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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
Algorithm to multiply two numbers
In 1960, Anatoly Karatsuba discovered Karatsuba multiplication, unleashing a flood of research into fast multiplication algorithms. This method uses
Multiplication_algorithm
Algorithms which recursively solve subproblems
efficient algorithms for many problems, such as sorting (e.g., quicksort, merge sort), multiplying large numbers (e.g., the Karatsuba algorithm), finding
Divide-and-conquer_algorithm
Russian mathematician (1937–2008)
went to two editions, 1975 and 1983. The Karatsuba algorithm is the earliest known divide and conquer algorithm for multiplication and lives on as a special
Anatoly_Karatsuba
Recursive algorithm for matrix multiplication
multiplication algorithm multiplies two complex numbers using 3 real multiplications instead of 4 Toom-Cook algorithm, a faster generalization of the Karatsuba algorithm
Strassen_algorithm
finding algorithm Cipolla's algorithm Tonelli–Shanks algorithm Multiplication algorithms: fast multiplication of two numbers Karatsuba algorithm Schönhage–Strassen
List_of_algorithms
Algorithm for multiplying large numbers
additions and multiplication by small constants (Knuth, p. 296). The Karatsuba algorithm is equivalent to Toom-2, where the number is split into two smaller
Toom–Cook_multiplication
Multiplication algorithm
1971 until 2007. It is asymptotically faster than older methods such as Karatsuba and Toom–Cook multiplication, and starts to outperform them in practice
Schönhage–Strassen_algorithm
Overview of and topical guide to algorithms
test Modular exponentiation Fast Fourier transform Karatsuba algorithm Schönhage–Strassen algorithm Gaussian elimination LU decomposition QR decomposition
Outline_of_algorithms
Greatest integer less than or equal to square root
an example. The Karatsuba square root algorithm applies the same divide-and-conquer principle as the Karatsuba multiplication algorithm to compute integer
Integer_square_root
Method for division with remainder
efficient multiplication algorithm such as the Karatsuba algorithm, Toom–Cook multiplication or the Schönhage–Strassen algorithm. The result is that the
Division_algorithm
Number, approximately 3.14
They include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform-based methods. The Gauss–Legendre iterative algorithm: Initialize a
Pi
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
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
Calculations where numbers' precision is only limited by computer memory
{\displaystyle \mathbb {Z} } . Fürer's algorithm Karatsuba algorithm Mixed-precision arithmetic Schönhage–Strassen algorithm Toom–Cook multiplication Little
Arbitrary-precision arithmetic
Arbitrary-precision_arithmetic
Branch of elementary mathematics
multiplication algorithms with a low computational complexity to be able to efficiently multiply very large integers, such as the Karatsuba algorithm, the Schönhage–Strassen
Arithmetic
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
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
Decomposition of a number into a product
efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty
Integer_factorization
Algorithmic technique
Transm. 27, No.4, 339-360 (1991); translation from Probl. Peredachi Inf. 27, No.4, 76–99 (1991). Ekatherina Karatsuba. Fast Algorithms and the FEE method
Binary_splitting
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
Arithmetical operation
Multiplication algorithm Karatsuba algorithm, for large numbers Toom–Cook multiplication, for very large numbers Schönhage–Strassen algorithm, for huge numbers
Multiplication
Algorithmic runtime requirements for common math procedures
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
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
Exponent of a power of two
divide and conquer algorithms, such as the Karatsuba algorithm for multiplying n-bit numbers in time O(nlog2 3), and the Strassen algorithm for multiplying
Binary_logarithm
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
Pollard's_rho_algorithm
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
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
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
Prize in Economics winner Anatoly Karatsuba, developed the Karatsuba algorithm (the first fast multiplication algorithm) David Kazhdan, Soviet, American
List of Russian mathematicians
List_of_Russian_mathematicians
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
Anatoly Karatsuba, developed the Karatsuba algorithm (the first fast multiplication algorithm) Leonid Khachiyan, developed the Ellipsoid algorithm for linear
List_of_Russian_IT_developers
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
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 by
Williams's_p_+_1_algorithm
Analytic function in mathematics
S2CID 250796539. Karatsuba, A. A. (1996). "Density theorem and the behavior of the argument of the Riemann zeta function". Mat. Zametki (60): 448–449. Karatsuba, A
Riemann_zeta_function
Anatoly Karatsuba, developed the Karatsuba algorithm (the first fast multiplication algorithm) Leonid Khachiyan, developed the Ellipsoid algorithm for linear
List_of_Russian_scientists
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
converting NFA into DFA published by Michael O. Rabin and Dana Scott 1960 – Karatsuba multiplication 1961 – CRC (Cyclic redundancy check) invented by W. Wesley
Timeline_of_algorithms
Algorithm in computational number theory
kangaroo algorithm (also Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced
Pollard's_kangaroo_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
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
Multiplication: Multiplication algorithm — general discussion, simple methods Karatsuba algorithm — the first algorithm which is faster than straightforward
List of numerical analysis topics
List_of_numerical_analysis_topics
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
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
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
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
first personal computer MIR Anatoly Karatsuba, developed the Karatsuba algorithm (the first fast multiplication algorithm) Yevgeny Kaspersky, developer of
List_of_Russian_people
