Although the RSA algorithm uses rings rather than fields, the Euclidean algorithm can still be used to find a multiplicative inverse where one exists. The Euclidean algorithm also has other applications in error-correcting codes ; for example, it can be used as an alternative to the Berlekamp-Massey algorithm for decoding BCH and Reed-Solomon codes , which are based on Galois fields Take the prime numbers 13 and 7. Their product gives us our maximum value of 91. Let's take our public encryption key to be the number 5. Then using the fact that we know 7 and 13 are the factors of 91 and applying an algorithm called the Extended Euclidean Algorithm, we get that the private key is the number 29. I can't seem to make sense out. Using Euclidean Algorithm in RSA? Ask Question Asked 2 years, 10 months ago. Active 2 years, 10 months ago. Viewed 401 times 2 $\begingroup$ it leverages multiplication and subtraction, which humans are fairly good at, to make fractions like 15996751/3870378 reducible. Also useful in. RSA: Common Modulus attack with extended Euclidean algorithm. Jan 17, 2015 • rsa. RSA common modulus attack using extended euclidean. RSA, a commonly used public key cryptosystem, is very secure if you use sufficiently large numbers for encryption. Even then there are attacks against it Using Extended Euclidean Algorithm to create RSA private key. Ask Question Asked 7 years, 1 month ago. Active 2 years, 7 months ago. Viewed 23k times 6. 1. This is for an assignment I'm doing through school. I am having trouble generating a private key. My main.

- The problem is that I'm doing this for RSA and when I test the algorithm with 5 or 6 digit numbers it takes about 4 seconds, so I need a way to optimise the algorithm to work with hundreds of digits. I found references to the Extended Euclidean Algorithm online, but to my understanding that is used for checking the modular inverses that you create
- For more detail on back substitution go to: http://bit.ly/1W5zJ2g Here is a link with help on relative primes: http://www.mathsisfun.com/definitions/relative..
- We then present the RSA cryptosystem and use Sage's built-in commands to encrypt and decrypt data via the RSA algorithm. Note that this tutorial on RSA is for pedagogy purposes only. For further details on cryptography or the security of various cryptosystems, consult specialized texts such as [MenezesEtAl1996] , [Stinson2006] , and [TrappeWashington2006]
- Extended Euclidean Algorithm: Extended Euclidean algorithm also finds integer coefficients x and y such that: ax + by = gcd(a, b) Examples: Input: a = 30, b = 20 Output: gcd = 10 x = 1, y = -1 (Note that 30*1 + 20*(-1) = 10) Input: a = 35, b = 15 Output: gcd = 5 x = 1, y = -2 (Note that 35*1 + 15*(-2) = 5

- The Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, without explicitly factoring the two integers. 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
- g, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that + = (,). This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this.
- Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube
- In this video I show how to run the extended Euclidean algorithm to calculate a GCD and also find the integer values guaranteed to exist by Bezout's theorem
- In this article, we will demonstrate Extended Euclidean Algorithm.For this, we will see how you can calculate the greatest common divisor in a naive way which takes O(N) time complexity which we can improve to O(log N) time complexity using Euclid's algorithm.Following it, we will explore the Extended Euclidean Algorithm which has O(log N) time complexity
- How to optimise Euclidean Algorithm for large numbers? -RSA. Hot Network Questions Can I sign into two Steam accounts on the same PC? Is there a way to remove old files without internet? How can invading aliens access the Internet to find out all about us? What techniques do you.
- The Extended Euclidean Algorithm is used during RSA key generation to (1) verify that a component (call it e) of one of the keys is a member of the multiplicative group of integers modulo (p-1)·(q-1) where p and q are primes; and to (2) find the inverse of e in that group, which will be a component of the other key

