Fastest Gcd Algorithm

This considers the simplyfying of (-3)-(-6) where gcd on -3 and -6 would result in 3, not -3 as with the other function. Let's find the LCM of 30 and 45. Estimate how many times faster it will be to find gcd(31415, 14142) by Euclid’s algorithm compared with the algorithm based on checking consecutive integers from min{m, n} down to gcd(m, n). The applet below is an interactive illustration to the process of finding gcd and lcm of two integers by first finding the prime factorization of each. Euclidean algorithm. Finally, Magma contains implementations of the fast classical Lehmer extended GCD ('XGCD') algorithm (which is about 5 times faster than the Euclidean XGCD algorithm) and the Schönhage recursive ("half-GCD") algorithm, yielding asymptotically-fast GCD and XGCD algorithms. See also Euclid's algorithm. After messing around a little I have created a pretty fast prime/factorization module. Computing gcd(a;b) by the Euclidean algorithm. Chor and Goldreich [1] have the fastest parallel GCD algorithm; it is based on the systolic array GCD algorithm of Brent and Kung. I would not recommend the Alternate 2 method for the following reasons: 1. gcd(a, gcd(b, gcd(c, ))) If so, then what is such an algorithm?. It perhaps is surprising to find out that this lemma is all that is necessary to compute a gcd, and moreover, to compute it very efficiently. The Euclidean Algorithm Having now shown that Z n is not a field whenever n is not prime, we want to show Z p is a field whenever p is prime. , greatest common divisor (factor) of two integers. Python has an inbuilt method to find out the GCD. If we subtract smaller number from larger (we reduce larger number), GCD doesn’t change. algorithm inside the McEliece cryptosystem; see also [36, Algorithm 10] and [30]. Goldreich, An improved parallel algorithm for integer GCD, Algorithmica, 5, 1990, 1--10 Google Scholar Cross Ref. As anotherexample,Bos[21]developedaconstant-timeversionofKaliski'salgorithm,and reportedin[21,Table1]thatthistook486000ARMCortex-A8cyclesforinversionmodulo a254-bitprime. ie gcd(a,b,c)=gcd(gcd(a,b),c). GCD of 29 and 0 is 0. It is mainly used for big integers that have a representation as a string of digits relative to some chosen numeral system base, say β = 1000 or β = 2 32. The best way to find the gcd of n numbers is indeed using recursion. A novel fast hybrid GCD computation algorithm. The GCD has a number of properties that allow us to express the GCD of a pair of larger numbers as the GCD of smaller numbers. The binary GCD algorithm, also known as Stein's algorithm, is an algorithm that computes the greatest common divisor of two nonnegative integers. It is based on Floyd’s cycle-finding algorithm and on the observation that two numbers x and y are congruent modulo p with probability 0. (b) Given integers m and n, state the de nition of the greatest common divisor of m and n. The two numbers share no prime factors, so the gcd is 1. Using the euclidean algorithm for $\gcd(121, 330)$, how many divisions are required for the process? Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. A novel fast hybrid GCD computation algorithm A novel fast hybrid GCD computation algorithm Mohamed, Faraoun Kamel 2014-01-01 00:00:00 We propose a novel algorithm for integer's greatest common divisor (GCD) computation that hybridises both Euclidian and binary algorithms according to a new schema, in order to accelerate the GCD computation especially in the case of large bit difference. For Int , GHC has a rewrite rule to use GMP's fast gcd, depending on hardware and/or GMP version, that can be faster or slower than the binary algorithm (on my 32-bit box, binary is faster, on my. The purpose of the program is to find the GCD(Greatest Common Denominator) of two integers that are inputed. This method says calculate LCM. Our main application results in a fast computational algorithm for entries of the Rational Interpolation Table and, in particular, the Padé Table. Still, it is a costly operation. Stein's algorithm or binary GCD algorithm is an algorithm that computes the greatest common divisor of two non-negative integers. Like all quantum computer algorithms, Shor's algorithm is probabilistic: it gives the correct answer with high probability, and the probability of failure can be decreased by repeating the algorithm. I would not recommend the Alternate 2 method for the following reasons: 1. Hopcroft, Fast parallel matrix and GCD computations, Information and Control, 52, 3, 1982, 241--256 Google Scholar Cross Ref; B. The (extended) generalized Euclidean algorithm 2. Opcode = 00 / 01 /10 /11 for gcd using Euclid’s algorithm / gcd using Stein’s algorithm / addition / subtraction. The Efficient algorithm is much better. Show your work. The Greatest Common Divisor (GCD) of two whole. Numerical analysis. 607927 That is, lim x→∞. This problem investigates the binary gcd algorithm, which avoids the remainder computations used in Euclid's algorithm. For gcd(x, 11^3) != 1, either x = 0 or shares a common factor with 11^3. The Euclidean algorithm is a fast al-. e, 1) find gcd of left half of given numbers (let it be a) 2) find gcd of right half of given numbers(let it be b) 3) find. Notation: d gcd(a, b) Example: ±2, ±7, and ±14 are the only integers that are common divisors of both 42 and 56. Pollard's rho algorithm is an algorithm for integer factorization. He devised a clever and fast algorithm to nd the greatest common divisor of two integers. Since 14 is the largest, gcd(42, 56) 14. - Smart input recognition. The greatest common divisor (GCD, or GCF (greatest common factor)) of two or more integers is the largest integer that is a divisor of all the given numbers. The source TypeScript code and compiled JavaScript code are available for viewing. It w orks o v er an y eld, but the nite elds that o ccur most in co ding theory are the most often used. With ( x) as input, Algorithm 5. The HCF or GCD of two integers is the largest integer that can exactly divide both numbers (without a remainder). Hence the Euclid algorithm: gcd(m,n) = n if m = 0 = gcd(r,m), r = n mod m where r m, of course. The basic idea of the algorithm is to use some information about the order of an. The (extended) generalized Euclidean algorithm 2. The greatest common divisor (GCD, or GCF (greatest common factor)) of two or more integers is the largest integer that is a divisor of all the given numbers. Gcd Of N Numbers Program In Prolog Codes and Scripts Downloads Free. You may write a program or function, taking input and returning output via any of our accepted standard methods (including STDIN/STDOUT, function parameters/return values, command-line arguments, etc. I also suspect that the overhead of using BigIntegers is likely to be small compared to using logic such as Fred's to eliminate