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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
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
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
division algorithm: for polynomials in several indeterminates Pollard's kangaroo algorithm (also known as Pollard's lambda algorithm): an algorithm for solving
List_of_algorithms
British mathematician
improved by others. His discrete logarithm algorithms include the rho algorithm for logarithms and the kangaroo algorithm. He received the RSA Award for Excellence
John_Pollard_(mathematician)
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
Quantum algorithm for integer factorization
Shor's algorithm is a quantum algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor
Shor's_algorithm
Offset logarithmic integral pH Plethystic logarithm Pollard's kangaroo algorithm Pollard's rho algorithm for logarithms Polylogarithm Polylogarithmic function
Index_of_logarithm_articles
Password cracking dataset
character NTLM passwords. A5/1 Brute-force attack DistrRTgen Pollard's kangaroo algorithm Oechslin, P. (2003). "Making a Faster Cryptanalytic Time-Memory
Rainbow_table
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
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
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
Largest integer that divides given integers
|a|. This case is important as the terminating step of the Euclidean algorithm. The above definition is unsuitable for defining gcd(0, 0), since there
Greatest_common_divisor
Algorithm for integer multiplication
The Karatsuba algorithm is a fast multiplication algorithm for integers. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. It is a
Karatsuba_algorithm
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
Ancient algorithm for generating prime numbers
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking
Sieve_of_Eratosthenes
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
Schoof's algorithm Elliptic curve cryptography Baby-step giant-step Public key cryptography Schoof–Elkies–Atkin algorithm Pollard rho Pollard kangaroo Elliptic
Counting points on elliptic curves
Counting_points_on_elliptic_curves
Algorithm to multiply two numbers
multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient
Multiplication_algorithm
Integer factorization algorithm
to Pollard's p − 1 algorithm. In fact, it is also able to find p if p − 1 is smooth, in which case it degenerates into a slow version of Pollard's algorithm
Williams's_p_+_1_algorithm
Method for computing the relation of two integers with their greatest common divisor
and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common
Extended_Euclidean_algorithm
Algorithm checking for prime numbers
test and the cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena
AKS_primality_test
Algorithm in computational number theory
Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Multiplication algorithm
The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers, published by Arnold Schönhage and Volker Strassen
Schönhage–Strassen_algorithm
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
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
On finding a repeating loop in a sequence
number-theoretic algorithms are based on cycle detection, including Pollard's rho algorithm for integer factorization and his related kangaroo algorithm for the
Cycle_detection
Best results achieved to date
Challenge. To set a new record, they used their own software based on the Pollard Kangaroo on 256x NVIDIA Tesla V100 GPU processor and it took them 13 days. Two
Discrete_logarithm_records
Method for division with remainder
A division algorithm is an algorithm which, given two integers N and D (respectively the numerator and the denominator), computes their quotient and/or
Division_algorithm
Algorithm used in modular arithmetic
The Tonelli–Shanks algorithm (referred to by Shanks as the RESSOL algorithm) is used in modular arithmetic to solve for r in a congruence of the form r2
Tonelli–Shanks_algorithm
In computational number theory, Cipolla's algorithm is a technique for solving a congruence of the form x 2 ≡ n ( mod p ) , {\displaystyle x^{2}\equiv
Cipolla's_algorithm
Multiplication algorithm
ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algorithm as long multiplication after the multiplier and multiplicand
Ancient Egyptian multiplication
Ancient_Egyptian_multiplication
Efficient algorithm to count points on elliptic curves
Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography
Schoof's_algorithm
Algorithm for computing the greatest common divisor
The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, is an algorithm that computes the greatest common divisor
Binary_GCD_algorithm
Special-purpose integer factorization algorithm
number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special
Special_number_field_sieve
Method in number theory
In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials
Berlekamp–Rabin_algorithm
Fast greatest common divisor algorithm
GCD algorithm, named after D. H. Lehmer, is a fast GCD algorithm for multiple-precision arithmetic, which improves on the simpler Euclidean algorithm by
Lehmer's_GCD_algorithm
Algorithm for checking if a number is prime
exponentiation algorithm like binary or addition-chain exponentiation). The algorithm can be written in pseudocode as follows: algorithm lucas_primality_test
Lucas_primality_test
Probabilistic primality test
or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar
Miller–Rabin_primality_test
Algorithm for generating prime numbers
