáp) : nickname for a tall or long-legged man, from Äáp ‘stork’.Southern French : from Occitan cap ‘head’ (Latin caput); probably a nickname for a person with something distinctive about his head. The word was often used in the metaphorical sense ‘chief’, ‘principal’, and the surname may also have denoted a leader or a village elder. In some cases it may also be a topographic name from the same word used in the sense of a promontory or headland.Americanized spelling of German Kapp.English : variant spelling of Capp.
Boy/Male
British, English
Shorn Head
Male
Hungarian
Hungarian form of English Philip, FÜLÖP means "lover of horses."
Surname or Lastname
English
English : nickname for a person with a large or unusually shaped head, from Middle English poll ‘head’ (Middle Low German polle ‘(top of the) head’) + the pejorative suffix -ard. The term pollard in the sense denoting an animal that has had its horns lopped is not recorded before the 16th century, and as applied to a tree the word is not recorded until the 17th century; so both these senses are almost certainly too late to have contributed to the surname.English : pejorative derivative of the personal name Paul. The surname has been established in Ireland since the 14th century.
Surname or Lastname
English
English : nickname from Middle English dull + -ard ‘dull or stupid person’. Compare Doll 5.Irish : either an importation to Ireland of the English name or, possibly, a reduced and altered form of de la Hyde (see Dollarhide).
Surname or Lastname
English and French
English and French : from the personal name Coll + the pejorative suffix -ard.
Girl/Female
Muslim
Serene, Tranquil (1)
Boy/Male
British, English, Teutonic
Short Haired
Boy/Male
Muslim
Deer name of a sahabi who p
Boy/Male
Shakespearean
King Henry IV, Part 1' Earl of March. Scroop.
Boy/Male
British, English
Shorn Head
Boy/Male
Indian
Deer name of a sahabi who p
Girl/Female
Australian, British, Danish, English, German
Mistress of All; Power of the Home; World Ruler; P
Surname or Lastname
English (Gloucestershire)
English (Gloucestershire) : from Middle English soler ‘solar’, ‘upper floor of a house’ (Old English solor), probably an occupational name for a servant whose duties were centered in the upper part of a house.
Surname or Lastname
Scottish spelling of Irish Morey 1.English and French
Scottish spelling of Irish Morey 1.English and French : from the personal name Amaury (see Morey 2).
Surname or Lastname
English
English : variant of Wolford.
Girl/Female
Muslim
Tall, Towering (1)
POLLARDS P-1-ALGORITHM
POLLARDS P-1-ALGORITHM
Female
Hebrew
(עלִיָּה) Variant spelling of Hebrew Aliya, AALIYAH means "to ascend, to go up."
Girl/Female
Tamil
Vishrutha | விஷà¯à®°à¯à®¤à®¾
Famous
Girl/Female
Indian
Gods name, King of beauty
Male
Iranian/Persian
Variant spelling of Persian Jamshid, possibly JAMSHED means "shining river."
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Lord Indra
Girl/Female
Arabic, Muslim
Mother of the Blessed
Female
Russian
(ÐатальÑ) Russian form of Latin Natalia, NATALYA means "birthday," or in Church Latin "Christmas day."Â
Boy/Male
Australian, Celtic, French, Gaelic, German, Irish
Little Champion
Girl/Female
Hindu
Girl/Female
Biblical
Bed-candle, changing.
POLLARDS P-1-ALGORITHM
POLLARDS P-1-ALGORITHM
POLLARDS P-1-ALGORITHM
POLLARDS P-1-ALGORITHM
POLLARDS P-1-ALGORITHM
p. pr. & vb. n.
of Pollard
n.
The doctrines or principles of the Lollards.
imp. & p. p.
of Pollard
p. a.
1 Having corners or angles.
v. t.
To lop the tops of, as trees; to poll; as, to pollard willows.