checks you don't need to make. Number Theory – Euclidian Algorithm for GCD (1) Number Theory – Euler's Totient Function (1) Number Theory – FindDigits of N! (1) Number Theory – Finding Divisors (1) Number Theory – Modular Multiplicative Inverse using ExtendedGCD (1) Number Theory – N! under modulo P (1) Number Theory – Prime Factorization (1). Note: On newer versions of the GCC compiler there is a __gcd(a, b) function in #include which might be faster on your computer. Michael Monagan, [email protected] * Additionally, you should have a fast gcd algorithm. GCD calculation in Alternate 3 is just a truncated version of the ORIGINAL one (using the same Euclid algorithm, slightly re-written in terms of vars naming), and it's lacking the safety/performance boost provided in ORIGINAL one by the thorough validating/checking of initial conditions: for example, there is no need to run the Euclid part if. int main() {. Read and learn for free about the following article: The Euclidean Algorithm If you're seeing this message, it means we're having trouble loading external resources on our website. It may not be "the fastest", or the "neatest", but doing it for yourself will teach you some. This algorithm originated in the ideas of Lehmer, Knuth and Schönhage. However, we believe that computing the prime factorization of large numbers is hard… Euclid’s algorithm provides a much faster way. Read and learn for free about the following article: The Euclidean Algorithm If you're seeing this message, it means we're having trouble loading external resources on our website. Numerical analysis. Here was the answer I was looking for. The Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, without explicitly factoring the two integers. The HCF or GCD of two integers is the largest integer that can exactly divide both numbers (without a remainder). Two fast GCD algorithms. Active 2 years, 11 months ago. If the gcd(m, N) = 1. Computing gcd(a;b) by the Euclidean algorithm. It can be shown that such an inverse exists if and only if a and m are coprime, but we will ignore this for this task. The Efficient algorithm is much better. Number Theory Operations. I also suspect that the overhead of using BigIntegers is likely to be small compared to using logic such as Fred's to eliminate checks you don't need to make. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity µ(n) and stop the recursion at a depth slightly smaller than lgn. Two fast GCD algorithms. (b) Given integers m and n, state the de nition of the greatest common divisor of m and n. euclid’s algorithm Greatest common divisor of two non- negative integers m and n denoted gcd(m,n),is defined as the largest integer that divides both m and n evenly,i. ) of two numbers a and b in locations named A and B. There is another algorithm, due to Euclid, as follows: 8085 = 1·7560+525 7560 = 14·525+210 525 = 2·210+105 210 = 2·105 The result of this computation is that 105 is the greatest common divisor of 8085 and 7560. This uses the division algorithm to:-find the greatest common divisor (gcd) [ aka highest common factor (hcf)] find the lowest common multiple (lcm. Euclidean algorithm. There are multiple methods to find GCD , GDF or HCF of two numbers but Euclid's algorithm is very popular. N2 - In this paper, we propose a new score function for subband decomposition ICA to improve the algorithm performance. Numerical. Here was the answer I was looking for. This is true, if the GCD is computed via the classical Euclidean algorithm (or via the binary Euclidean algorithm), and leads to a quadratic algorithm for computing the Jacobi symbol. Input will be two non-negative integers. Then at the end of the output, the greatest will be displayed. gcd(N 1,N 2 …N m) = gcd(N 1, (N 1 *N 2 *…*N m mod N 1 2)/N 1) The secret sauce is in making this run fast--note that the first step is to compute the product of all the keys, a 729 million. It is the fastest completely deterministic factorization algorithm, but is one of the slower class of algorithms (factoring N takes O( N 1/4) steps). von zur Gathen and J. The Fast Euclidean Algorithm computes the same GCD in O ⁢ (𝖬 ⁢ (n) ⁢ log ⁡ (n)) field operations, where 𝖬 ⁢ (n) is the time to multiply two n-degree polynomials; with FFT multiplication the GCD can thus be computed in time O ⁢ (n ⁢ log 2 ⁡ (n) ⁢ log ⁡ (log ⁡ (n))). GCD and LCM by Plain Factorization. It perhaps is surprising to find out that this lemma is all that is necessary to compute a gcd, and moreover, to compute it very efficiently. The Euclidean Algorithm. Luckily, we can compute inverses in a generic way, using an algorithm called the extended Euclidean algorithm. wikiHow is a "wiki," similar to Wikipedia, which means that many of our articles are co-written by multiple authors. Euclid's Algorithm computes the greatest common divisor of two positive integers, and it can be written in a recursive form as follows. Let us discuss what this algorithm says and understand its proof of correctness. 3 Greatest Common Divisor • Binary GCD:Binary GCD: • Lehmer's Algorithm: • Subquadratic GCD: • Extended GCD: • Jacobi Symbol:Jacobi Symbol:. The most direct method of calculating the greatest common divisor of two numbers b and c would be to make a list of the common divisors, and note the value of the largest common divisor. But I was getting timeouts in certain programs when I did this. Basic Euclidean Algorithm for GCD. GCD’’ (Euclid’s Algorithm) gcd'' is Euclid’s or the Euclidean Algorithm. 100 Trade Center Dr. Once again we check if Y is zero, if yes then we have our greatest common divisor or GCD otherwise we keep continue like this until Y becomes zero. It can be shown that such an inverse exists if and only if a and m are coprime, but we will ignore this for this task. 4 Queen's problem and solution using backtracking algorithm. Theorem: Euclid’s algorithm computes the greatest common divisor of x and y if x y. Sparse primes allow very fast Figure 1. 