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
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
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
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
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
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
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
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
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
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
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
Mathematical operation on points on an elliptic curve
providing security just over 200 bits was proposed in which a variant of Karatsuba strategy was used to implement the field multiplication needed for the
Elliptic curve point multiplication
Elliptic_curve_point_multiplication
Conjecture on zeros of the zeta function
conjecture. The estimates of Selberg and Karatsuba can not be improved in respect of the order of growth as T → ∞. Karatsuba (1992) proved that an analog of the
Riemann_hypothesis
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
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
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
Problem of inverting exponentiation in groups
Index calculus algorithm Number field sieve Pohlig–Hellman algorithm Pollard's rho algorithm for logarithms Pollard's kangaroo algorithm (aka Pollard's
Discrete_logarithm
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
Finite-state machine whose output values are determined only by its current state
N.J.(1956). Karatsuba A. A. Solution of one problem from the theory of finite automata. Usp. Mat. Nauk, 15:3, 157–159 (1960). Karatsuba A. A. Experimente
Moore_machine
Constants of the mathematical zeta function
2006) below. A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba. In general, for negative
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
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
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
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
Division with remainder of integers
Haining Fan; Ming Gu; Jiaguang Sun; Kwok-Yan Lam (2012). "Obtaining More Karatsuba-Like Formulae over the Binary Field". IET Information Security. 6 (1):
Euclidean_division
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
Branch of pure mathematics
equation f ( x , y ) = 0 {\displaystyle f(x,y)=0} . Long 1972, p. 1. Karatsuba, A.A. (2020). "Number theory". Encyclopedia of Mathematics. Springer.
Number_theory
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 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
Fast summation method in mathematics
of series of a special form. It was constructed in 1990 by Ekaterina Karatsuba and is so-named because it makes fast computations of the Siegel E-functions
FEE_method
possible to adapt fast integer multiplication algorithms such as the Karatsuba and Toom-Cook algorithms to work with carry-less multiplications. The definition
Carry-less_product
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
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
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
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
polynomial Karatsuba multiplication Lenstra–Lenstra–Lovász lattice basis reduction algorithm (for polynomial factorization) Lindsey–Fox algorithm Remez algorithm
List_of_polynomial_topics
Mathematical problem in number theory
A. A. Karatsuba, "Trigonometric sums in number theory and analysis". Berlin–New-York: Walter de Gruyter, (2004). G. I. Arkhipov, A. A. Karatsuba, V. N
Waring's_problem
Mathematical lemma
square forms Trial division Shor's Multiplication Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's Euclidean division Binary Chunking
Bhaskara's_lemma
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
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
Variant of fast Fourier transform
accelerate eliptic curve cryptography over F(2521-1), the P-521. This is a Karatsuba-like technique featuring a cyclic convolution similar to IBDWT. For examples
Irrational base discrete weighted transform
Irrational_base_discrete_weighted_transform
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
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
Extension of the factorial function
A fast algorithm for calculation of the Euler gamma function for any algebraic argument (including rational) was constructed by E.A. Karatsuba. For arguments
Gamma_function
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
Russian mathematician (born 1939)
589.{{cite journal}}: CS1 maint: untitled periodical (link) Anatolii A. Karatsuba and Yu. P. Ofman (1962), "Умножение многозначных чисел на автоматах" ("Multiplication
Yuri_Ofman
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
Form of interpolation
essential to perform sub-quadratic multiplication and squaring, such as Karatsuba multiplication and Toom–Cook multiplication, where interpolation through
Polynomial_interpolation
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
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
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
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, approximately 0.916
Ramanujan, for the second formula. The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba. Using these series, calculating
Catalan's_constant
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
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
Boy/Male
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Elixir of the Truth
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Guardian of the Earth
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Earth; Heavenly; Celestial
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English : topographic name for someone who lived on a slope, from Middle English side ‘slope’ (Old English sīde), or a habitational name from Syde in Gloucestershire, named with this word. This name is also established in Ireland.
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Darling, dearly loved, from the Old english 'deorling'. Also a.
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
n.
Alt. of Algorithm
n.
The art of calculating with any species of notation; as, the algorithms of fractions, proportions, surds, etc.
n.
The art of calculating by nine figures and zero.