Sundaram is a variant of the sieve of Eratosthenes, a simple deterministic algorithm for finding all the prime numbers up to a specified integer. It was discovered
Sieve_of_Sundaram
Algorithm for integer factorization
elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
converse is not necessarily true. Grantham's stated goal when developing the algorithm was to provide a test that primes would always pass and composites would
Quadratic_Frobenius_test
Number-theoretic algorithm
In computational number theory, Cornacchia's algorithm is an algorithm for solving the Diophantine equation x 2 + d y 2 = m {\displaystyle x^{2}+dy^{2}=m}
Cornacchia's_algorithm
Pocklington's algorithm is a technique for solving a congruence of the form x 2 ≡ a ( mod p ) , {\displaystyle x^{2}\equiv a{\pmod {p}},} where x and
Pocklington's_algorithm
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
Factorization algorithm
the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity
General_number_field_sieve
Algorithm for solving the discrete logarithm problem
branch of mathematics, the baby-step giant-step is a meet-in-the-middle algorithm for computing the discrete logarithm or order of an element in a finite
Baby-step_giant-step
Primality test for certain numbers
based on the Lucas–Lehmer primality test. It is the fastest deterministic algorithm known for numbers of that form.[citation needed] For numbers of the form
Lucas–Lehmer–Riesel_test
System of rapid mental calculation
This article presents some methods devised by Trachtenberg. Some of the algorithms Trachtenberg developed are for general multiplication, division and addition
Trachtenberg_system
Methods to test or prove primality
Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin in the same year. The algorithm was altered and improved by several collaborators
Elliptic_curve_primality
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
Greatest integer less than or equal to square root
y {\displaystyle y} and k {\displaystyle k} be non-negative integers. Algorithms that compute (the decimal representation of) y {\displaystyle {\sqrt {y}}}
Integer_square_root
Probabilistic primality test
no value. Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log2n log log
Fermat_primality_test
Algorithm for computing logarithms
theory, the Pohlig–Hellman algorithm, sometimes credited as the Silver–Pohlig–Hellman algorithm, is a special-purpose algorithm for computing discrete logarithms
Pohlig–Hellman_algorithm
Standard division algorithm for multi-digit numbers
In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit numbers that is simple enough to perform by hand. It breaks
Long_division
Mathematical procedure
a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}=0.\,} An integer relation algorithm is an algorithm for finding integer relations. Specifically, given a set of real
Integer_relation_algorithm
Algorithm for determining whether a number is prime
Adleman–Pomerance–Rumely primality test is an algorithm for determining whether a number is prime. Unlike other, more efficient algorithms for this purpose, it avoids the
Adleman–Pomerance–Rumely primality test
Adleman–Pomerance–Rumely_primality_test
Integer factorization algorithm
most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer n
Trial_division
Korkine–Zolotarev (KZ) lattice basis reduction algorithm or Hermite–Korkine–Zolotarev (HKZ) algorithm is a lattice reduction algorithm. For lattices in R n {\displaystyle
Korkine–Zolotarev lattice basis reduction algorithm
Korkine–Zolotarev_lattice_basis_reduction_algorithm
Algorithms to generate prime numbers
In computational number theory, a variety of algorithms make it possible to generate prime numbers efficiently. These are used in various applications
Generation_of_primes
Integer factorization algorithm
x-y} will give a non-trivial factor of N {\displaystyle N} . A practical algorithm for finding pairs ( x , y ) {\displaystyle (x,y)} which satisfy x 2 ≡
Shanks's square forms factorization
Shanks's_square_forms_factorization
Exponentation in modular arithmetic
multiplicative inverse d of b modulo m (for instance by using extended Euclidean algorithm). More precisely: c = be mod m = d−e mod m, where e < 0 and b ⋅ d ≡ 1
Modular_exponentiation
Probabilistic primality test
composite return probably prime Using fast algorithms for modular exponentiation, the running time of this algorithm is O(k·log3 n), where k is the number
Solovay–Strassen primality test
Solovay–Strassen_primality_test
Algorithm to solve the discrete logarithm problem
In mathematics, the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has
Function_field_sieve
Algorithm in number theory
(also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method
Dixon's_factorization_method
Algorithm for generating prime numbers
In mathematics, the sieve of Atkin is a modern algorithm for finding all prime numbers up to a specified integer. Compared with the ancient sieve of Eratosthenes
Sieve_of_Atkin
Algorithm for generating prime numbers
In mathematics, the sieve of Pritchard is an algorithm for finding all prime numbers up to a specified bound. Like the ancient sieve of Eratosthenes,
Sieve_of_Pritchard
Test if a Mersenne number is prime
odd prime. The primality of p can be efficiently checked with a simple algorithm like trial division since p is exponentially smaller than Mp. Define a
Lucas–Lehmer_primality_test
Probabilistic primality testing algorithm
primality test is a probabilistic or possibly deterministic primality testing algorithm that determines whether a number is composite or is a probable prime.