345--348] (which is about 5 times faster than the Euclidean XGCD algorithm) to compute the extended GCD of two integers. If the gcd(m, N) = 1. The GCD of 1920 and 1080 are 120, GCDd/vsors=divisors(120)=[1,2,3,4,5,6,8,10,12,15, 20,24,30,40,60,120], total number of elements is 16 hence there are only 16 distinct splits in this case as compared to the number of choices for maximum desired blocks which is 1 to 1920*1080. (1) Apply the division algorithm: a= bq+ r, 0 r a. 1 gives a proper factorization of ˜(x) or E F = n. This algorithm originated in the ideas of Lehmer, Knuth and Schönhage. Stein in 1967. The Euclidean algorithm is a fast al-. Your goal in this problem is to implement the Euclidean algorithm for computing the greatest common divisor. It is used in countless applications, including computing the explicit expression in Bezout's identity, constructing continued fractions, reduction of fractions to their simple forms, and attacking the RSA cryptosystem. scan-conversion and antialiasing process. Euclid outlined an algorithm for solving this problem. Euclidean algorithm for computing the greatest common divisor; Manacher's Algorithm - Finding all sub-palindromes in O(N). Stein's algorithm or binary GCD algorithm is an algorithm that computes the greatest common divisor of two non-negative integers. Then multiply each factor the greatest number of times it occurs in. It's based on a few properties: If both numbers are even, then we can factor out a two of both and compute the GCD of the remaining numbers: $\gcd(2a, 2b) = 2 \gcd(a, b)$. This algorithm can be useful if is a fixed number in your program (so, you can hardcode a precomputed value of ()), or if is a prime number, in which case () = −. All common denominators have to be displayed. Using the euclidean algorithm for $\gcd(121, 330)$, how many divisions are required for the process? Because the algorithm is fast (Max[data] shows that any two numbers between 0 and 599 will resolve in at most 12 steps), the value jumps rapidly between only a few discrete values (0-12), which makes a regular 3D plot very noisy. Hi! gcd6 has a typo: x & y should be x && y, otherwise gcd(11,4) returns 15, which is obviously wrong (damn you, binary and!) In my tests, compiling with -O3, Ram’s was the fastest by about 1. gcd? By Euclid’s algorithm, perhaps the ﬁrst algorithm ever invented: algorithm gcd(x,y) if y = 0 then return(x) else return(gcd(y,x mod y)) Note: This algorithm assumes that x ‚y ‚0 and x >0. Parallel algorithms. Chor and O. Lehmer noted that that most of the quotient s from each step of the division part of the standard algorithm are small. The greatest common divisor (GCD, or GCF (greatest common factor)) of two or more integers is the largest integer that is a divisor of all the given numbers. C++ Program to Find GCD Examples on different ways to calculate GCD of two integers (for both positive and negative integers) using loops and decision making statements. * * So, we have three dependencies here: * * - isprime(int) (typically look up bitset sieve) * - std::vector primes (vector of prime numbers, typically sieved) * - binary_gcd(int, int) or similar, fast, gcd function. ; Divide 30 by 15, and get the result 2 with remainder 0, so 30. For our example, 24 and 60, below are the steps to find GCD using Euclid's algorithm. GCD - Greatest Common Divisor Keywords Greatest Common Divisor ,Magnitude Comparator, Multiplexer, Full Subtractor, Euclidean Algorithm. Lehmer's GCD algorithm. Within the classical range, Magma uses the fast classical Accelerated GCD algorithm of Kenneth Weber to compute the GCD of two integers, and the fast classical Lehmer extended GCD (XGCD') algorithm [Knu97, pp. Our interpolation algorithm requires nding roots of a polynomial in GF(p)[x], which in turn requires an e cient polynomial GCD algorithm. This implements the binary GCD algorithm which removes the branch mispredictions. Have you ever thought about the fastest way to sort N numbers. Motivated by this observation, we review the Fast Extended Euclidean algorithm for univariate polynomials, which recursively. A simple way to find GCD is to factorize both numbers and multiply common factors. Computing gcd(a;b) by the Euclidean algorithm. As concrete applications, this paper saves time in (1) modular. The Euclidean algorithm for finding the greatest common divisor is applicable. Lehmer noted that most of the quotients from each step of the division part of the. The subject of computing the GCD was brought up a couple of times lately, and we assumed that the straightforward divide-and-remained implementation was the most efficient. To understand this example, you should have the knowledge of the following C++ programming topics:. gcd stands for the "Greatest Common Divisor" and lcm for the "Least Common Multiple". It may not be "the fastest", or the "neatest", but doing it for yourself will teach you some. Also say the algorithm solves the problem Problems often have multiple algorithms that solve them - example, GCD: The obvious algorithm Euclid's GCD algorithm (300 B. It turns out, that you can design a fast GCD algorithm that avoids modulo operations. First I use the Euclid algorithm to calculate: So I know the GCD is 1. 1 Euclid's Algorithm. For any integers a;b, gcd(a;b) is a linear combination of a and b, i. Mathematical analysis. Pollard's rho algorithm is an algorithm for integer factorization. The best way to find the gcd of n numbers is indeed using recursion. Recall: The greatest common divisor (GCD) of m and n is the largest integer that divides both m and n with no remainder. gcd is a regular Euclid's GCD implementation (binary Euclid's GCD is much slower then the regular. This is true, if the GCD is computed via the classical Euclidean algorithm (or via the binary Euclidean algorithm), and leads to a quadratic algorithm for computing the Jacobi symbol. * * This function is placed in the public domain by the author,. Your goal in this problem is to implement the Euclidean algorithm for computing the greatest common divisor. The applet below is an interactive illustration to the process of finding gcd and lcm of two integers by first finding the prime factorization of each. load= if ‘1’ , loads input 8-bit data in A and B. By using recursion, this leads to a solution. The GCD is sometimes called the greatest common factor (GCF). I understand this , but when testing large numbers , how to get the factors ,, numbers like 78 , 132. If the integers are the size of normal native int lengths (e. Thus every two steps, the numbers shrink by at least one bit. def xgcd(a, b): """return (g, x, y) such that a*x + b*y = g = gcd (a, b)""" x0, x1, y0, y1 = 0, 1, 1, 0 while a != 0: (q, a), b = divmod(b, a), a. Using gcd(a,b,c)=gcd(gcd(a,b),c) is the best method, much faster in general than using for example factorization. According to Wikipedia, it is 60% faster than more common ways to compute the GCD. Interestingly, algorithms 2 and 3 both substantially improve on algorithm 1 with Python arithmetic at least up to n = 10000, but both are worse than algorithm 1 with gmpy. In 1994, Sorenson's right- and left-shift k-ary algorithms [10] match Chor and Goldreich's performance. ) of two numbers a and b in locations named A and B. For a similar project, that translates the collection of articles into Portuguese, visit https://cp-algorithms-brasil. Read and learn for free about the following article: The Euclidean Algorithm If you're seeing this message, it means we're having trouble loading external resources on our website. Example: if a=2103859 andb=21735then GCD(a,b)=21035. Euclid's Algorithm I Intuition Consider nding the gcd(184 ;1768). of Computer