I had an assignment to implement RSA algorithm, according to our textbook, the steps are like below. I have made my form so that user can enter the values of the prime numbers, p, q, and the public key, e, or the user can press on a buttons to generate these numbers randomly, I have made it very simple, with no validations, and encrypting only numbers below the length n of the RSA algorithm ** The Extended Euclidean Algorithm**.** The Extended Euclidean Algorithm** is just a fancier way of doing what we did Using the Euclidean algorithm above. It involves using extra variables to compute ax + by = gcd(a, b) as we go through the Euclidean algorithm in a single pass The extended Euclidean algorithm is particularly useful when a and b are co-prime, since x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Both extended Euclidean algorithms are widely used in cryptography How to solve 17x ≡ 3 (mod 29) using Euclid's Algorithm. If you want to see how Bézout's Identity works, see https://www.youtube.com/watch?v=9PRPr6J_btM 0:00. The Euclid's algorithm (or Euclidean Algorithm) is a method for efficiently finding the greatest common divisor (GCD) of two numbers. Implementation available in 10 languages along wth questions, applications, sample calculation, complexity, pseudocode

This site already has The greatest common divisor of two integers, which uses Euclidean algorithm.As it turns out (for me), there exists Extended Euclidean algorithm. This algorithm computes, besides the greatest common divisor of integers a and b, the coefficients of Bézout's identity, that is integers x and y such tha RSA Algorithm is widely used in secure data transmission. It was invented by Rivest, Shamir, and Adleman in the year 1978 and hence the name is RSA.It is an asymmetric cryptography algorithm which basically means this algorithm works on two different keys i.e. Public Key and Private Key.Here Public key is distributed to everyone while the Private key is kept private Euclidean algorithm, procedure for finding the greatest common divisor (GCD) of two numbers, described by the Greek mathematician Euclid in his Elements (c. 300 bc). The method is computationally efficient and, with minor modifications, is still used by computers. The algorithm involve

The Euclidean algorithm is arguably one of the oldest and most widely known algorithms. It is a method of computing the greatest common divisor (GCD) of two integers a a a and b b b.It allows computers to do a variety of simple number-theoretic tasks, and also serves as a foundation for more complicated algorithms in number theory Problem with simple RSA encryption algorithm. Extended Euclidean algorithm is used to generate the private key. The problem with multiplicative_inverse(e, phi) method. It is used for finding the multiplicative inverse of two numbers. The function does not return private key correctly. It returns None value Extended Euclidean Algorithm explained with examples Before you read this page This page assumes that you have read the explanation about the Euclidean Algorithm (click here), the non-extended version of the algorithm.If you have not read that page, please consider reading it. It is not very complicated, but if you skip it, this page will become more difficult to understand RSA algorithm is the most popular asymmetric key cryptographic algorithm based on the mathematical fact that it is easy to find and multiply large prime numbers but difficult to factor their product. In this type of attack, the attacker can find out the plain text from cipher text using the extended euclidean algorithm. 3

** The computation of the modular multiplicative inverse is an essential step in the derivation of key-pairs in the RSA public-key encryption method**. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extended Euclidean algorithm) that can be used for the calculation of modular multiplicative inverses Using Euclidean Algorithm for the RSA example. Say you have e = 5 , phi(377) = 336. How do you find d? Show full work. d should be = 26 The Euclidean Algorithm is to find the greatest common divisor of two given integers. In this article, you will see this critical algorithm proven in detail. Further, I will show you how to use these computations to solve linear congruence equations and linear Diophantine equations. While this algorithm has been around a while, it is the key to much success

** The mathematics behind RSA algorithm**. This algorithm uses a set of complex mathematics rules to find out the encryption and decryption key. The required mathematics for this include: prime factorisation, Euler totient function, Euclidean algorithm (for finding GCD) and modulus The Euclidean algorithm (also called Euclid's algorithm) is an algorithm to determine the greatest common divisor of two integers

RSA algorithm is a public key encryption technique and is considered as the most secure way of encryption. It was invented by Rivest, Shamir and Adleman in year 1978 and hence name RSA algorithm. Algorithm. The RSA algorithm holds the following features − RSA algorithm is a popular exponentiation in a finite field over integers including. RSA algorithm is asymmetric cryptography algorithm. Asymmetric actually means that it works on two different keys i.e. Public Key and Private Key. As the name describes that the Public Key is given to everyone and Private key is kept private ** The Euclidean Algorithm After explaining these ingredients, we turn to How the RSA algorithm works About factoring and nding prime numbers Uses of RSA algorithm 1**. Modular arithmetic Recall from elementary school how to divide a positive integer a (the dividend) by another positive integer b (the divisor) new Euclidean algorithm that we call Approximate Euclidean algorithm is to compute a good approximation of quotient by simple 64-bit division and to use it for reducing the number of iterations of the Euclidean algorithm. It runs much faster than original Euclidean algorithm and Binary Euclidean algorithm

