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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
Algorithm for computing greatest common divisors
mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest
Euclidean_algorithm
Largest integer that divides given integers
This again gives gcd(48, 18) = 6. The binary GCD algorithm is a variant of Euclid's algorithm that is specially adapted to the binary representation of
Greatest_common_divisor
Topics referred to by the same term
Look up gcd in Wiktionary, the free dictionary. GCD may refer to: Greatest common divisor Binary GCD algorithm Polynomial greatest common divisor Lehmer's
GCD
Bowyer–Watson algorithm: create voronoi diagram in any number of dimensions Fortune's Algorithm: create voronoi diagram Binary GCD algorithm: Efficient way
List_of_algorithms
Two numbers without shared prime factors
coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm. The number of integers coprime
Coprime_integers
Algorithmic runtime requirements for common math procedures
"Two Fast GCD Algorithms". Journal of Algorithms. 16 (1): 110–144. doi:10.1006/jagm.1994.1006. Crandall, R.; Pomerance, C. (2005). "Algorithm 9.4.7 (Stehlé-Zimmerman
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Numbers obtained by adding the two previous ones
That is, gcd ( F n , F n + 1 ) = gcd ( F n , F n + 2 ) = gcd ( F n + 1 , F n + 2 ) = 1 {\displaystyle \gcd(F_{n},F_{n+1})=\gcd(F_{n},F_{n+2})=\gcd(F_{n+1}
Fibonacci_sequence
Quadratic homogeneous polynomial in two variables
many classes of binary quadratic forms with discriminant D. Their number is the class number of discriminant D. He described an algorithm, called reduction
Binary_quadratic_form
On finding a repeating loop in a sequence
In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any
Cycle_detection
Use of functions that call themselves
The Euclidean algorithm, which computes the greatest common divisor of two integers, can be written recursively. Function definition: gcd ( x , y ) = {
Recursion_(computer_science)
Method in number theory
O(n^{2}\log p)} . Binary exponentiation works in O ( n 2 log p ) {\displaystyle O(n^{2}\log p)} . Taking the gcd {\displaystyle \gcd } of two polynomials
Berlekamp–Rabin_algorithm
Family of related bitwise operations on machine words
Gosper's loop-detection algorithm, which can find the period of a function of finite range using limited resources. The binary GCD algorithm spends many cycles
Find_first_set
Property of a mathematical operation
common multiple functions act associatively. gcd ( gcd ( x , y ) , z ) = gcd ( x , gcd ( y , z ) ) = gcd ( x , y , z ) lcm ( lcm ( x , y )
Associative_property
Concept in modular arithmetic
multiplicative inverse modulo m, this gcd must be 1. The last of several equations produced by the algorithm may be solved for this gcd. Then, using a method called
Modular multiplicative inverse
Modular_multiplicative_inverse
Decomposition of a number into a product
factorization of Δ and by taking a gcd, this ambiguous form provides the complete prime factorization of n. This algorithm has these main steps: Let n be
Integer_factorization
Algorithm for fast modular multiplication
division by R is easy, significantly improving the speed of the algorithm. In binary computers, R is always a power of two, since division by powers of
Montgomery modular multiplication
Montgomery_modular_multiplication
Property of operations
{\displaystyle x\in \{0,1\}} . In a GCD domain (for instance in Z {\displaystyle \mathbb {Z} } ), the operations of GCD and LCM are idempotent. In a Boolean
Idempotence
Digital signature scheme
