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A third approach consists in extending the algorithm of subresultant pseudo-remainder sequences in a way that is similar to the extension of the Euclidean algorithm to the extended Euclidean algorithm. This allows that, when starting with polynomials with integer coefficients, all polynomials that are computed have integer coefficients.
A modular multiplicative inverse of a modulo m can be found by using the extended Euclidean algorithm. The Euclidean algorithm determines the greatest common divisor (gcd) of two integers, say a and m. If a has a multiplicative inverse modulo m, this gcd must be 1. The last of several equations produced by the algorithm may be solved for this gcd.
Modular exponentiation can be performed with a negative exponent e by finding the modular multiplicative inverse d of b modulo m using the extended Euclidean algorithm. That is: c = b e mod m = d −e mod m, where e < 0 and b ⋅ d ≡ 1 (mod m). Modular exponentiation is efficient to compute, even for very large integers.
When that occurs, that number is the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252).
The value is the encryption of . Decryption An encrypted message can be decrypted with ... Using the Extended Euclidean Algorithm, compute the inverse of ...
Euclidean algorithm; Coprime; Euclid's lemma; Bézout's identity, Bézout's lemma; Extended Euclidean algorithm; Table of divisors; Prime number, prime power. Bonse's inequality; Prime factor. Table of prime factors; Formula for primes; Factorization. RSA number; Fundamental theorem of arithmetic; Square-free. Square-free integer; Square-free ...
Let us run extended Euclidean algorithm for locating least common divisor of polynomials () and . The goal is not to find the least common divisor, but a polynomial r ( x ) {\displaystyle r(x)} of degree at most ⌊ ( d + k − 3 ) / 2 ⌋ {\displaystyle \lfloor (d+k-3)/2\rfloor } and polynomials a ( x ) , b ( x ) {\displaystyle a(x),b(x)} such ...
2 Encryption Algorithm. Toggle Encryption Algorithm subsection. 2.1 Key generation. ... Use the extended Euclidean algorithm to find and such that + =. Use the ...