While researching the concept of modular multiplicative inverse, I came across some interesting insights:
ax = 1 (mod m)
=> This implies that m divides ax - 1, and x is the sought-after inverse
=> Expressing it as ax - 1 = q*m (where q is an integer)
An essential condition here is gcd(a, m) = 1, indicating that a and m are co-prime.
In your scenario:
ed = 1 mod((p-1)(q-1)) // Given values for p, q, and e
=> Simplified to ed - 1 = z*((p-1)(q-1)) // where z is an unknown integer representing d
Referencing the Wikipedia article, one can employ the extended Euclidean GCD Algorithm to calculate the modular inverse through:
ax + by = g, with g being the gcd(a,b), assuming a and b are co-prime
// The extended GCD algorithm provides the values of x and y.
For this specific case, the equation would resemble:
ed - z*((p-1)(q-1)) = 1; // Aligning it with the structure mentioned above
a -> e
x -> d
b -> (p-1)(q-1)
y -> z
By implementing this algorithm, we can determine the values of d and z.
The extended gcd algorithm for ax + by = gcd(a,b) might follow a pattern similar to (source):
function xgcd(a, b) {
if (b == 0) {
return [1, 0, a];
}
temp = xgcd(b, a % b);
x = temp[0];
y = temp[1];
d = temp[2];
return [y, x-y*Math.floor(a/b), d];
}
This algorithm operates in time complexity O(log(m)^2), providing efficiency compared to exponentiation.
In javascript, there may not be a built-in function for this process. Encouraging exploration of algorithms, you could experiment with adapting the code to accommodate your data range, helping you progress in the right direction.