Extended Euclidean Algorithm The Euclidean algorithm works by successively dividing one number (we assume for convenience they are both positive) into another and computing the integer quotient and remainder at each stage. This produces a strictly decreasing sequence of remainders, which terminates at zero, and the las encryption RSA algorithm. Many of these algorithms in turn rely on the Euclidean Algorithm as an algorithm acting on the ring of integers or as an algorithm acting on a ring of polynomials. Here we introduce the Euclidean algorithm for the integers. The Euclidean Algorithm on the set of polynomials is similar Learn about RSA algorithm in Java with program example. The term RSA is an acronym for Rivest-Shamir-Adleman who brought out the algorithm in 1977. RSA is an asymmetric cryptographic algorithm which is used for encryption purposes so that only the required sources should know the text and no third party should be allowed to decrypt the text as it is encrypted For the Euclidean Algorithm and the Extended Euclidean Algorithm, we'll show two versions: . A non-recursive version, which is easier to understand; A recursive version, because it's a lot shorter (but harder to understand if you don't know what's going on)

Introduction: 3 - Euclidean Algorithm You are advised not study this section until you are satisfied that you understand the contents of Section 2 - Modular arithmetic. This section contains all the remaining prerequisite basic mathematical tools that you need in order to understand cipher systems such as the RSA public key cipher system 21-110: The extended Euclidean algorithm. The Euclidean algorithm, which is used to find the greatest common divisor of two integers, can be extended to solve linear Diophantine equations.(Our textbook, Problem Solving Through Recreational Mathematics, describes a different method of solving linear Diophantine equations on pages 127-137.

* Note that although the RSA algorithm uses rings rather than fields, the Euclidean algorithm can still be used to find a multiplicative inverse where one exists*. The Euclidean algorithm also has other applications in error-correcting codes ; for example, it can be used as an alternative to the Berlekamp-Massey algorithm for decoding BCH and Reed-Solomon codes , which are based on Galois fields Public-Key Encryption by RSA Algorithm Objective The purpose of this page is to demonstrate step by step how a public-key encryption system works. We use the RSA algorithm (named after the inventors Rivest, Shamir, Adleman) with very small primes. The basic functions are implemented in JavaScript and can be viewed in the source This d can always be determined (if e was chosen with the restriction described above)—for example with the extended Euclidean algorithm.. Encryption and decryption. Internally, this method works only with numbers (no text), which are between 0 and n.. Encrypting a message m (number) with the public key (n, e) is calculated: . m':= m e (mod n) Decrypting with the private key (n, d) is done. The Euclidean Algorithm We are given aand b. We can divide aby bto get a= n 1b+ r 1 for some integers n 1 and r 1 with 0 ≤r ≤b−1. We can then divide b by r 1 to get b= n 2r 1 + r RSA practical (at least in the published literature). On the other hand, there is no proo

The Euclidean Algorithm. Welcome to advancedhighermaths.co.uk A sound understanding of the Euclidean Algorithm is essential to ensure exam success. To access a wealth of additional AH Maths free resources by topic please use the above Search Bar or click on any of the Topic Links at the bottom of this page as well as the Home Page HERE.. Study at Advanced Higher Maths level will provide. RSA's main security foundation relies upon the fact that given two large prime numbers, a composite number (in this case \(n\)) can very easily be deduced by multiplying the two primes together.But, given just \(n\), there is no known algorithm to efficiently determining \(n\) 's prime factors. In fact, it is considered a hard problem This article describes Euclidean algorithm and its programming in Python & R Posts about Extended Euclidean algorithm written by Dan Ma. The title of the blog post is the answer to a decryption challenge problem in connection to a factoring problem of a 129-digit number that is known as RSA-129 The extended Euclidean algorithm will give us a method for calculating p efficiently (note that in this application we do not care about the value for s, so we will simply ignore it.) The Extended Euclidean Algorithm for finding the inverse of a number mod n. We will number the steps of the Euclidean algorithm starting with step 0