Baillie–PSW_primality_test
Cyclic algorithm to solve indeterminate quadratic equations
The chakravala method (Sanskrit: चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly
Chakravala_method
factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning that it is suitable for factoring any integer
Continued fraction factorization
Continued_fraction_factorization
Algorithm for multiplying large numbers
introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers
Toom–Cook_multiplication
Algorithm for generating numbers coprime with first few primes
list of initial prime numbers constitute complete parameters for the algorithm to generate the remainder of the list. These generators are referred to
Wheel_factorization
Study of algorithms for performing number theoretic computations
mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating
Computational_number_theory
Mathematical lemma
Newton-Raphson Long Short SRT Discrete logarithm Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve Greatest common
Bhaskara's_lemma
Primality test for numbers of a certain form
in contrast to the probably prime results typical of other Monte Carlo algorithms such as the Miller-Rabin test. An approximate upper bound error probability
Proth's_theorem
Mathematical for factoring integers
made Euler's factorization method disfavoured for computer factoring algorithms, since any user attempting to factor a random integer is unlikely to know
Euler's_factorization_method
Primality test for Fermat numbers
F_{n}} by repeated squaring. This makes the test a fast polynomial-time algorithm. However, Fermat numbers grow so rapidly that only a handful of Fermat
Pépin's_test
Factorization method based on the difference of two squares
of the quadratic sieve and general number field sieve, the best-known algorithms for factoring large semiprimes, which are the "worst-case". The primary
Fermat's_factorization_method
Number-theoretic algorithm
Newton-Raphson Long Short SRT Discrete logarithm Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve Greatest common
Pocklington_primality_test
British government recognitions
Mont Albert, Victoria. For public service. Bernard Curren Masterson, of Kangaroo Flat, Victoria. For public service. Victor James Meehan, of Shelley, Western
1977 Silver Jubilee and Birthday Honours
1977_Silver_Jubilee_and_Birthday_Honours
POLLARDS KANGAROO-ALGORITHM
POLLARDS KANGAROO-ALGORITHM
Boy/Male
British, English
Shorn Head
Female
English
The origin of the American southern "Dixie" is uncertain; however, Louisiana dollars had the French word dix printed on them, DIXIE means "tenth," and this may have been what inspired the song about "the land of dixies," and later the name itself.
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 : nickname from Middle English loller ‘indolent fellow’, a derivative of lolle ‘to droop, dangle, or loll’.English : nickname from Middle English lollere ‘mumbler’, bestowed on a pious person or on a Lollard (a follower of the 14th-century religious reformer John Wyclif).
Surname or Lastname
English
English : variant of Wolford.
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
English
English : unexplained; possibly a variant of Dollard. The name was in VA by 1698.
Surname or Lastname
English
English : nickname for someone with close-cropped hair or a large head, Middle English bolling ‘pollard’, or for a heavy drinker, from Middle English bolling ‘excessive drinking’.German (Bölling) : from a pet form of a personal name formed with Germanic bald ‘bold’, ‘brave’ (see Baldwin).Swedish : either an ornamental name composed of Boll + the suffix -ing ‘belonging to’, or possibly a habitational name from a place named Bolling(e).