Rabin signature algorithm is a method of digital signature originally published by Michael O. Rabin in 1979. The Rabin signature algorithm was one of the
Rabin_signature_algorithm
Algorithm checking for prime numbers
the binary logarithm, and φ ( r ) {\displaystyle \varphi (r)} is Euler's totient function of r. Step 3 is shown in the paper as checking 1 < gcd(a,n)
AKS_primality_test
Integer factorization algorithm
= gcd ( 194 , 1649 ) ⋅ gcd ( 34 , 1649 ) = 97 ⋅ 17 {\displaystyle 1649=\gcd(194,1649)\cdot \gcd(34,1649)=97\cdot 17} using the Euclidean algorithm to
Quadratic_sieve
Error correction code
popular algorithms for this task are: Peterson–Gorenstein–Zierler algorithm Berlekamp–Massey algorithm Sugiyama Euclidean algorithm Peterson's algorithm is
BCH_code
Condition under which an odd prime is a sum of two squares
{\displaystyle 2\leq a\leq p-2} the gcd of a {\displaystyle a} and p {\displaystyle p} may be expressed via the Euclidean algorithm yielding a unique and distinct
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Integer factorization algorithm
{\displaystyle R_{j}=R_{j-1}.} [citation needed] Then if f = gcd ( N , R j ) {\displaystyle f=\gcd(N,R_{j})} is not equal to 1 {\displaystyle 1} and not equal
Shanks's square forms factorization
Shanks's_square_forms_factorization
Probabilistic primality test
''probably prime''. T When n is odd and composite, at least half of all a with gcd(a,n) = 1 are Euler witnesses. We can prove this as follows: let {a1, a2,
Solovay–Strassen primality test
Solovay–Strassen_primality_test
Methods to test or prove primality
{p}}+1\right)^{2}\leq \left({\sqrt[{4}]{N}}+1\right)^{2}<q} and thus gcd ( q , m p ) = 1 {\displaystyle \gcd(q,m_{p})=1} and there exists an integer u with the property
Elliptic_curve_primality
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
Division with remainder of integers
Presently, most division algorithms, including long division, are based on this numeral system or its variants, such as binary numerals. A notable exception
Euclidean_division
Result on periodic sequences
least p + q − gcd ( p , q ) {\displaystyle p+q-\gcd(p,q)} , then w {\displaystyle w} also has period gcd ( p , q ) {\displaystyle \gcd(p,q)} . Theorem—Let
Fine_and_Wilf's_theorem
Special kind of semigroup in mathematics
a2, a3} where a1 < a2 < a3 and gcd ( a1, a2, a3) = 1. Its worst-case complexity is not as good as Greenberg's algorithm but it is much simpler to describe
Numerical_semigroup
Mathematical problem
condition that the greatest common divisor (GCD) is equal to 1. Indeed, the potential sums are multiples of the GCD in all cases. Hence, if it is not 1, then
Coin_problem
Class in computational complexity theory
inverse, rank; Polynomial GCD, by a reduction to linear algebra using Sylvester matrix Finding a maximal matching. Often algorithms for those problems had
NC_(complexity)
Mathematical construct in computer algebra
{lm} (g)}{\mathrm {gcd} }}\,f-{\frac {1}{\operatorname {lc} (g)}}\,{\frac {\operatorname {lm} (f)}{\mathrm {gcd} }}\,g;} where gcd denotes the greatest
Gröbner_basis
Algebraic structure modeling logical operations
common divisor (gcd) and the least common multiple (lcm) of a and b, respectively. The ring addition a + b is given by lcm(a, b) / gcd(a, b). The picture
Boolean_algebra_(structure)
Digital signal generator
(GRR) given by GRR = 2 N GCD ( Δ F , 2 N ) {\displaystyle {\mbox{GRR}}={\frac {2^{N}}{{\mbox{GCD}}(\Delta F,2^{N})}}} where GCD is the greatest common divisor
Numerically controlled oscillator
Numerically_controlled_oscillator
rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his