Euclidean algorithm The Euclidean algorithm is one of the oldest numerical algorithms still to be in common use. It solves the problem of computing the greatest common divisor (gcd) of two positive integers. 12.1. Euclidean algorithm by subtraction The original version of Euclid's algorithm is based on subtraction: we recursively subtrac Number-Theoretic Algorithms (RSA and related algorithms) Chapter 31, CLRS book. p2. Outline • Modular arithmetic Extended Euclidean Algorithm a b x y d a b ax by! t u u u u u 299 221) 221 299 221 (1 34 u u. p17. Given 0, compute , , such that gcd( , ) Start with any two elements and of a Euclidean Domain. If , then . Otherwise take the remainder when is divided by , and find . Repeat this until the remainder is 0. Then . Usually the Euclidean algorithm is written down just as a chain of divisions with remainder: for and so . Example. To see how it works, just take an example. Say . We have.

* Javascript RSA Algorithm Demonstration*. I created this demonstration of the RSA algorithm for a course that I took about data structures and algorithms. Javascript has some limitations on handling large numbers - I compensated for some of these with the Extended Euclidean and Russian Peasant algorithms, but there are still some problems, depending on the choice of P and Q How Euclidean algorithm works? September 14, 2018 | No Comments. History. Euclid was a great Greek mathematician who lived around 300 B.C. He has been regarded as Father Of Geometry. According to Wikipedia, in his famous work called The Elements, he has left the following notes about finding GCD of 2 numbers In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor (GCF) or highest common factor (HCF). It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Elements.. In its simplest form, Euclid's algorithm starts with a pair of. CS 124 Section #8 RSA, Random Walks, Linear Programming 3/30/20 1Extended Euclidean Algorithm Given a;b, the extended Euclidean algorithm calculates x;ysuch that ax+ by= dwhere dis the GCD of a;b. This algorithm is necessary to implement RSA encryption. The algorithm is recursive. Algorithm 1 ExtendedEuclideanAlgorithm 1: procedure findgcd(a;b

Python []. Both functions take positive integers a, b as input, and return a triple (g, x, y), such that ax + by = g = gcd(a, b). Iterative algorithm [ RSA encryption, type of public-key cryptography widely used for data encryption of e-mail and other digital transactions over the Internet. RSA is named for its inventors, Ronald L. Rivest, Adi Shamir, and Leonard M. Adleman, who created it while on the faculty at the Massachusetts Institute o Download this app from Microsoft Store for Windows 10, Windows 8.1, Windows 10 Mobile, Windows Phone 8.1. See screenshots, read the latest customer reviews, and compare ratings for extended euclidean algorithm By Euclidean algorithm, we know that gcd(168, 64) = 8. So In this way, you don't need to reduce the fraction anymore because the denominator is the least common multiple of 64 and 168 Understanding Euclidean Algorithm for Greatest Common Divisor. Basic Version - Subtraction Based The basic algorithm given by Euclid simplifies the GCD determination process by using the principle that the greatest common divisor of two numbers does not change if the larger of the two numbers is replaced by the difference of the two

- The Euclidean algorithm is one of the oldest algorithms in common use. It appears in Euclid's Elements (c. 300 BC), specifically in Book 7 (Propositions 1-2) and Book 10 (Propositions 2-3). Centuries later, Euclid's algorithm was discovered independently both in India and in China, primarily to solve Diophantine equations that arose in astronomy and making accurate calendars
- The extended
**Euclidean****algorithm**is particularly useful when a and b are co-prime, since x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Both extended**Euclidean****algorithms**are widely used in cryptography - rsa python3 image-encryption privatekey extended-euclidean-algorithm rsa-encryption rsa-algorithm miller-rabin publickey-cryptosystem Updated Jul 21, 2020 Jupyter Noteboo
- The Extended Euclidean Algorithm finds the Modular Inverse . The following explanations are more of a technical nature. Read them if intend to implement the Euclidean Algorithm, skip them if you don't and go straight to the bottom of this page to view the Extended Euclidean Algorithm in action
- The RSA algorithm relies on the following facts as well: * It is extremely difficult to factor a large number. * Nevertheless, using the Euclidean algorithm it is extremely simple to calculate the gcd of two (even very large numbers. Computing the GCD: We shall start with an example. Let a = 792 and b = 75. 792 = 10.75 + 42 75 = 1.42 + 3