Science and Institute for Advanced Computer Studies Univ. e, 1) find gcd of left half of given numbers (let it be a) 2) find gcd of right half of given numbers(let it be b) 3) find. The greatest common divisor may be computed by the Euclidean algorithm, which is very fast, because it does not require factorization of the given numbers. Goldreich, An improved parallel algorithm for integer GCD, Algorithmica, 5, 1990, 1--10 Google Scholar Cross Ref. It can be shown that such an inverse exists if and only if a and m are coprime, but we will ignore this for this task. We present a unified framework for the asymptotically fast Half-GCD (HGCD) algorithms, based on properties of the norm. gcd (a, 0) = a and gcd (0, b) = b because everything divides 0. euclid’s algorithm Greatest common divisor of two non- negative integers m and n denoted gcd(m,n),is defined as the largest integer that divides both m and n evenly,i. If both a and b are 0, gcd is zero gcd(0, 0) = 0. ⍺=⍵: ⍺ } Binary GCD ----- This algorithm, which takes a pair of non-negative whole (unsigned integer) numbers as argument, is suitable for implementing GCD in a low-level compiled or assembly language. And this is a quite fast algorithm, because I keep dividing the numbers that I have by each other, and it gets small fast. I intend to write a program which calculates the LCM of inputed integers. The Euclidean algorithm (also called Euclid's algorithm) is an efficient method for computing the greatest common divisor (GCD), also known as the greatest common factor (GCF) or highest common factor (HCF). Lehmer's GCD algorithm, named after Derrick Henry Lehmer, is a rather fast GCD algorithm, an improvement on the simpler but slower Euclidean algorithm. 4 Wiedemann's. Evaluating the Greatest Common Divisor function in C# Available Routines. Instead, we can find the GCD of the problem using a very fast algorithm known as Euclid's algorithm and then use that result to find the LCM as LCM ( a , b ) = (a * b) / GCD ( a , b ). That’s a good thing!. Examples of Iterative and Recursive Algorithms Fast Exponentiation Recursive Integers a, n, and m, with 0 ≤ n and 0 ≤ a < m. 30 = 2 × 3 × 5. (For example, Knuth observed that the quotients 1, 2, and 3 comprise 67. In mathematics, amongst the natural numbers greater than 1, a prime number (or a prime) is such that has no divisors other than itself (and 1). Fast algorithm for modular division (residue) 3. The problem with this method is that there is no efficient algorithm to factor integers. greatest common divisor Although it does not appear to be as efficient as the Miller-Rabin algorithm, in 2002 a relatively simple deterministic algorithm that efficiently determines whether a given large number is a prime was developed. The most direct method of calculating the greatest common divisor of two numbers b and c would be to make a list of the common divisors, and note the value of the largest common divisor. GCD of Polynomials Using Division Algorithm : If f (x) and g(x) are two polynomials of same degree then the polynomial carrying the highest coefficient will be the dividend. (b) Use the \magic box" algorithm to nd integers u and v such that au+ bv = d. Brute force approach of trying every number and seeing if it is the LCM is not the best approach. Initially (x) = x; then (x) is iteratively replaced by ’(x) until either Algorithm 5. Calculating GCD Using Euclid Algorithm In Python For any two positive integer number m and n, GCD ( greatest common divisor) is the largest integer number which divides them evenly. Fortunately, all of these algorithms are already implemented for us in GMP, the C++ big-integer library. If gcd(a, N) ≠ 1, then there is a nontrivial factor of N, so we are done. 1 Introduction 279 7. As we will see, the Euclidean. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity µ(n) and stop the recursion at a depth slightly smaller than lgn. org are unblocked. 100 Trade Center Dr. T(n) = (1:2)n) considered \slow". 1 Binary GCD. e, 1) find gcd of left half of given numbers (let it be a) 2) find gcd of right half of given numbers(let it be b) 3) find. The greatest common divisor may be computed by the Euclidean algorithm, which is very fast, because it does not require factorization of the given numbers. Yang (UIC/RUB + Sinica) Fast Safe GCD + Inversions 2019. Lehmer's GCD algorithm, named after Derrick Henry Lehmer, is a fast GCD algorithm, an improvement on the simpler but slower Euclidean algorithm. Euclid's algorithm is probably fine. This uses the division algorithm to:-find the greatest common divisor (gcd) [ aka highest common factor (hcf)] find the lowest common multiple (lcm. Examples of Iterative and Recursive Algorithms Fast Exponentiation Recursive Integers a, n, and m, with 0 ≤ n and 0 ≤ a < m. Stein's algorithm replaces division with arithmetic shifts, comparisons, and subtraction. The algorithm is compared with the new (1 + i)-ary algorithm of Weilert and found to be somewhat faster if properly implemented. This is the extended Euclid algorithm. Euclid, a Greek mathematician in 300 B. This may be done using the Euclidean algorithm. For very large integers, the fastest GCD algorithms [2, 6, 10, 11] are all based on half-gcd procedure and computes the GCD in O(nlog2 nloglogn) time. Fundamentals. In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. The Institute is located at 17 Gauss Way, on the University of California, Berkeley campus, close to Grizzly Peak, on the. The GCD in the previous step should be 1. 35 Component Pascal. (For example, Knuth observed that the quotients 1, 2, and 3 comprise 67. The run time complexity is O((log 2 u v)²) bit operations. In fact, for polynomials one uses gcd with the derivative first to find factors which occurs more than once. A new version of the Euclidean algorithm is developed for computing the greatest common divisor of two Gaussian integers. In modern terms ,Euclid’s algorithm is based on applying repeatedly the equality. Euclidean algorithm. This allows us to write , where are some elements from the same Euclidean Domain as and that can be determined using the algorithm. In the integer case algorithm design and proof of correctness are complicated by the effect of carries. It was invented by John Pollard in 1975. 345--348] (which is about 5 times faster than the Euclidean XGCD algorithm) to compute the extended GCD of two integers. The smallest prime is thus 2. Extended Euclidean Algorithm. Then we can read the message. Beyond the basic arithmetic operations Python has few natively implemented number theory functions. 