Gauss_composition_law
Matrix form in linear algebra
algorithm (intermediate numbers) is bounded by a polynomial in the binary encoding size of the numbers in the input matrix. One class of algorithms is
Hermite_normal_form
Cryptography framework
is a large prime. For any encryption exponent e in the range 1..p-1 with gcd(e,p-1) = 1. The corresponding decryption exponent d is chosen such that de
Three-pass_protocol
C++ programming library
C++ template libraries Parallel Patterns Library Grand Central Dispatch (GCD) Software Architecture Building Blocks "oneAPI Threading Building Blocks
Threading_Building_Blocks
Algorithm for public key cryptography
randomly and independently of each other such that gcd ( p q , ( p − 1 ) ( q − 1 ) ) = 1 {\displaystyle \gcd(pq,(p-1)(q-1))=1} . This property is assured if
Paillier_cryptosystem
Integer that is a perfect square modulo some integer
(a|p) roots (i.e. zero if a N p, one if a ≡ 0 (mod p), or two if a R p and gcd(a, p) = 1.) In general, if a composite modulus n is written as a product
Quadratic_residue
{\displaystyle i_{1},i_{2},j_{1},j_{2}} such that gcd ( j 1 , j 2 ) = 1 {\displaystyle {\text{gcd}}(j_{1},j_{2})=1} , and for t ≥ 0 , x t = w i 1 + j
Square-free_word
Cryptographic algorithm created by Adi Shamir
Shamir's secret sharing (SSS) is an efficient secret sharing algorithm for distributing private information (the "secret") among a group, first developed
Shamir's_secret_sharing
Branch of abstract algebra studying divisibility relations
algorithm. If: a = b q + r {\displaystyle a=bq+r} then: gcd ( a , b ) = gcd ( b , r ) {\displaystyle \operatorname {gcd} (a,b)=\operatorname {gcd}
Divisibility_theory
Increasing sequence of reduced fractions
value holds: gcd ( ‖ a c b d ‖ , ‖ a e b f ‖ ) = gcd ( ‖ a c b d ‖ , ‖ c e d f ‖ ) = gcd ( ‖ a e b f ‖ , ‖ c e d f ‖ ) {\displaystyle \gcd
Farey_sequence
Calculation of complex statistical distributions
In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution
Markov_chain_Monte_Carlo
Composite number that passes Fermat's probable primality test
n=341=11\cdot 31} , this product is gcd ( 10 , 340 ) ⋅ gcd ( 30 , 340 ) = 100 {\displaystyle \gcd(10,340)\cdot \gcd(30,340)=100} . For n = 341 {\displaystyle
Fermat_pseudoprime
Number-theoretic algorithm
efficiently using binary exponentiation: a 2 N − 1 ≡ 2 27456 ≡ 1 ( mod 27457 ) {\displaystyle a_{2}^{N-1}\equiv 2^{27456}\equiv 1{\pmod {27457}}} gcd ( a 2 ( N
Pocklington_primality_test
Cross-platform reverse-Polish calculator program
implementation of the Euclidean algorithm to find the GCD: dc -e '??[dSarLa%d0<a]dsax+p' # shortest dc -e '[a=]P?[b=]P?[dSarLa%d0<a]dsax+[GCD:]Pp' # easier-to-read
Dc_(computer_program)
Error-correcting code
In coding theory, rank codes (also called Gabidulin codes) are non-binary linear error-correcting codes over not Hamming but rank metric. They described
Rank_error-correcting_code
Geometry problem on grid points
points that can be chosen with no three in line is at most 2 gcd ( m , n ) {\displaystyle 2\gcd(m,n)} . When both dimensions are equal, and prime, it is not
No-three-in-line_problem
Sorting algorithm which uses multiple comparison intervals
variants, determining their time complexity remains an open problem. The algorithm was first published by Donald Shell in 1959, and has nothing to do with
Shellsort
Method for representing or encoding numbers
010101\dots _{2}} 0.2 6 {\displaystyle 0.2_{6}} For integers p and q with gcd (p, q) = 1, the fraction p/q has a finite representation in base b if and
Positional_notation
Numbers that contain only the digit 1