Greek mathematicians later used algorithms in the sieve of Eratosthenes for finding prime numbers, and the Euclidean algorithm for finding the greatest common divisor of two numbers. In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor of integers a and b, also the. Example of Extended Euclidean Algorithm Recall that gcd(84,33) = gcd(33,18) = gcd(18,15) = gcd(15,3) = gcd(3,0) = 3 We work backwards to write 3 as a linear combination of 84 and 33: For randomized algorithms we need a random number generator. • Most languages provide you with a function rand The RSA Algorithm Under the Hood Part 1, Prime Numbers, Euclidean Algorithm, & Greatest Common Divis; Posted in Video: Thursday, May 23, 201

Euclidean Algorithm for Greatest Common Divisor (GCD) The Euclidean Algorithm finds the GCD of 2 numbers. You will better understand this Algorithm by seeing it in action. Assuming you want to calculate the GCD of 1220 and 516, lets apply the Euclidean Algorithm-Assuming you want to calculate the GCD of 1220 and 516, lets apply the Euclidean. Could someone please explain to me the Extended Euclidean algorithm? I need to use it to find d: ed - 1 mod phi = 0 in order to get the private key for RSA encryption * The Euclidean Algorithm is one of the oldest numerical algorithms still in use today*. Attributed to ancient Greek mathematician Euclid in his book Elements written approximately 300 BC, th RSA Algorithm. To generate a key pair, mod (p - 1)(q - 1) = 1. In accordance with the Euclidean algorithm, the private key is now {d, n}. Encryption of plaintext m to ciphertext c is defined as c = (m ^ e) mod n. Decryption would then be defined as m = (c ^ d) mod n

- For more information and examples using the Euclidean Algorithm see our GCF Calculator and the section on Euclid's Algorithm. References. The Math Forum: LCD, LCM. Bureau 42: The Euclidean Algorithm: Greatest Common Factors Through Subtraction
- 12.2 The Rivest-Shamir-Adleman (RSA) Algorithm for 8 Public-Key Cryptography — The Basic Idea 12.2.1 The RSA Algorithm — Putting to Use the Basic Idea 12 12.2.2 How to Choose the Modulus for the RSA Algorithm 14 12.2.3 Proof of the RSA Algorithm 17 12.3 Computational Steps for Key Generation in RSA 2
- rsa rsa-cryptography extended-euclidean-algorithm rsa-encryption rsa-key euclidean-algorithm rsa-algorithm miller-rabin Updated May 8, 2020; Java; Mehul2205 / Cryptography-Assignments Star 0 Code Issues Pull requests These are the lab assignments performed during the course of cryptography. brute-force-attacks.
- RSA - Given p,q and e.. recover and use private key w/ Extended Euclidean Algorithm - crypto150-what_is_this_encryption @ alexctf 2017 - rsa_egcd.p
- ed such that: i1c + i2d =
- Euclidean algorithm definition is - a method of finding the greatest common divisor of two numbers by dividing the larger by the smaller, the smaller by the remainder, the first remainder by the second remainder, and so on until exact division is obtained whence the greatest common divisor is the exact divisor —called also Euclid's algorithm

Euclidean Algorithm to find GCD of Two numbers: If we recall the process we used in our childhood to find out the GCD of two numbers, it is something like this: This process is known as Euclidean algorithm. The idea behind this algorithm is, GCD(a,b) = GCD(b,r 0) where, a = bq 0 + r 0 and a> Busque trabalhos relacionados com Extended euclidean algorithm rsa ou contrate no maior mercado de freelancers do mundo com mais de 18 de trabalhos. É grátis para se registrar e ofertar em trabalhos Erm The textbook (?) is really confusing, since it does not give you examples. I believe that the photo you uploaded is referring to the same thing as that below.

