Fastest modular exponentiation in JavaScript

Question

My problem is to compute (g^x) mod p quickly in JavaScript, where ^ is exponentiation, mod is the modulo operation. All inputs are nonnega

Answer 1

I use "%" for modular (mod) and "/" for integer division. Let function f(p,g,x,r) calculate (r*g^x)%p on the condition that r

bigint_t f(p,g,x,r) {
  bigint_t i, z = g, y;
  for (i = 1; i < x; ++i) {
    y = z; z *= g;
    if (z > p) break;
  }
  if (i >= x - 1) return r*z%p; // g^x*r%p = g^x*r
  else return f(p,y,x/i,g^(x%i)*r%p); // reduce to (r*g^(x%i)%p)*(g^i)^(x/i)%p
}

This routine involves a little more calculation, but each integer is less than 4096 bits which is usually much smaller than g^x. I believe this could be more efficient than the direct calculation. Also note that g^(x%i) can be calculated in a faster manner because we have calculated g^(i+1).

EDIT: see this post. Mehrdad gives the right (and better) solution.