Euclidean Algorithm is based on gcd(m, n) = gcd(m − n, n) for m > n. Similarly, using Rm(b) − Rn(b) × bm−n = Rm−n(b), it can be easily shown that gcd(Rm(b)
Repunit
Equivalence class in mathematics
{1}{n}}\sum _{d\mid n}\varphi (d)k^{n/d}={\frac {1}{n}}\sum _{i=1}^{n}k^{\,{\rm {gcd}}(i,n)}} different k-ary necklaces of length n, where φ {\displaystyle \varphi
Necklace_(combinatorics)
Algebraic study of differential equations
+ y 2 , ∂ y ( p ) = 2 ⋅ y , gcd ( p , ∂ y ( p ) ) = 1 {\textstyle p(y)=1+y^{2},\ \partial _{y}(p)=2\cdot y,\ \gcd(p,\partial _{y}(p))=1} q ( z ) =
Differential_algebra
One over a whole number
that Bézout's identity is satisfied: a x + b y = gcd ( x , y ) = 1. {\displaystyle \displaystyle ax+by=\gcd(x,y)=1.} In modulo- y {\displaystyle y} arithmetic
Unit_fraction
Unique positive real number which when multiplied by itself gives 2
where a , b ∈ Z {\displaystyle a,b\in \mathbb {Z} } and gcd ( a , b ) = 1 {\displaystyle \gcd(a,b)=1} Squaring both sides, 2 = a 2 b 2 {\displaystyle
Square_root_of_2
Algebraic structure
{\displaystyle P'=-1} , implying that g c d ( P , P ′ ) = 1 {\displaystyle \mathrm {gcd} (P,P')=1} , which in general implies that the splitting field is a separable
Finite_field
Proof that a number is prime
(2) holds. This requires calculation of gcd, done for large numbers usually using the Extended Euclidean algorithm, over the number of primes provided. Each
Primality_certificate
Decimal representation of a number whose digits are periodic
) := max { ord n ( b ) ∣ gcd ( b , n ) = 1 } {\displaystyle \lambda (n):=\max\{\operatorname {ord} _{n}(b)\,\mid \,\gcd(b,n)=1\}} which again divides
Repeating_decimal
(Mathematical) decomposition into a product
every integral domain in which greatest common divisors exist (known as a GCD domain) is a UFD. Every principal ideal domain is a UFD. A Euclidean domain
Factorization
Arithmetic function related to the divisors of an integer
multiplicative (since each divisor c of the product mn with gcd ( m , n ) = 1 {\displaystyle \gcd(m,n)=1} distinctively correspond to a divisor a of m and
Divisor_function
Algebraic structure with addition and multiplication
mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted like
Ring_(mathematics)
Mathematical group that can be generated as the set of powers of a single element
That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of
Cyclic_group
Type of pseudoprime
{\displaystyle (P,Q)} pseudoprime if and only if ( 1 ) gcd ( n , 2 Q D ) = 1 , {\displaystyle (1)\qquad \gcd(n,2QD)=1,} ( 2 ) U n − δ ( P , Q ) ≡ 0 ( mod n )
Frobenius_pseudoprime
Composite number which passes Miller–Rabin primality test
strong pseudoprime, this even gives us a factorization: 31697 = gcd(28419+1, 31697) × gcd(28419−1, 31697) = 29 × 1093. For another example, pick n = 47197
Strong_pseudoprime
Fraction with denominator a power of two
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example,
Dyadic_rational
Codes intended to correct short, contiguous errors in a communications channel
\ell } -burst-error correcting code. Lemma 1— gcd ( p ( x ) , x 2 ℓ − 1 + 1 ) = 1. {\displaystyle \gcd \left(p(x),x^{2\ell -1}+1\right)=1.} Proof Let
Burst_error-correcting_code
Composite number in number theory
Carmichael numbers satisfy the following equality: gcd ( ∑ x = 1 n − 1 x n − 1 , n ) = 1. {\displaystyle \gcd \left(\sum _{x=1}^{n-1}x^{n-1},n\right)=1.} A
Carmichael_number
Algebraic ring that need not have additive negative elements
al. in 1992.) A semiring is a set R {\displaystyle R} equipped with two binary operations + {\displaystyle +} and ⋅ , {\displaystyle \cdot ,} called addition
Semiring
Integer side lengths of a right triangle