1 General idea of the algorithm The p − 1 algorithm was developped by J. The Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, without explicitly factoring the two integers. If there is a need to calculate the Greatest Common Divisor for more than two numbers, you can calculate the GCD for the first. The Institute is located at 17 Gauss Way, on the University of California, Berkeley campus, close to Grizzly Peak, on the. You may wish to look into Strassen's factorization algorithm, which does the gcd that you are interested in, and in a clever way. The structure of the recursive algorithm is very close to the one of the well-known Knuth-Schönhage fast gcd algorithm, but. Step 1: use the classical greatest common divisor (gcd) algorithm on N and m, where N is the number you are trying to factor, and m is a random positive integer less than N. Definition: An element is called a greatest common divisor (gcd). GCD calculation in Alternate 3 is just a truncated version of the ORIGINAL one (using the same Euclid algorithm, slightly re-written in terms of vars naming), and it's lacking the safety/performance boost provided in ORIGINAL one by the thorough validating/checking of initial conditions: for example, there is no need to run the Euclid part if. For large inputs, the GCD can be computed even faster, as ﬁrst shown by Knuth (1970) and Schonhage (1971). We suggest improvements on the average complexity of the latter algorithm and also. Within the classical range, Magma uses the fast classical Accelerated GCD algorithm of Kenneth Weber to compute the GCD of two integers, and the fast classical Lehmer extended GCD (XGCD') algorithm [Knu97, pp. Then we can read the message. scan-conversion and antialiasing process. One of the earliest known numerical algorithms is that developed by Euclid (the father of geometry) in about 300 B. Use the predetermined mapping scheme to obtain the plaintext. Over the past few decades several variations on a "half GCD" algorithm for obtaining the pair of terms in the middle of a Euclidean sequence have been proposed. It matches the best existing parallel integer GCD algorithms of Sorenson and Chor and Goldreich, since it can be achieved in Oɛ(n / log n) time using at most n 1+ɛ processors on CRCW PRAM. 3 The Sylvester Matrix and Subresultants 285 7. As it turns out, if the greatest common divisor (gcd) of 2 numbers a and b is 1 (i. Father of Geometry, Euclid, came up with an out of box algorithm to find GCD of two numbers in a very cost effective way. Lehmer's uses matrix multiplication to improve upon the standard Euclidian algorithms. An algorithm is correct if the output it produces satisfies the problem definition. And this (recursive) computation continues until the first argument is 0, at which point the result is the final value of the second argument. Fast Exponentiation Greatest common divisor Extended Euclidean Algorithm obtain gcd(a,b)and x,y,s. Euclid's algorithm is probably fine. To create this article, 40 people, some anonymous, worked to edit and improve it over time. 2 Recursive Euclid algorithm. We show that parallel versions of both algorithms match the complexity of the best previous parallel GCD algorithm due to. A parallel extended GCD algorithm Sidi Mohamed Sedjelmaci LIPN CNRS UMR 7030, Université Paris-Nord, 99 Avenue J. To understand this example, you should have the knowledge of the following C++ programming topics:. Analysis and Design of Algorithms Efficient algorithm Take 5 steps to solve gcd_fast( ( 3918848, 1653264 ) ) Time complexity = O(log(n)) n depend on a,b 63. Do the division. Y1 - 2010/12/1. Stein's algorithm uses simpler arithmetic operations than the conventional Euclidean algorithm; it replaces division with arithmetic shifts, comparisons, and subtraction. Input will be two non-negative integers. This algorithm does not require factorizing numbers, and is fast. Estimate how many times faster it will be to find gcd(31415, 14142) by Euclid’s algorithm compared with the algorithm based on checking consecutive integers from min{m, n} down to gcd(m, n). The greatest common divisor GCD(a,b) of two non-negative integers a and b (which are not both equal to 0) is the greatest integer d that divides both a and b. Our interpolation algorithm requires finding roots of a polynomial in GF(p)[x], which in turn requires an efficient polynomial GCD algorithm. Michael Monagan, [email protected] Tag: algorithm,math,greatest-common-divisor,number-theory,clrs This is the pseudo code for calculating integer factorisation took from CLRS. We show that parallel versions of both algorithms match the complexity of the best previous parallel GCD algorithm due to. Suppose that remainder(a,b) returns the remainder when ais divided by b. HP15c program: Euclidean algorithm, Greatest Common Divisor (GCD), Highest Common Factor (HCF) command display f LBL B 001-42,21,12 // LBL B: Calculate GCD(a,b), a->x, b->y g ABS 002- 43 16 g X=0 003- 43 20 GTO 7 004- 22 7 x>y. The binary algorithm has so far been found to be faster than the Euclidean algorithm everywhere. This uses the division algorithm to:-find the greatest common divisor (gcd) [ aka highest common factor (hcf)] find the lowest common multiple (lcm. A more reasonable algorithm is Euclid's algorithm. Example: Euclidean Algorithm If you have 2 numbers, then you subtract the smaller number from the larger number, the GCD of these two number stays the same. 3 Iterative binary algorithm. That's a good thing! But Speed Eventually Matters. The following algorithm calculates the Greates Common Divisor (GCD), which is the largest integer number able to divide two numbers with no remainder. I can calculate φ(N) = 40, but my lecturer then says to use the extended Euclidean algorithm to compute d. // Fundamental idea of Euclid's algorithm (one of the oldest known algorithms) // for finding the greatest common divisor of two integers: // gcd(a, 0) = a // gcd(a, b) = gcd(b, a % b) // % means modulo, that is, the remainder of a/b. 2) For two numbers a and b, if gcd(a, b) is 1, then Φ(ab) = Φ(a) * Φ(b). 2: Example of a gcd algorithm for integers using this paper’s division steps. Also say the algorithm solves the problem Problems often have multiple algorithms that solve them - example, GCD: The obvious algorithm Euclid's GCD algorithm (300 B. Specifically, it takes quantum gates of order using fast multiplication,. A new parallel extended GCD algorithm is proposed. gcd(0, 0) is not typically defined, but it is convenient to set gcd(0, 0) = 0. For our example, 24 and 60, below are the steps to find GCD using Euclid's algorithm. gcd(0, 2) = 2. Euclid's division algorithm has the following steps: 1. rst = for resetting operation. Using Bernstein's quasi-linear bulk GCD algorithm [Ber08], we obtained RSA moduli with shared primes. 