- Take the Euclidean Algorithm for instance. It takes two arguments, commonly called a and b. Please tell me a better name for them? \$\endgroup\$ - northerner Apr 10 at 10:15 | show 4 more comments. Your Answer Thanks for contributing an answer to Code Review Stack Exchange! Please be sure to answer.
- The Euclidean Algorithm and Multiplicative Inverses Lecture notes for Access 2011 The Euclidean Algorithm is a set of instructions for ﬁnding the greatest common divisor of any two positive integers. Its original importance was probably as a tool in construction and measurement; the algebraic problem of ﬁnding gcd(a,b) is equivalent to the.
- 6, as guaranteed previously. The Extended Euclidean algorithm is useful for the RSA encryption method, as we will soon see. encryption method, as we will soon see
- Extended Euclidean Algorithm. version 1.0.0.0 (23.4 KB) by Michael Chan. Extended Euclidean algorithm is particularly useful when a and b are coprime, since x is the multip. 2.0. 3 Ratings. 6 Downloads. Updated 11 Sep 2011. View.
- # Author: Sam Erickson # Date: 2/23/2016 # # Program Description: This program gives the integer coefficients x,y to the # equation ax+by=gcd(a,b) given by the extended Euclidean Algorithm. def extendedEuclid (a, b): Preconditions - a and b are both positive integers
- Implementation of RSA Signature in SageMath Finding the points on Elliptic Curve Cryptography in SageMath Finding the inverse of (x^2+1) modulo (x^4+x+1) using Extended Euclidean Algorithm in SageMath [GF(2^4)
- In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two numbers, the largest number that div

- Python Math: Exercise-76 with Solution. Write a Python program to implement Euclidean Algorithm to compute the greatest common divisor (gcd). Note: In mathematics, the Euclidean algorithm[a], or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two numbers, the largest number that divides both of them without leaving a remainder
- Every Machine Learning engineer wants to achieve accurate predictions with their algorithms. Such learning algorithms are generally broken down into two types - supervised and unsupervised. K-means clustering is one of the unsupervised algorithms where the available input data does not have a labeled response
- e d because d is the inverse of e mod n. We can use the Extended Euclidean algorithm (in Mathematica, ExtendedGCD[integer, integer]) to deter
- The Euclidean algorithm is one of the oldest algorithms known, since it appeared in Euclid's Elements around 300 BC (7th book, Proposition 2). Euclid originally formulated the problem geometrically, as the problem of finding the greatest common measure for two line lengths.

- 1 RSA Algorithm 1.1 Introduction This algorithm is based on the diﬃculty of factorizing large numbers that have 2 and only 2 factors (Prime numbers). The system works on a public and private key system. The public key is made available to everyone. With this key a user can encrypt data but cannot decrypt it, the only person wh
- Here we will see the Euclidean algorithm to find the GCD of two numbers. The GCD (Greatest Common Divisor) can easily be found using Euclidean algorithm. There are two different approach. One is iterative, another one is recursive. Here we are going to use the recursive Euclidean algorithm. Algorithm EuclideanAlgorithm(a, b
- Extended Euclidean algorithm calculator . Given two integers \(a\) and \(b\), the extended Euclidean algorithm computes integers \(x\) and \(y\) such that \(ax + by.

In 1977, Rivest, one of the creators of the RSA algorithm, estimated that factoring a 125-digit number that is a product of two 63-digit prime numbers would require at least 40 quadrillion years using the best factoring algorithm available. Extended Euclidean algorithm RSA . The most commonly used public-key cryptosystem is RSA, which is named after its three developers Ron Rivest (b. 1947), Adi Shamir, and Leonard Adleman (b. 1945). At the time of the algorithm's development (1977), the three were researchers at the MIT Laboratory for Computer Science The **Euclidean** **Algorithm** We are given aand b. We can divide aby bto get a= n 1b+ r 1 for some integers n 1 and r 1 with 0 ≤r ≤b−1. We can then divide b by r 1 to get b= n 2r 1 + r **RSA** practical (at least in the published literature). On the other hand, there is no proo 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. To do this, we establish that whenever gcd(a,n)=1 then a has a multiplicative inverse (mod n). Our proof will be by giving an algorithm for constructing the inverse of a