triangle is given by ( a − 1 ) ( b − 1 ) − gcd ( a , b ) + 1 2 ; {\displaystyle {\tfrac {(a-1)(b-1)-\gcd {(a,b)}+1}{2}};} for primitive Pythagorean triples
Pythagorean_triple
Positive integer of the form (2^(2^n))+1
+ b 2 n g c d ( a + b , 2 ) {\displaystyle {\frac {a^{2^{n}}+b^{2^{n}}}{gcd(a+b,2)}}} with a, b any coprime integers, a > b > 0, are called generalized
Fermat_number
Features in Haskell programming language
s+p..q-1] ] ) ] The shortest possible code is probably nubBy (((>1) .) . gcd) [2..]. It is quite slow. Haskell allows indentation to be used to indicate
Haskell_features
Dialect of Lisp
for functional programming and associated techniques such as recursive algorithms. It was also one of the first programming languages to support first-class
Scheme_(programming_language)
Number of subsets of a given size
from this that ( n k ) {\displaystyle {\tbinom {n}{k}}} is divisible by n/gcd(n,k). In particular therefore it follows that p divides ( p r s ) {\displaystyle
Binomial_coefficient
Type of block code
q^{m}-1} for some m {\displaystyle m} and G C D ( n , b ) = 1 {\displaystyle GCD(n,b)=1} . The only vector in G F ( q ) n {\displaystyle GF(q)^{n}} of weight
Cyclic_code
1984 video game
original (PDF) on 13 September 2013. "Classic Game Postmortem - ELITE". GCD.com. "Elite - Review", Zzap!64 (1), Newsfield Publications Ltd: 16–17, May
Elite_(video_game)
Family of RISC-based computer architectures
greatest common divisor. In the C programming language, the algorithm can be written as: int gcd(int a, int b) { while (a != b) // We enter the loop when
ARM_architecture_family
Set with associative invertible operation
Bézout's identity and the fact that the greatest common divisor gcd ( a , p ) {\displaystyle \gcd(a,p)} equals 1 {\displaystyle 1} . In the case p = 5 {\displaystyle
Group_(mathematics)
Indian inventions
procedure for finding integers x and y satisfying the condition ax + by = gcd(a, b). Formal grammar / Formal systems – In his treatise Astadhyayi, Panini
List of Indian inventions and discoveries
List_of_Indian_inventions_and_discoveries
projection in line and column A direction is composed of two integers (p, q) with gcd (p, q) = 1 An angle is always between 0 and 180 °, which means that q is
Mojette_transform
Cryptographic scheme
2 {\displaystyle e>N^{2}} and g c d ( e , ϕ ( N 2 ) ) = 1 {\displaystyle gcd(e,\phi (N^{2}))=1} . Alice then computes a public number g m {\displaystyle
Commitment_scheme
matrix but with arbitrary entries in one column below the main diagonal. GCD matrix The n × n {\displaystyle n\times n} matrix ( S ) {\displaystyle (S)}
List_of_named_matrices
Algebra based on a vector space with a quadratic form
Operators, Springer Haile, Darrell E. (Dec 1984). "On the Clifford Algebra of a Binary Cubic Form". American Journal of Mathematics. 106 (6). The Johns Hopkins
Clifford_algebra
BINARY GCD-ALGORITHM
BINARY GCD-ALGORITHM
Male
Hindi/Indian
Variant spelling of Hindi Vijay, BIJAY means "victory."
Surname or Lastname
English
English : variant spelling of Gadd.Danish : from a medieval nickname Gad meaning ‘sting’, ‘point’, or from the Biblical male personal name Gad.Muslim : from a personal name based on Arabic jÄd ‘serious’, ‘earnest’.
Male
English
Short form of English Gideon, GID means "cutter down; hewer," i.e. "mighty warrior."
Male
English
Pet form of English Gerard, GED means "spear strong."
Male
Scandinavian
Scandinavian form of Old Norse Einarr, EINAR means "lone warrior."
Boy/Male
Hindu
God
Boy/Male
Indian
An intimate particle of the God of heaven
Male
Native American
Native American Navajo name GAD means "juniper tree."
