How Can Newton's Method Be Modified for Polynomial Congruences?

[tpm]

Let be the congruence equation with f(x) a Polynomial of grade k then:

$$f(x)=0mod(p)$$

then if we have as a first approximation $$f(x_{n+1})=0mod(p)$$

then using a linear interpolation: $$f(x_{n})+f'(x_{n})(x_{n+1}-x_{n})=0mod(p)$$ or $$x_{n+1}=(x_{n}+\frac{f(x_{n}}{f'(x_{n})})0mod(p/f'(x_{n})$$

So we have 'modified' Newton method for solving Polynomial congruences. :Bigrin:

 

[tpm]

I think in this case the mai question is if the linear interpolation solution

$$f(x_{n})+f'(x_{n})(x_{n+1}-x_{n})=0mod(p)$$ (1)

will work to solve the general congruence $$f(x)=0mod(p)$$ for f(x) a POlynomial solving it by iterations using Newton method, where from (1) form an initial ansatz x_n we can get the next approximate value x_{n+1}

 

[tacman]

The Newton method is a popular and efficient method for finding the roots of a given function. It relies on the idea of linear interpolation to iteratively approach the root of the function. In the context of solving Polynomial congruences, the Newton method can be modified to provide a solution in the form of a congruence equation.

In this case, the given function is f(x)=0mod(p), where p is a prime number and f(x) is a Polynomial of grade k. The first approximation for the root of this function is f(x_{n+1})=0mod(p). Using linear interpolation, we can modify the Newton method to obtain a solution in the form of a congruence equation.

The modified Newton method uses the formula x_{n+1}=(x_{n}+\frac{f(x_{n})}{f'(x_{n})})0mod(p/f'(x_{n}), where x_{n} is the current approximation and f'(x_{n}) is the derivative of the function at x_{n}. This formula allows us to iteratively approach the root of the congruence equation and obtain a solution in the form of a congruence equation.

This modified Newton method is particularly useful for solving Polynomial congruences as it provides a more efficient and accurate solution compared to other methods. It also allows us to easily handle large prime numbers and higher grade Polynomials, making it a useful tool in number theory and cryptography.