Surname or Lastname
German (of Slavic origin)
German (of Slavic origin) : from a pet form of the personal name Pavel or Paweł, respectively the Czech and Polish forms of Paul, or from a Sorbian cognate.German (of Slavic origin) : nickname for a small man, from Slavic palac ‘thumb’.Irish : MacLysaght ascribes the origin of this surname in Ireland to the arrival there in the 15th century of a Lombard family of bankers named de Palatio.English : from Old French palis, paleis ‘palisade’, ‘fence’, hence a topographic name for someone who lived by a palisade or a metonymic occupational name for a maker of fences.English : possibly a metonymic occupational name for someone who worked at a palace (bishop’s, archbishop’s, or royal), from Old French, Middle English palais, paleis.English : metonymic occupational name for a worker at a straw stack, from Old French paille ‘straw’ + Middle English hous ‘house’.Greek : ornamental name or nickname from Albanian pallë ‘sword’.Catalan (Pallà s) : variant spelling of Pallars, a regional name from the Catalan district of Pallars, in the Pyrenees.
Surname or Lastname
Catalan and Southern French (Rodés)
Catalan and Southern French (Rodés) : habitational name from any of several places named Rodés, mainly those in El Pallars and El Conflent districts, in northern Catalonia. This has the same origin as Occitan Rodés (Rodez in French), in Avairon department (southern France), which is first recorded in the 6th century in the Latin form Rutensis, apparently from the name of the Gaulish tribal name Ruteni.Catalan : variant of Roda, from Catalan rodes, the plural of roda ‘wheel’.English : variant of Rhodes.
Boy/Male
British, English, Teutonic
Short Haired
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 and French
English and French : from the personal name Coll + the pejorative suffix -ard.
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
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’.
Boy/Male
British, English
Shorn Head
POLLARDS KANGAROO-ALGORITHM
POLLARDS KANGAROO-ALGORITHM
Boy/Male
Egyptian American Greek Biblical Persian
Name of a pharaoh.
Male
English
French name, LEROY means "the king."Â In use by the English.
Boy/Male
Hindu
Girl/Female
Indian
To try, Desire
Female
Japanese
(鈴) Japanese name SUZU means "bell."
Boy/Male
Tamil
Lord Indra
Girl/Female
Tamil
Urjika | உரà¯à®œà¯€à®•à®¾
Boy/Male
Arabic
The Biblical Jesus is the English Language Equivalent; A Prophet's Name
Girl/Female
Spanish
Pure.
Boy/Male
Australian, Gaelic
Of the Strange Gauls
POLLARDS KANGAROO-ALGORITHM
POLLARDS KANGAROO-ALGORITHM
POLLARDS KANGAROO-ALGORITHM
POLLARDS KANGAROO-ALGORITHM
POLLARDS KANGAROO-ALGORITHM
v. t.
To lop the tops of, as trees; to poll; as, to pollard willows.
n.
genus of marsupials including the common kangaroo.
n.
A bollard timber. See under Bollard.
n.
A stag that has cast its antlers.
n.
The hare kangaroo.
n.
The doctrines or principles of the Lollards.
n.
Any one of numerous species of jumping marsupials of the family Macropodidae. They inhabit Australia, New Guinea, and adjacent islands, They have long and strong hind legs and a large tail, while the fore legs are comparatively short and feeble. The giant kangaroo (Macropus major) is the largest species, sometimes becoming twelve or fourteen feet in total length. The tree kangaroos, belonging to the genus Dendrolagus, live in trees; the rock kangaroos, of the genus Petrogale, inhabit rocky situations; and the brush kangaroos, of the genus Halmaturus, inhabit wooded districts. See Wallaby.
n.
A hornless animal (cow or sheep).
n.
A Lollard.
imp. & p. p.
of Pollard
n.
A small, leaping Australian marsupial of the genus Bettongia; the jerboa kangaroo.
n.
A fish, the chub.
n. pl.
Young cabbage, used as "greens"; esp. a kind cultivated for that purpose; colewort.
n.
Any small kangaroo belonging to Hypsiprymnus, Bettongia, and allied genera, native of Australia and Tasmania. Called also kangaroo rat.
n.
A species of kangaroo (Macropus Brunii), inhabiting New Guinea.
n.
A large male kangaroo.
n.
Any Australian kangaroo of the genus Petrogale, as the rock wallaby (P. penicillata).
n.
Alt. of Lollardy
p. pr. & vb. n.
of Pollard