Female
English
English pet form of German Belinda, possibly BINDY means "bright serpent" or "bright linden tree."
Female
Hebrew
Variant spelling of Hebrew Bina, BINAH means "intelligence, wisdom."Â
Surname or Lastname
English (chiefly South Yorkshire)
English (chiefly South Yorkshire) : topographic name for someone who lived on land enclosed by a bend in a river, from Old English binnan ēa ‘within the river’, or a habitational name from places in Kent called Binney and Binny, which have this origin.Scottish : habitational name from Binney or Binniehill near Falkirk, named in Gaelic as Beinnach, from beinn ‘hill’ + the locative suffix -ach.
Male
Hindi/Indian
(विनय) Hindi name VINAY means "leading asunder."
Male
English
English unisex form of Latin Hilarius and Hilaria, HILARY means "joyful; happy."Â Originally, this was strictly a masculine name.
Boy/Male
American, Australian, French, German, Greek, Latin, Polish, Swedish
Cheerful; Happy; Joyful; Similar to Hilary
Male
Hebrew
(גָּד) Hebrew name GAD means "troop." In the bible, this is the name of a prophet and the seventh son of Jacob by Zilpah. Compare with other forms of Gad.
Boy/Male
Indian, Punjabi, Sikh
Blessing
Female
Hebrew
(×‘Ö¼Ö´×™× Ö¸×”) Hebrew name BINA means "intelligence, wisdom."Â
Female
Turkish
Turkish name PINAR means "spring."
Girl/Female
English
Originally a diminutive used for names ending in -bina, like Albina, Columbina, and Robina, now...
Male
Greek
(Γάδ) Greek form of Hebrew Gad, GAD means "troop." In the bible, this is the name of a tribe descended from Gad, mentioned in the New Testament in Rev vii. 5. Compare with other forms of Gad.
BINARY GCD-ALGORITHM
BINARY GCD-ALGORITHM
Boy/Male
Australian, Lebanese
Choreographer
Girl/Female
Indian
Hope
Boy/Male
Hindu, Indian
Brilliance
Boy/Male
Tamil
Fragrance
Male
Greek
(ΧÏÏσης) Greek myth name of a priest of Apollo, derived from the word khrysos, KHRYSES means "golden."
Girl/Female
Muslim
Olive, Fiery, Sower of seeds
Girl/Female
Indian
Intelligent, Wise
Surname or Lastname
English
English : variant of Blakely.
Boy/Male
Indian
Lord of Heart
Boy/Male
Hindu, Indian, Telugu
Power of Sun Rise
BINARY GCD-ALGORITHM
BINARY GCD-ALGORITHM
BINARY GCD-ALGORITHM
BINARY GCD-ALGORITHM
BINARY GCD-ALGORITHM
a.
lasting for one day; as, a diary fever.
a.
Of or pertaining to the Canary Islands; as, canary wine; canary birds.
n.
A pale yellow color, like that of a canary bird.
n.
That which is constituted of two figures, things, or parts; two; duality.
a.
Of a pale yellowish color; as, Canary stone.
n.
A binary compound of silicon, or one regarded as binary.
v. t.
To treat as a god; to idolize.
n.
A register of daily events or transactions; a daily record; a journal; a blank book dated for the record of daily memoranda; as, a diary of the weather; a physician's diary.
n.
A binary compound of selenium, or a compound regarded as binary; as, ethyl selenide.
a.
Relating or belonging to bile; conveying bile; as, biliary acids; biliary ducts.
n.
A binary compound of phosphorus.
n.
See Finery.
n.
A binary compound of iodine, or one which may be regarded as binary; as, potassium iodide.
a.
Of or pertaining to the urine; as, the urinary bladder; urinary excretions.
n.
A binary compound of zinc.
v. i.
To perform the canary dance; to move nimbly; to caper.
a.
Containing ten; tenfold; proceeding by tens; as, the denary, or decimal, scale.
n.
Wine made in the Canary Islands; sack.
n.
A binary compound of hydrogen; a hydride.
n.
A canary bird.