Posted: May 22nd, 2023

The p-1 method is used to factor primes for instance p & q such that n = p*q and avoid weaknesses in the implementation of the algorithm.

The method works well when q-1 or p-1 can be factored as a product of given small primes.

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Therefore to factor 618240007109027021, we first compute;

2^(100!) mod n using Power (2,100!) mod n

In our case;

nÂ := 618240007109027021

a1:Â = 100!;

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t1Â :Â = Power (2,a1) mod n

a1 = 9.3326215443944152681699238856267e+157

t1 = 78737314835659020

Then we find a factor of n, test if itâ€™s prime, and evaluate how p-1 factors;

factor1:= igcd (t1 â€“ 1, n); // use gcd to find the factor

isprime(factor1) //test if factor is prime

ifactor(factor1-1) //if the factor is prime, test how p-1 factors

so factor1 = 250387201

And itâ€™s true (2)^{8}Â (3)^{5}Â (5)^{2}Â (23) (7)

To find the second factor q, we simply divide n by p and also evaluate how q-1 factors;

Factor2:= n/factor1; // divide n by factor1 to obtain q (factor2)

isprime(factor2) //test if q is prime

ifactor(factor2-1) //if the factor is prime, test how q-1 factors

factor2 = 2469135821

And itâ€™s true, (2)^{2}Â (5) (123456791)

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