2 Recursive Euclid algorithm. Chor and O. The computation of the greatest common divisor (GCD) of a pair of polynomials is an important issue in computational mathematics and also plays a crucial role in many science and engineering fields. There is another routine be Stein which uses binary arithmatic. We will use the notation gcd (a, b) to mean the greatest common divisor of a and b. The best discrete log algorithms known are faster than trying every element, but are not polynomial time. takes the algorithm to complete ! Sorting 100,000 elements can take much more time than sorting 1,000 elements • and more than 10 times longer ! the variable n suggests the "number of things" ! If an algorithm requires 0. Stein in 1967. , greatest common divisor (factor) of two integers. If a and b are both even, gcd(a, b) = 2*gcd(a/2, b/2) because 2 is a common divisor. To create this article, 40 people, some anonymous, worked to edit and improve it over time. The best way to find the gcd of n numbers is indeed using recursion. The (extended) generalized Euclidean algorithm 2. The gcd is the only number that can simultaneously satisfy this equation and. For a similar project, that translates the collection of articles into Portuguese, visit https://cp-algorithms-brasil. (Berman & Paul 2005). Khan Academy. Tersian in 1962 and published by G. The best discrete log algorithms known are faster than trying every element, but are not polynomial time. GCD of two numbers is the largest number that divides both of them. (c) Suppose that m and n are two integers such that m jn. If we subtract smaller number from larger (we reduce larger number), GCD doesn’t change. As concrete applications, this paper saves time in (1) modular. Fundamentals. The most direct method of calculating the greatest common divisor of two numbers b and c would be to make a list of the common divisors, and note the value of the largest common divisor. Khan Academy. shows a fast quantum algorithm for order-ﬁnding. Most importantly, however, this algorithm is suita-. takes the algorithm to complete ! Sorting 100,000 elements can take much more time than sorting 1,000 elements • and more than 10 times longer ! the variable n suggests the "number of things" ! If an algorithm requires 0. [23, 28Jan-20] Fast matrix multiplication. size(),str2. Let a = bq + r, where a, b, q, and r are integers. For very large integers, the fastest GCD algorithms [2, 6, 10, 11] are all based on half-gcd procedure and computes the GCD in O(nlog2 nloglogn) time. The computation of the greatest common divisor (GCD) of a pair of polynomials is an important issue in computational mathematics and also plays a crucial role in many science and engineering fields. shows a fast quantum algorithm for order-ﬁnding. Euclidean Algorithm. As we will see, the Euclidean. * * So, we have three dependencies here: * * - isprime(int) (typically look up bitset sieve) * - std::vector primes (vector of prime numbers, typically sieved) * - binary_gcd(int, int) or similar, fast, gcd function. The greatest common divisor of two positive integers a and b is the largest integer which evenly divides both numbers (with no remainder). To create this article, 40 people, some anonymous, worked to edit and improve it over time. load= if ‘1’ , loads input 8-bit data in A and B. Here’s the mathematics. The number of steps to reach this inverse will be the same as the number of steps to reach GCD(a,p)=1 using the Euclidean algorithm with a and p. Much more effective way to get the least common multiple of two numbers is to multiply them and divide the result by their greatest common divisor. (a) De ne the phrase m divides n. This algorithm does not require factorizing numbers, and is fast. It is also know as Greatest Common Factors (GCF), Greatest Common Measure (GCM), Highest Common Divisor (HCD), or Highest Common Factor (HCF). ISSN 1088-6842(online) ISSN 0025-5718(print). Algorithm 6 has the same specifications as the EEA and runs in operations in for input polynomials in degree. If a does divide b (as in a%b == 0) then the GDC is abs(b). So if the input is uniform equally distributed, it will return values for the first few (e. Together, they cited 9 references. This is described in many textbooks, for example Knuth section 4. The GCD of two numbers is the largest positive integer that divides both the numbers fully i. The smallest prime is thus 2. Two other benefits of our approach are (a) a simplified correctness proof of the polynomial HGCD algorithm and (b) the first explicit integer HGCD algorithm. 4 The Modular GCD Algorithm 300 7. It is very important in number theory and in computing. This remarkable fact is known as the Euclidean Algorithm. Before giving the algorithm we need to prove the following properties gcd(a,b) = 2·gcd(a 2, b 2) if a,b are even gcd(a, b 2) if a is odd, b is even gcd(a−b 2,b) if a, b are odd. Stein's Algorithm Stein's algorithm provides an enhanced version of the Euclidean algorithm for calculating the greatest common divisor (GCD) for a pair of integers. ; Divide 45 by 30, and get the result 1 with remainder 15, so 45=1·30+15. ) Its very likely that on other. The Binary GCD Algorithm Euclid's algorithm is so fast that there really is not much point in trying to improve it. This thesis provides a theoretical study on the MCP of two algorithms for ﬁnding the greatest common divisor (GCD) of univariate polynomials with real coeﬃcients: the Euclidean algorithm, and an algorithm based on QR-factorization given in [14]. To do this, we establish that whenever gcd(a,n)=1 then a has a multiplicative inverse (mod n). It is used in countless applications, including computing the explicit expression in Bezout's identity, constructing continued fractions, reduction of fractions to their simple forms, and attacking the RSA cryptosystem. 7% of all quotients Knuth, "The Art of. univariate gcd is performed using euclidean algorithm, which causes explosion of coefficients and is slow but for trivial examples. When we say cost, we are referring to time here. in the integer case with fast gcd algorithms; see (Brent, 1976; Sch onhage, 1971; Stehl e and Zimmermann, 2004). Solution: (a) The integer m divides n if there exists an integer r such. The gcd is the only number that can simultaneously satisfy this equation and. T1 - Generalized Cauchy Distribution (GCD)-based score functions for a fast and flexible subband decomposition ICA. Euclid's GCD Algorithm. And this is a quite fast algorithm, because I keep dividing the numbers that I have by each other, and it gets small fast. It is easy to prove in the same way as asymptotic of Euclid algorithm is proven. 最大公约数(GCD, Greatest Common Divisor). (For example, Knuth observed that the quotients 1, 2, and 3 comprise 67. Brute force approach of trying every number and seeing if it is the LCM is not the best approach. Stein’s Algorithm for finding GCD. swift file contains three different algorithms of how to calculate the greatest common divisor. , realized that the greatest common divisor of a and b is one of the following:. von zur Gathen and J. The basic idea of the algorithm is to use some information about the order of an. The idea is pretty simple. This algorithm has been known since ancient times. 7 A Heuristic Polynomial GCD Algorithm 320 Exercises 331. But there is a ﬁfth operation which I would argue is just as fundamental — and that is the operation of taking greatest common divisors. 3 Iterative binary algorithm. GCD of Polynomials Using Division Algorithm : If f (x) and g(x) are two polynomials of same degree then the polynomial carrying the highest coefficient will be the dividend. This may be done using the Euclidean algorithm. The Euclidean algorithm is a fast al-. Borodin, J. The Euclidean algorithm. swift file contains three different algorithms of how to calculate the greatest common divisor. This problem investigates the binary gcd algorithm, which avoids the remainder computations used in Euclid's algorithm. If you do not consier a or b as possible negative numbers, a GCD funktion may return a negative GCD, wich is NOT a greatest common divisor, therefore a funktion like this may be better. Fast Fourier Transform Splitting Algorithm. (a) Use the Euclidean algorithm to compute d = gcd(a;b). Although the algorithm was first published by the Israeli physicist and. As the name implies, the Euclidean Algorithm was known to Euclid, and appears in The Elements; see section 2. 1 Introduction 279 7. Non-prime numbers are known as composite, i. For large inputs, the GCD can be computed even faster, as ﬁrst shown by Knuth (1970) and Schonhage (1971). 2260 ÷ 816 = 2 R 628 (2260 = 2 × 816 + 628) 816 ÷ 628 = 1 R 188 (816 = 1 × 628 + 188). Analysis and Design of Algorithms Naive algorithm is too slow. gcf (a,b) = gcf (a,b-a), if b>a. Usual answer: Need constant-time algorithm. Silver and J. Since 2012, several papers applied the method to keys found on the Internet and elsewhere resulting in tens of thousands of broken keys. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity µ(n) and stop the recursion at a depth slightly smaller than lgn. Consider the problem of finding LCM of a number. The gcd is the only number that can simultaneously satisfy this equation and. An algorithm is correct if the output it produces satisfies the problem definition. It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Elements. 2 Application: fast modular composition of polynomials 316 12. cpp, Maggie Johnson // Description: An implementation of Euclid's algorithm for computing gcd. Sender and Receiver have public and private key and they can only understand message. And thus the only thing you need to do is to place. He devised a clever and fast algorithm to nd the greatest common divisor of two integers. If the integers are the size of normal native int lengths (e. In case, if both have the same coefficient then compare the next least degree’s coefficient and proceed with the division. First let me show the computations for a=210 and b=45. gcd(0, 0) is not typically defined, but it is convenient to set gcd(0, 0) = 0. Greatest Common Divisor d is the greatest common divisor of integers a and b if d is the largest integer which is a common divisor of both a and b. 2 Stein's Algorithm (Binary GCD) 33 CoffeeScript. e the remainder is 0). (b) Compute gcd(22 35;56 7). The following is the product of and. Euclidean algorithm. 7% of all quotients Knuth, "The Art of. Euclid's Algorithm GCF Calculator. Is there any algorithm faster than Euclid's algorithm for finding if gcd of two numbers is one answer 1 >>accepted The Binary GCD algorithm tends to outperform the Euclidean algorithm. Algorithm Used to find 1. 7 A Heuristic Polynomial GCD Algorithm 320 Exercises 331. 1 gives a proper factorization of ˜(x) or E F = n. algorithms make on the input given : number of divisions made is 10 7. GCD – Greatest Common Divisor Keywords Greatest Common Divisor ,Magnitude Comparator, Multiplexer, Full Subtractor, Euclidean Algorithm. Forcomparison,BernsteinandSchwabe[17]hadreported527102ARM. Fast arithmetic is available today in the computer algebra system Magma [21]. Parallel algorithms. Chor and O. We can use the Extended Euclidean algorithm (in Mathematica, ExtendedGCD[integer, integer]) to determine e. Brute force approach of trying every number and seeing if it is the LCM is not the best approach. Fastest? One you don’t implement yourself. Given an integer a, and an integer b, gcd(a,b)=d will be the last non-zero remainder of a set of equations resulting from the division algorithm. Tersian in 1962 and published by G. After messing around a little I have created a pretty fast prime/factorization module. In Euclid's algorithm, we start with two numbers X and Y. In mathematics, GCF or HCF or GCD of two or more numbers is the largest positive integer that divides the numbers without a remainder. This is true, if the GCD is computed via the classical Euclidean algorithm (or via the binary Euclidean algorithm), and leads to a quadratic algorithm for computing the Jacobi symbol. The Fast Euclidean Algorithm computes the same GCD in O ⁢ (𝖬 ⁢ (n) ⁢ log ⁡ (n)) field operations, where 𝖬 ⁢ (n) is the time to multiply two n-degree polynomials; with FFT multiplication the GCD can thus be computed in time O ⁢ (n ⁢ log 2 ⁡ (n) ⁢ log ⁡ (log ⁡ (n))). If N = M×s, with N, M, s, positive integers, then any divisor of M is also a divisor of N, making M their greatest common divisor:. algorithm is more efficient for multidigit GCD compu-tation than the straight forward adaptation of the plus– minus algorithm (experimentally 2. Information Processing Letters 110 :5, 198-201. 1 Answer to GCD algorithm in Blockly In Chapter 0 of Computer Science: An Overview, Brookshear lists Euclid’s algorithm for computing the Greatest Common Divisor (GCD). These inverses let us solve modular equations. Greatest Common Divisor (Euclid's Algorithm). It is optimised for use in computing, utilising fast bitwise shifts rather than the usually slower repeated subtraction, division or modulus operations. { distinguish easy problems (ones that have \fast" algorithms) from hard problems (ones that do not have any \fast. SO my thought was to write a recursive program looking for the GCD of two numbers, if the remainder is not 0 then change n to m and m to r and then calculate the GCD. Khan Academy. The gcd function can be computed by hand as a succession of modular reductions. Suppose we want to calculate the GCD of a and b. Mathematics of computing. Euclidean algorithm. I have honestly never written a program where computing the GCD was the bottleneck. 33 % 5 is 3. Math 110 Homework # 1. Stein's algorithm replaces division with arithmetic shifts, comparisons, and subtraction. It's based on a few properties: If both numbers are even, then we can factor out a two of both and compute the GCD of the remaining numbers: $\gcd(2a, 2b) = 2 \gcd(a, b)$. gcd(a, b, c, ) vs. Since 2012, several papers applied the method to keys found on the Internet and elsewhere resulting in tens of thousands of broken keys. Design and analysis of algorithms. Strassen’s Algorithm. This is an extremely fast solution for finding GCD. There are multiple methods to find GCD , GDF or HCF of two numbers but Euclid's algorithm is very popular. This algorithm can be useful if is a fixed number in your program (so, you can hardcode a precomputed value of ()), or if is a prime number, in which case () = −. Reminder: To say that d ja means that 9c 2Z such that a = d c. 2 algorithm B. I use gcd a lot in my primes-utils gem, and in cryptography and Number Theory problems. * * This function is placed in the public domain by the author,. Algorithm 6 has the same specifications as the EEA and runs in operations in for input polynomials in degree. A generalized coordinate descent (GCD) algorithm for computing the solution path of the hybrid Huberized support vector machine (HHSVM) and its generalization, including the LASSO and elastic net (adaptive) penalized least squares, logistic regression, HHSVM, squared hinge loss SVM and expectile regression. But in practice you can code this algorithm in various ways. The source TypeScript code and compiled JavaScript code are available for viewing. But there is a ﬁfth operation which I would argue is just as fundamental — and that is the operation of taking greatest common divisors. Over the past few decades several variations on a "half GCD" algorithm for obtaining the pair of terms in the middle of a Euclidean sequence have been proposed. , there are integers s;t such that gcd(a;b) = sa+ tb. Binary Euclidean algorithm This algorithm ﬁnds the gcd using only subtraction, binary representation, shifting and parity testing. Key features: - Calculates both GCD (aka GCF, HCF, GCM) and LCM with a single key press. Lehmer's uses matrix multiplication to improve upon the standard Euclidian algorithms. * * So, we have three dependencies here: * * - isprime(int) (typically look up bitset sieve) * - std::vector primes (vector of prime numbers, typically sieved) * - binary_gcd(int, int) or similar, fast, gcd function. cpp, Maggie Johnson // Description: An implementation of Euclid's algorithm for computing gcd. Most computers can perform the operations of subtraction, testing the parity (odd or even) of a binary integer, and halving more quickly than computing remainders. We present a unified framework for the asymptotically fast Half-GCD (HGCD) algorithms, based on properties of the norm. The Greatest Common Divisor Also known as "the greatest common factor", " the greatest common measure" of a number. (Berman & Paul 2005). int main() {. Fast Modular Composition algorithm. exponential time (e. And there it is. Reference T&W, p. If you're behind a web filter, please make sure that the domains *. Using Bernstein's quasi-linear bulk GCD algorithm [Ber08], we obtained RSA moduli with shared primes. Cryptology ePrint Archive: Report 2019/266. Find the GCD (Greatest Common Divisor) of two numbers using EUCLID'S ALGORITHM; Compute the value of A raise to the power B using Fast Exponentiation ; Algorithms Implementation. * Additionally, you should have a fast gcd algorithm. Otherwise, use the period-finding subroutine (below) to find r, the period of the following function:. My assignment is to calculate the GCD of two numbers n and m using the Euclidean Algorithm which basically states that if the remainder = 0 the GCD is the 2nd of the two numbers. While the worst-case complexity of Weber’s “Accelerated integer GCD algorithm” is O logφ (k) 2, we show that the worst-case number of iterations of the while loop is exactly 1 2 logφ (k), where φ:= 1 2 1 + √ 5. This paper will demonstrate a variant with a relatively simple proof of correctness. JAVA Program import java. This is useful for finding the multiplicative inverse (i. Parallel computing methodologies. SO my thought was to write a recursive program looking for the GCD of two numbers, if the remainder is not 0 then change n to m and m to r and then calculate the GCD. of Maryland, College Park, MD 20742 [email protected] In Euclid's algorithm, we start with two numbers X and Y. Luckily, we can compute inverses in a generic way, using an algorithm called the extended Euclidean algorithm. Since 14 is the largest, gcd(42, 56) 14. In 1969, Volker Strassen came up with an algorithm whose asymptotic bound beat cubic. It is very important in number theory and in computing. JAVA Program import java. Creating connections. In modern terms ,Euclid’s algorithm is based on applying repeatedly the equality. In general case, however, computing ϕ ( m ) {\displaystyle \phi (m)} is equivalent to factoring, which is a hard problem, so prefer using the extended GCD algorithm. Euclid's Algorithm. Euclidean algorithm. Estimate approximately how many times faster it will be to find gcd (213486, 5423) with the help of the Euclid's algorithm compared with the alroithm based on checking consecutive integers from min {m,n} down to gcd(m,n). 2 Binary GCD algorithm Using the idea that division and multiplication by powers of two is a quick'' operation for computers (based on binary arithmetic) and the fact that the difference between two odd numbers is even, we can write an algorithm for GCD that does not use division (except by 2) at all. Until the late 1960s, it was believed that computing the product C of n x n matrices requires essentially a cubic number of operations, as the fastest known algorithm was the naive algorithm which indeed runs in $$O(n^{3})$$ time. 3) For any two prime numbers p and q, Φ(pq) = (p-1)*(q-1). 345--348] (which is about 5 times faster than the Euclidean XGCD algorithm) to compute the extended GCD of two integers. Fast Exponentiation Greatest common divisor Extended Euclidean Algorithm obtain gcd(a,b)and x,y,s. , real line, or line with floating point end points) scan-conversion with antialiasing. Most importantly, however, this algorithm is suita-. Once again, however, researchers have found a better algorithm, running in time. Parallel algorithms. Over the past few decades several variations on a "half GCD" algorithm for obtaining the pair of terms in the middle of a Euclidean sequence have been proposed. clk = generating clock. org are unblocked. Input will be two non-negative integers. That is why a bias in the key generation algorithm is fatal for the security of the key. Khan Academy. you can use divide and conquer approach ,i. We present a quasi-linear time recursive algorithm that computes the greatest common divisor of two integers by simulating a slightly modi ed version of the binary algorithm. The most common algorithm for finding the greatest common divisor of two numbers is the Euclid's algorithm.